Answer:
a) Approximate the change in R is 0.5 ohm.
b) The actual change in R is 0.04 ohm.
Step-by-step explanation:
Given : The total resistance R of two resistors connected in parallel circuit is given by [tex]\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}[/tex]
To find :
a) Approximate the change in R ?
b) Compute the actual change.
Solution :
a) Approximate the change in R
[tex]R_1=12\ ohm[/tex] and [tex]R_2=10\ ohm[/tex]
[tex]R_1[/tex] is decreased from 12 ohms to 11 ohms.
i.e. [tex]\triangle R_1=21-11=1\ ohm[/tex]
[tex]R_2[/tex] is increased from 10 ohms to 11 ohms.
i.e. [tex]\triangle R_2=11-10=1\ ohm[/tex]
The change in R is given by,
[tex]\frac{1}{\triangle R}=\frac{1}{\triangle R_1}+\frac{1}{\triangle R_2}[/tex]
[tex]\frac{1}{\triangle R}=\frac{\triangle R_2+\triangle R_1}{(\triangle R_1)(\triangle R_2)}[/tex]
[tex]\triangle R=\frac{(\triangle R_1)(\triangle R_2)}{\triangle R_2+\triangle R_1}[/tex]
[tex]\triangle R=\frac{(1)(1)}{1+1}[/tex]
[tex]\triangle R=\frac{1}{2}[/tex]
[tex]\triangle R=0.5\ ohm[/tex]
b) The actual change in Resistance
[tex]\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}[/tex]
[tex]\frac{1}{R}=\frac{1}{12}+\frac{1}{10}[/tex]
[tex]R=\frac{10\times 12}{10+12}[/tex]
[tex]R=\frac{120}{22}[/tex]
[tex]R=5.46\ ohm[/tex]
When resistances are charged, [tex]R_1=R_2=11[/tex]
[tex]\frac{1}{R'}=\frac{1}{R_1}+\frac{1}{R_2}[/tex]
[tex]\frac{1}{R'}=\frac{1}{11}+\frac{1}{11}[/tex]
[tex]R'=\frac{11}{2}[/tex]
[tex]R'=5.5\ ohm[/tex]
Change in resistance is given by,
[tex]C=R'-R[/tex]
[tex]C=5.5-5.46[/tex]
[tex]C=0.04\ ohm[/tex]
Final answer:
The actual change in total resistance R of a parallel circuit as R_1 is decreased from 12 ohms to 11 ohms and R_2 is increased from 10 ohms to 11 ohms is calculated using the formula 1/R = 1/R_1 + 1/R_2.
Explanation:
The question is asking for the change in total resistance R of a parallel circuit when the resistances R_1 and R_2 are changed from 12 ohms to 11 ohms and 10 ohms to 11 ohms, respectively. To calculate the actual change, we can use the formula:
1/R = 1/R_1 + 1/R_2
Before the change, the total resistance is:
1/R_initial = 1/12 + 1/10
After the change, it becomes:
1/R_final = 1/11 + 1/11
By calculating both 1/R_initial and 1/R_final, we determine R_initial and R_final separately and find the difference between them to get the actual change in resistance.
Farmer Pickles wants Bob to paint the circular fence which encloses his sunflower field. If the parametric equations x = 18 cos(θ) and y = 18 sin(θ) describe the base of the fence (in yards) and the height of the fence is given by the equation h(x, y) = 12 + (2x − y)/6, then how many gallons of paint will Bob need to complete the project. Assume that one gallon of paint covers three hundred square feet of fence.
Answer:
40.72 gallons
Step-by-step explanation:
Since [tex]x = 18cos(\theta)[/tex] and [tex]y = 18sin(\theta)[/tex]. We can say this base has a shape of a circle of radius r = 18.
We can also calculate h in term of angle θ:
[tex]h(\theta) = 12 + \frac{36cos(\theta) - 18sin(\theta)}{6}[/tex]
[tex]h(\theta) = 12 + 6cos(\theta) - 3sin(\theta)[/tex]
We can calculate the area of the fence if we integrate this h function over θr, which is the chord length. θ ranges from 0 to 2π
A = ∫h(θ)rdθ
A = ∫(12 + 6cos(θ) - 3sin(θ))18d
A = (12θ + 6sin(θ) + 3cos(θ))18
As θ ranges from 0 to 2π as a full circle
A = (12*0 + 6sin(0) + 3cos(0))18 - (12*2π + 6sin(2π) + 3cos(2π))18
A = 3*18 - 18(24π + 3) = 1357.17 square yard
As 1 yard = 3 feet then 1 square yard = 9 square feet
1 gallon of paint can cover 300 square feet or 300 / 9 = 33.33 square yard
So to cover square yard Bob would need
1357.17 / 33.33 = 40.72 gallons of paint.
According to the National Association of Realtors, it took an average of three weeks to sell a home in 2017. Suppose data for the sale of 39 randomly selected homes sold in Greene County, Ohio, in 2017 showed a sample mean of 3.6 weeks with a sample standard deviation of 2 weeks. Conduct a hypothesis test to determine whether the number of weeks until a house sold in Greene County differed from the national average in 2017. Useα = 0.05for the level of significance, and state your conclusion.(a)State the null and alternative hypothesis. (Enter != for ≠ as needed.)H0:_____________Ha:__________(b)Find the value of the test statistic. (Round your answer to three decimal places.)Find the p-value. (Round your answer to four decimal places.)p-value =___________.
(a) Hypotheses:
[tex]\[ H_0: \mu = 3 \]\\H_a: \mu \neq 3 \][/tex]
(b) **Test Statistic and P-Value:
[tex]\[ t \approx 1.878 \][/tex]
(c) [tex]\[ \text{P-Value} \approx 0.0678 \][/tex]
Let's perform a hypothesis test to determine whether the number of weeks until a house is sold in Greene County differs from the national average. The average number of weeks to sell a home nationally is 3 weeks.
Hypotheses:
- Null hypothesis [tex](\(H_0\))[/tex]: The average number of weeks to sell a home in Greene County is equal to the national average [tex](\(μ = 3\) weeks)[/tex].
[tex]\[ H_0: \mu = 3 \][/tex]
- Alternative hypothesis [tex](\(H_a\))[/tex]: The average number of weeks to sell a home in Greene County differs from the national average.
[tex]\[ H_a: \mu \neq 3 \][/tex]
Given Information:
- Sample mean [tex](\(\bar{x}\))[/tex] = 3.6 weeks
- Sample standard deviation s = 2 weeks
- Sample size n = 39
- Level of significance [tex](\(\alpha\))[/tex] = 0.05
Test Statistic:
The test statistic for a one-sample t-test is given by:
[tex]\[ t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} \][/tex]
Calculation:
[tex]\[ t = \frac{3.6 - 3}{\frac{2}{\sqrt{39}}} \][/tex]
Let's calculate the value of \(t\) and the p-value.
Calculation:
[tex]t = \frac{3.6 - 3}{\frac{2}{\sqrt{39}}} \]\\t \approx \frac{0.6}{\frac{2}{\sqrt{39}}} \]\\t \approx \frac{0.6}{0.319439} \]\\t \approx 1.878 \][/tex]
Degrees of Freedom:
The degrees of freedom for a one-sample t-test is \(n-1\), where \(n\) is the sample size.
[tex]\[ \text{Degrees of Freedom} = 39 - 1 = 38 \][/tex]
P-Value:
Now, we need to find the p-value associated with this t-statistic and degrees of freedom. Since it's a two-tailed test, we're interested in the probability that the t-statistic is greater than 1.878 or less than -1.878.
You can use a t-table or statistical software to find this p-value. For [tex]\(df = 38\)[/tex] and [tex]\(t \approx 1.878\)[/tex], the p-value is approximately 0.0678 (two-tailed).
Conclusion:
Since the p-value (0.0678) is greater than the significance level (\(\alpha = 0.05\)), we do not reject the null hypothesis.
(a) Hypotheses:
[tex]\[ H_0: \mu = 3 \]\\H_a: \mu \neq 3 \][/tex]
(b) Test Statistic and P-Value:
[tex]\[ t \approx 1.878 \][/tex]
[tex]\[ \text{P-Value} \approx 0.0678 \][/tex]
(c) Conclusion:
At a 5% level of significance, there is not enough evidence to conclude that the average number of weeks until a house is sold in Greene County differs from the national average in 2017.
(a)State the null and alternative hypothesis. (Enter != for ≠ as needed.)H0:H0: μ = 3 (weeks) Ha: μ ≠ 3 (weeks)(b) the value of the test statistic. (Round your answer to three decimal places.) the p-value. (Round your answer to four decimal places.)p-value =1.702
ExplanationTo conduct the hypothesis test, calculate the test statistic and the p-value. The test statistic for a hypothesis test comparing a sample mean to a population mean when the population standard deviation is known is calculated using the formula:
[tex]\[ Z = \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}} \][/tex]
Where:
[tex]\(\bar{X}\)[/tex]n (3.6 weeks)
\(\mu\) = population mean (3 weeks)
[tex]\(\sigma\)[/tex]n standard deviation (2 weeks)
[tex]\(n\)[/tex]mple size (39)
Plugging in the values:
[tex]\[ Z = \frac{3.6 - 3}{\frac{2}{\sqrt{39}}} \][/tex]
[tex]\[ Z ≈ 1.702 \][/tex]
The p-value associated with this test statistic can be found using a standard normal distribution table or a statistical calculator. With a Z-score of 1.702, the corresponding two-tailed p-value is approximately 0.0883.
The calculated test statistic, \( Z ≈ 1.702 \), falls within the critical region for a two-tailed test at a significance level of α = 0.05. This means that the p-value (0.0883) is greater than the chosen level of significance (0.05). Consequently, there is insufficient evidence to reject the null hypothesis.
Thus, based on the sample data from Greene County, Ohio, in 2017, we do not have enough evidence to conclude that the average number of weeks until a house sold differs significantly from the national average in 2017. This suggests that, statistically, the time it takes for a house to sell in Greene County during that year might not be significantly different from the national average.
Overall, with a p-value of 0.0883 and at a significance level of 0.05, the analysis does not provide enough evidence to reject the null hypothesis, suggesting that the average number of weeks until a house sold in Greene County, Ohio, in 2017 may not differ significantly from the national average in the same year.
Two vertical poles, one 4 ft high and the other 16 ft high, stand 15 feet apart on a flat field.
A worker wants to support both poles by running rope from the ground to the top of each post.
If the worker wants to stake both ropes in the ground at the same point, where should the stake be placed to use the least amount of rope?
Answer:
The rope should be staked at 3ft distance from bottom of 4 ft pole.
Step-by-step explanation:
Given that the poles are 4 ft and 16 ft, separated by distance of 15 ft.
Refering the given figure, let the distance from 4 ft pole be "x".
consequently, distance from 16 ft pole is (15-x).
Refering the figure, in right angled triangle DCO, by pythagoras theorm, the the length OD is [tex]\sqrt{16-x^{2} }[/tex].
Similarly, the length OA is [tex]\sqrt{481+x^{2}-30x}[/tex].
thus the length of rope is AO + OD = L = f(x) =
[tex]\sqrt{16-x^{2} }[/tex] + [tex]\sqrt{481+x^{2}-30x}[/tex]
now, for a function of one variable, to have maximum or minimum, differentiate once, and find value of x.
Now, f'(x)= [tex]x(\sqrt{16+x^{2} }^{-1} )+ (x-15)(\sqrt{481+x^{2}-30x }^{-1})[/tex]
f'(x)=0 gives,
x=3 or -5
but the length cannot be negetive.
thus x=3.
By applying the pythagroean theorem and differentiation, the distance from the bottom of the pole that the rope should be staked at is: 3 ft.
What is the Pythagorean Theorem?The pythagorean theorem states that the square of the hypotenuse of a right triangle equals the sum the square of the otehr two legs of the right triangle.
The sketch of the problem is given in the diagram that is attached below.
Applying the pythagorean theorem in right triangle DCO, we would have:
DO = √(16 - x²)
AO would also be found using the pythagorean theorem:
AO = √(481 + x² - 30x)
Therefore, the length of the rope = AO + DO = f(x), which is:
f(x) = √(481 + x² - 30x) + √(16 - x²)
To find the value of x, take derivative and set equal to 0.
Differentiating, f(x) = √(481 + x² - 30x) + √(16 - x²), we would have:
f'(x) = 0 which will give us, x = 3 or -5.
Since the length cannot be negative, therefore, the distance from the bottom of the pole that the rope should be staked at is: 3 ft.
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The velocity function (in meters per second) is given for a particle moving along a line.v(t) = 3t − 8, 0 ≤ t ≤ 3(a) Find the displacement.(b) Find the distance traveled by the particle during the given time interval.
(a) The displacement of the particle is [tex]\(-\frac{21}{2}\)[/tex] meters.
(b) The distance travelled by the particle during the given time interval is [tex]\(\frac{169}{6}\)[/tex] meters.
(a) To find the displacement of the particle over the time interval [0, 3], we need to calculate the definite integral of the velocity function v(t) over this interval. The displacement is given by the integral of velocity with respect to time.
The velocity function is v(t) = 3t - 8.
The displacement S over the interval [a, b] is given by:
[tex]\[ S = \int_{a}^{b} v(t) \, dt \][/tex]
For our interval [0, 3], we have:
[tex]\[ S = \int_{0}^{3} (3t - 8) \, dt \][/tex]
To compute this integral, we find the antiderivative of the integrand:
[tex]\[ \int (3t - 8) \, dt = \frac{3}{2}t^2 - 8t + C \][/tex]
Now, we evaluate this antiderivative at the endpoints of the interval and subtract:
[tex]\[ S = \left(\frac{3}{2}(3)^2 - 8(3)\right) - \left(\frac{3}{2}(0)^2 - 8(0)\right) \][/tex]
[tex]\[ S = \left(\frac{3}{2} \cdot 9 - 24\right) - (0) \][/tex]
[tex]\[ S = \frac{27}{2} - 24 \][/tex]
[tex]\[ S = \frac{27}{2} - \frac{48}{2} \][/tex]
[tex]\[ S = -\frac{21}{2} \][/tex]
So the displacement of the particle is [tex]\(-\frac{21}{2}\)[/tex] meters.
(b) To find the distance travelled by the particle, we need to consider the intervals where the velocity is positive and where it is negative. The distance travelled is the sum of the distances travelled during each interval, without regard to direction.
First, we find the times when the velocity is zero by setting v(t) equal to zero:
[tex]\[ 3t - 8 = 0 \][/tex]
[tex]\[ 3t = 8 \][/tex]
[tex]\[ t = \frac{8}{3} \][/tex]
Since [tex]\(0 \leq t \leq 3\),[/tex] the velocity changes sign at [tex]\(t = \frac{8}{3}\)[/tex]. This divides the interval [0, 3] into two subintervals: [0, 8/3] and [8/3, 3].
We calculate the distance travelled in each subinterval and sum them:
For [0, 8/3]:
[tex]\[ d_1 = \int_{0}^{\frac{8}{3}} |3t - 8| \, dt \][/tex]
Since the velocity is negative on this interval, we have:
[tex]\[ d_1 = \int_{0}^{\frac{8}{3}} (8 - 3t) \, dt \][/tex]
[tex]\[ d_1 = \left(8t - \frac{3}{2}t^2\right) \Bigg|_{0}^{\frac{8}{3}} \][/tex]
[tex]\[ d_1 = \left(8\left(\frac{8}{3}\right) - \frac{3}{2}\left(\frac{8}{3}\right)^2\right) - (0) \][/tex]
[tex]\[ d_1 = \frac{64}{3} - \frac{3}{2}\left(\frac{64}{9}\right) \][/tex]
[tex]\[ d_1 = \frac{64}{3} - \frac{32}{3} \][/tex]
[tex]\[ d_1 = \frac{32}{3} \][/tex]
For [8/3, 3]:
[tex]\[ d_2 = \int_{\frac{8}{3}}^{3} |3t - 8| \, dt \][/tex]
Since the velocity is positive on this interval, we have:
[tex]\[ d_2 = \int_{\frac{8}{3}}^{3} (3t - 8) \, dt \][/tex]
[tex]\[ d_2 = \left(\frac{3}{2}t^2 - 8t\right) \Bigg|_{\frac{8}{3}}^{3} \][/tex]
[tex]\[ d_2 = \left(\frac{3}{2}(3)^2 - 8(3)\right) - \left(\frac{3}{2}\left(\frac{8}{3}\right)^2 - 8\left(\frac{8}{3}\right)\right) \][/tex]
[tex]\[ d_2 = \left(\frac{27}{2} - 24\right) - \left(\frac{32}{3} - \frac{64}{3}\right) \][/tex]
[tex]\[ d_2 = \frac{27}{2} - 24 + \frac{32}{3} \][/tex]
[tex]\[ d_2 = \frac{81}{6} - \frac{144}{6} + \frac{64}{6} \][/tex]
[tex]\[ d_2 = -\frac{105}{6} \][/tex]
[tex]\[ d_2 = -\frac{35}{2} \][/tex]
Since we cannot have a negative distance, we take the absolute value:
[tex]\[ d_2 = \frac{35}{2} \][/tex]
The total distance travelled is the sum of the distances in each interval:
[tex]\[ D = d_1 + d_2 \][/tex]
[tex]\[ D = \frac{32}{3} + \frac{35}{2} \][/tex]
[tex]\[ D = \frac{64}{6} + \frac{105}{6} \][/tex]
[tex]\[ D = \frac{169}{6} \][/tex]
So the distance travelled by the particle during the given time interval is [tex]\(\frac{169}{6}\)[/tex] meters.
How do I solve this?
Answer:
[tex]t=\frac{\pm\sqrt{dk}}{k}[/tex]
Step-by-step explanation:
Given relation:
[tex]d=k\ t^2[/tex]
We need to solve for [tex]t[/tex]
The given relation can be rearranged by isolating [tex]t[/tex] on one side.
Swapping the sides of the equation we have,
[tex]k\ t^2=d[/tex]
Dividing both sides by [tex]k[/tex] on order to cancel out [tex]k[/tex] on left side.
[tex]\frac{k\ t^2}{k}=\frac{d}{k}[/tex]
[tex]t^2=\frac{d}{k}[/tex]
We have got an expression for [tex]t^2[/tex] but we need to solve for [tex]t[/tex]
So, we take square root both sides to change [tex]t^2[/tex] to [tex]t[/tex]
[tex]\sqrt{t^2}=\sqrt{\frac{d}{k}}[/tex]
[tex]t=\pm\sqrt{\frac{d}{k}}[/tex]
So, we have successfully isolated [tex]t[/tex] on left side.
But the expression we got is a fraction with a square root in the denominator. Thus we need to rationalize it to make it in simplest form.
The expression can be written by taking square root separately for numerator and denominator :
[tex]t=\pm{\frac{\sqrt{d}}{\sqrt{k}}}[/tex]
Multiplying the numerator and denominator by [tex]\sqrt{k}[/tex]
[tex]t=\pm\frac{\sqrt{d}}{\sqrt{k}}\times\frac{\sqrt{k}}{\sqrt{k}} [/tex]
[tex]t=\pm\frac{\sqrt{dk}}{(\sqrt{k})^2}[/tex]
Square of a square root will remove the square root. Thus we have,
∴ [tex]t=\frac{\pm\sqrt{dk}}{k}[/tex]
Thus we have successfully got the expression in the simplest form.
While her husband spent 2½ hours picking out new speakers, a statistician decided to determine whether the percent of men who enjoy shopping for electronic equipment is higher than the percent of women who enjoy shopping for electronic equipment. The population was Saturday afternoon shoppers. Out of 66 men, 23 said they enjoyed the activity. Eight of the 23 women surveyed claimed to enjoy the activity. Interpret the results of the survey. Conduct a hypothesis test at the 5% level. Let the subscript m = men and w = women.
State the distribution to use for the test. (Round your answers to four decimal places.)
Answer:
[tex]z=\frac{p_{M}-p_{W}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{M}}+\frac{1}{n_{W}})}}[/tex] (1)
[tex]z=\frac{0.34848-0.34782}{\sqrt{0.34831(1-0.34831)(\frac{1}{66}+\frac{1}{23})}}=0.0057[/tex]
[tex]p_v =P(Z>0.0057)=0.4977[/tex]
The p value is a very high value and using the significance level [tex]\alpha=0.05[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can say the the percent of men who enjoy shopping for electronic equipment is NOT significantly higher than the percent of women who enjoy shopping for electronic equipment.
Step-by-step explanation:
1) Data given and notation
[tex]X_{M}=23[/tex] represent the number of men that said they enjoyed the activity of Saturday afternoon shopping
[tex]X_{W}=8[/tex] represent the number of women that said they enjoyed the activity of Saturday afternoon shopping
[tex]n_{M}=66[/tex] sample of male selected
[tex]n_{W}=23[/tex] sample of demale selected
[tex]p_{M}=\frac{23}{66}=0.34848[/tex] represent the proportion of men that said they enjoyed the activity of Saturday afternoon shopping
[tex]p_{W}=\frac{8}{23}=0.34782[/tex] represent the proportion of women with red/green color blindness
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the value for the test (variable of interest)
2) Concepts and formulas to use
We need to conduct a hypothesis in order to check if the percent of men who enjoy shopping for electronic equipment is higher than the percent of women who enjoy shopping for electronic equipment , the system of hypothesis would be:
Null hypothesis:[tex]p_{M} \leq p_{W}[/tex]
Alternative hypothesis:[tex]p_{M} > p_{W}[/tex]
We need to apply a z test to compare proportions, and the statistic is given by:
[tex]z=\frac{p_{M}-p_{W}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{M}}+\frac{1}{n_{W}})}}[/tex] (1)
Where [tex]\hat p=\frac{X_{M}+X_{W}}{n_{M}+n_{W}}=\frac{23+8}{66+23}=0.34831[/tex]
3) Calculate the statistic
Replacing in formula (1) the values obtained we got this:
[tex]z=\frac{0.34848-0.34782}{\sqrt{0.34831(1-0.34831)(\frac{1}{66}+\frac{1}{23})}}=0.0057[/tex]
4) Statistical decision
Using the significance level provided [tex]\alpha=0.05[/tex], the next step would be calculate the p value for this test.
Since is a one side right tail test the p value would be:
[tex]p_v =P(Z>0.0057)=0.4977[/tex]
So the p value is a very high value and using the significance level [tex]\alpha=0.05[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can say the the percent of men who enjoy shopping for electronic equipment is NOT significantly higher than the percent of women who enjoy shopping for electronic equipment.
A two-proportion z-test shows there is insufficient evidence at the 5% significance level to conclude that a higher proportion of men enjoy shopping for electronic equipment compared to women. The p-value is greater than the alpha value, leading to not rejecting the null hypothesis.
A hypothesis test can be used to determine if the percent of men who enjoy shopping for electronic equipment is higher than the percent of women who enjoy the same activity. We will perform a two-proportion z-test for this purpose. The hypotheses are as follows:
Null Hypothesis (H0): pm = pw
Alternative Hypothesis (Ha): pm > pw
Here, pm is the proportion of men who enjoy shopping for electronic equipment, and pw is the proportion of women who enjoy shopping for electronic equipment.
Given:
Out of 66 men, 23 enjoy shopping (pm = 23/66 ≈ 0.3485).Out of 23 women, 8 enjoy shopping (pw = 8/23 ≈ 0.3478).The test statistic for the difference in proportions is calculated as:
z = (pm - pw) / [tex]\sqrt{(p*(1 - p*)(1/nm + 1/nw))}[/tex]
Where p* = (xm + xw) / (nm + nw)
Substituting the values:
p* = (23 + 8) / (66 + 23) ≈ 0.3484z = (0.3485 - 0.3478) / [tex]\sqrt{(0.3484 * (1 - 0.3484) * (1/66 + 1/23))}[/tex] ≈ 0.0130The calculated z-value is approximately 0.0130.
The corresponding p-value for a z-value of 0.0130 is approximately 0.4948. Since p-value > alpha (0.4948 > 0.05), we do not reject the null hypothesis.
Conclusion:
At the 5% significance level, there is insufficient evidence to conclude that the proportion of men who enjoy shopping for electronic equipment is significantly higher than that of women.
Researchers claim that 40 tissues is the average number of tissues a person uses during the course of a cold. The company who makes Puffs brand tissues thinks that fewer of their tissues are needed. What are their null and alternative hypotheses?Group of answer choices1. H0: μ < 40 vs. H1: μ = 402. H0: μ = 40 vs. H1: μ < 403. H0: = 40 vs. H1: < 404. H0: μ = 40 vs. H1: μ > 40
Answer: 2. H0: μ = 40 vs. H1: μ < 40
Step-by-step explanation:
Null hypothesis [tex](H_0)[/tex] : It is a statement about population parameter by concerning the specific idea. It contains ≤,≥ and = signsAlternative hypothesis [tex](H_a)[/tex] : It is a statement about population parameter but against null hypothesis. It contains < , > and ≠ signs.Let [tex]\mu[/tex] be the average number of tissues a person uses during the course of a cold.
Given : Researchers claim that 40 tissues is the average number of tissues a person uses during the course of a cold.
i.e. [tex]\mu=40[/tex]
The company who makes Puffs brand tissues thinks that fewer of their tissues are needed.
i.e. [tex]\mu<40[/tex]
Thus , the set of hypothesis would be :-
[tex]H_0: \mu=40[/tex]
[tex]H_a: \mu<40[/tex]
Thus , the correct answer is option 2. H0: μ = 40 vs. H1: μ < 40
The null hypothesis (H0) is that an average person uses 40 tissues during a cold (H0: μ = 40). The alternative hypothesis (H1), which Puffs brand wants to prove, is that less than 40 of their tissues are needed during a cold (H1: μ < 40).
Explanation:The question is related to forming a null hypothesis and an alternative hypothesis in the context of statistics. Here, we are talking about the utilization of tissues during a cold. The researchers claim that on average a person uses 40 tissues (let's call this number 'μ') throughout a cold's duration. However, the Puffs brand makers, as per their assumption, think that less tissues of their brand are needed.
The null hypothesis, denoted by H0, is usually a claim of no effect or no difference. In this case, it is H0: μ = 40, denoting that the average number of tissues used is 40. This is also the claim that the Puffs brand intends to challenge or nullify.
On the other hand, the alternative hypothesis (H1) is the claim that the party interested in the outcome (here, the Puffs brand) wants to validate. Here, it would be H1: μ < 40, which signifies that fewer than 40 Puffs tissues are needed during the course of a cold.
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A survey randomly selected 250 top executives. The average height of these executives was 66.9 inches with a standard deviation of 6.2 inches. What is a 95% confidence interval for the mean height, μ, of all top executives? a. 63.5 < μ < 66.1 b. 65.3 < μ < 68.5 c. 62.8 < μ < 66.8 d. 66.1 < μ < 67.7
Answer: Option 'd' is correct.
Step-by-step explanation:
Since we have given that
n = 250
Average = 66.9 inches
standard deviation = 6.2 inches
We need to find the 95% confidence interval for the mean.
So, z = 1.96
Interval would be
[tex]\bar{x}\pm z\dfrac{\sigma}{\sqrt{n}}\\\\=66.9\pm 1.96\times \dfrac{6.2}{\sqrt{250}}\\\\=66.9\pm 0.77\\\\=(66.9-0.77,66.9+0.77)\\\\=(66.13,67.67)[/tex]
Hence, option 'd' is correct.
Find angle x on the given figure
Answer:
41 is correct
Step-by-step explanation:
90 + 49 = 139
180 - 139 = 41
There are five Oklahoma State Officials: Governor (G), Lieutenant Governer (L), Secretary of State (S), Attorney General (A), and Treasurer (T). Take all possible samples without replacement of size 3 that can be obtained from the population of five officials. (Note, there are 10 possible samples!)
(a) What is the probability that the governor is included in the sample?
(b) What is the probability that the governor and the attorney general are included in the sample?
So, the required probabilities are,
Part(a):P(A)=0.6
Part(b):P(B)=0.1
Given that:
There are five Oklahoma state officials,
Governor (G), Lieutenant Governor (L), Secretary of State (S), Attorney General (A), and Treasurer (T).
All possible samples of size 3 are obtained from the population of five officials.
Here order does not matter so we use the combinations.
[tex]5_C_3=10[/tex] possible samples.
So, S={GLS,GLA,GLT,GSA,GST,GAT,LSA,LST,LAT,SAT}
Hence, n(s)=10
Part(a):
Let A denotes the event that the governor is included in the sample.
A={GLS,GLA,GLT,GSA,GST,GAT}
That is n(A)=6
So, the probability that the governor is included in the sample,
[tex]P(A)=\frac{n(A)}{n(S)} \\=\frac{6}{10}\\ =0.6[/tex]
Part(b):
Let B denotes the event that the government attorney general and the treasure are included in the sample.
B={GAT}
That is n(B)=1
Hence, the probability that the government attorney general and the treasure are included in the sample is,
[tex]P(B)=\frac{n(B)}{n(S)} \\=\frac{1}{10} \\=0.1[/tex]
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(a) Probability of governor included: 6/10 = 0.6.
(b) Probability of governor and attorney general included: 1/10 = 0.1.
let's break it down in detail:
(a) Probability that the governor is included in the sample:
To find this probability, we need to count how many of the possible samples include the governor. From the list of all possible samples:
1. {G, L, S}
2. {G, L, A}
3. {G, L, T}
4. {G, S, A}
5. {G, S, T}
6. {G, A, T}
We see that in 6 out of 10 samples, the governor is included. Thus, the probability of the governor being included in the sample is [tex]\( \frac{6}{10} = 0.6 \).[/tex]
(b) Probability that the governor and the attorney general are included in the sample:
To find this probability, we need to count how many of the possible samples include both the governor and the attorney general. Looking at the list of all possible samples again:
1. {G, L, S}
2. {G, L, A}
3. {G, L, T}
4. {G, S, A}
5. {G, S, T}
6. {G, A, T}
7. {L, S, A}
8. {L, S, T}
9. {L, A, T}
10. {S, A, T}
We see that only in one out of the 10 samples, both the governor and the attorney general are included (sample number 2). Thus, the probability of both the governor and the attorney general being included in the sample is [tex]\( \frac{1}{10} = 0.1 \).[/tex]
So, to summarize:
(a) Probability that the governor is included in the sample: 0.6
(b) Probability that the governor and the attorney general are included in the sample: 0.1
What is the result when the number 30 is decreased by 10%
The result when the number 30 is decreased by 10% is 27, calculated by finding 10% of 30, which is 3, and subtracting it from the original number.
Explanation:When the number 30 is decreased by 10%, this implies we need to find 10% of 30 and then subtract it from 30. To calculate 10% of a number, you can simply divide the number by 10. Hence, 10% of 30 is 30 ÷ 10 = 3. Now, to decrease 30 by this value, we do the subtraction: 30 - 3 = 27. Therefore, the result when the number 30 is decreased by 10% is 27.
The number 27 is obtained by subtracting 10% from 30.
Step 1: Identify the initial number.
Given that the initial number is 30.
Step 2: Determine the percentage to be decreased.
The percentage to be decreased is 10%.
Step 3: Calculate the amount to be decreased.
To find 10% of 30, multiply 30 by 0.10.
10% of 30 = 30 × 0.10 = 3.
Step 4: Subtract the amount to be decreased from the initial number.
30 - 3 = 27.
Step 5: Interpret the result.
So, when 30 is decreased by 10%, the result is 27.
Therefore, the result of decreasing the number 30 by 10% is 27.
A password can be any string of length 7, 8, or 9. Each character in the password can be any capital letter or special character from the set {*, $, &, #, %}. There are no other restrictions on the password. How many possible different passwords are there?
Answer:
744
Step-by-step explanation:
The alphabet has 26 letters. Since each character of the password can be a capital letter or one of the four character of the set {*, $, &, #, %}, then each space in the password has 26+5=31 options to select a character.
a) If the password has length 7, then there are 31*7=217 different passwords.
b) If the password has length 8, then there are 31*8=248 different passwords and,
c) if the password has length 9, then there are 31*9=279 different passwords.
Then the total of different passwords is 217+248+279=744.
There are different numbers of possible passwords based on the length of the password.
Explanation:To determine the number of possible different passwords, we need to consider the different options for each character in the password and multiply them together.
For a password of length 7, each character has 5 possible options (capital letters or special characters). Therefore, there are 5 options for each of the 7 characters, giving us a total of 5^7 = 78,125 possible passwords.
Similarly, for a password of length 8, there are 5 options for each of the 8 characters, giving us a total of 5^8 = 390,625 possible passwords.
Finally, for a password of length 9, there are 5 options for each of the 9 characters, giving us a total of 5^9 = 1,953,125 possible passwords.
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A baseball player bunts a ball down the first base line. It rolls 33 ft at an angle of 25 Degree with the first base path. The pitcher's mound is 60.5 ft from home plate. How far must he travel to get to the ball? Note that a baseball diamond is a square. The pitcher must run feet. (Round to the nearest foot.)
Final answer:
The distance the pitcher must travel to get to the ball is approximately 36 ft.
Explanation:
The distance the pitcher must travel to get to the ball can be found using trigonometry. We can break down the motion of the ball into horizontal and vertical components. The horizontal component of the motion is the distance the ball rolls down the first base line, which is given as 33 ft. The vertical component of the motion can be found using the angle of 25 degrees. We can use the sine function to find the vertical distance:
Vertical distance = sin(25 degrees) * 33 ft = 14.04 ft
The distance the pitcher must travel is the hypotenuse of a right triangle formed by the horizontal and vertical distances. Using the Pythagorean theorem, we can find the distance:
Distance = sqrt((33 ft)^2 + (14.04 ft)^2) = 35.88 ft
Therefore, the pitcher must run approximately 36 ft to get to the ball.
Discrete MathA coin is flipped four times. For each of the events described below, express the event as a set in roster notation. Each outcome is written as a string of length 4 from {H, T}, such as HHTH. Assuming the coin is a fair coin, give the probability of each event.( a ) The first and last flips come up heads.( b ) There are at least two consecutive flips that come up heads.( c) There are at least two consecutive flips that are the same
Answer:
a) 25%
b) 50%
c) 87.5%
Step-by-step explanation:
There are [tex]2^4 = 16[/tex] number of outcome in total.
a)First and last flips come up heads. There are 4 outcomes here. Probability of the event is 4/16 = 1/4, or 25%.
- HHHH
- HHTH
- HTHH
- HTTH
b) at least 2 Consecutive flips that is heads. There are 8 outcomes. Probability of event is 8/16 = 1/2, or 50%
- HHHH
- HHHT
- HHTH
- HHTT
- THHH
- THHT
- HTHH
- TTHH
c) At least 2 consecutive flips that are the same. There are 14 outcomes. Probability of 14/16 = 7/8, or 87.5%
- HHHH
- HHHT
- HHTH
- HHTT
- THHH
- THHT
- HTHH
- TTHH
- TTTT
- TTTH
- TTHT
- HTTT
- HTTH
- THTT
Final answer:
In a) The probability is 1/4 and the event can be expressed as {HH**}, b) The probability is 7/16 and the event can be expressed as {HH**, *HH*, **HH}, c) The probability is 9/16 and the event can be expressed as {HH**, TT**, *HH*, *TT*}
Explanation:
To express the events as sets in roster notation, we need to list all the outcomes that satisfy each event.
(a) The first and last flips come up heads:
This event can be expressed as {HH**} (where ** represents any outcome for the two middle flips). The probability of this event is 1/2 * 1 * 1/2 = 1/4, since the first flip must be heads (1/2 probability), the second and third flips can be either heads or tails (1/2 probability each), and the last flip must be heads (1/2 probability).
(b) There are at least two consecutive flips that come up heads:
This event can be expressed as {HH**, *HH*, **HH}, where * represents any outcome for a single flip. The probability of this event is 1/2 * 1 * 1/2 * 1/2 + 1/2 * 1/2 * 1 * 1/2 + 1/2 * 1/2 * 1/2 * 1 = 7/16, since there are three possible patterns that satisfy this event.
(c) There are at least two consecutive flips that are the same:
This event can be expressed as {HH**, TT**, *HH*, *TT*}, where * represents any outcome for a single flip. The probability of this event is 1/2 * 1 * 1 + 1/2 * 1 * 1 + 1/2 * 1/2 * 1 + 1/2 * 1/2 * 1 = 9/16, since there are four possible patterns that satisfy this event.
In a random sample of 8 people, the mean commute time to work was 33.5 minutes and the standard deviation was 7.2 minutes. A 95% confidence interval using the t-distribution was calculated to be (27.5 comma 39.5 ). After researching commute times to work, it was found that the population standard deviation is 9.5 minutes. Find the margin of error and construct a 95% confidence interval using the standard normal distribution with the appropriate calculations for a standard deviation that is known. Compare the results.
Answer: Margin of error = 6.58
Confidence interval = (26.91, 40.08)
Step-by-step explanation:
Since we have given that
Sample size n = 8
Sample mean = 33.5 minutes
Population standard deviation = 9.5 minutes
At 95% confidence interval,
α = 0.05
t = 1.96
So, Margin of error is given by
[tex]t\times \dfrac{\sigma}{\sqrt{n}}\\\\=1.96\times \dfrac{9.5}{\sqrt{8}}\\\\=6.58[/tex]
Confidence interval would be
Lower limit:
[tex]\bar{x}-6.58\\\\=33.5-6.58\\\\=26.91[/tex]
Upper limit:
[tex]\bar{x}+6.58\\\\=33.5+6.58\\\\=40.08[/tex]
Hence, the interval would be (26.91, 40.08)
But at standard deviation 7.2 minutes, the confidence interval was (27.5, 39.5)
Confidence interval using the standard normal distribution is wider than the confidence interval using t distribution.
The information regarding the statistics shows that the margin of error is 6.58.
How to calculate the margin of errorFrom the information given, the following can be deduced:
Sample size = 8Standard deviation of population = 9.5Sample mean = 33.5Therefore, the margin of error will be:
= 1.96 × 9.5/✓8
= 6.58
Also, the lower limit will be:
= 33.5 - 6.58
= 26.91
The upper limit will be:
= 33.5 + 6.58
= 40.08
Therefore, the interval will be (26.91, 40.08)
When the standard deviation is 7.2 minutes, the confidence interval is (27.5, 39.5).
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Ackerman & Goldsmith (2011) found that students who studied text from printed hard copy had better test scores than students who studied text presented on a screen. In a related study, a professor noticed that several students in a large class had purchased the e-book version of the course textbook. For the final exam, the overall average for the entire class was μ = 85, but the n = 10 students who purchased the e-book had a mean of M = 77 with a standard deviation of s = 6. Do these data indicate a significant decrease in test scores on the final exam due to the use of a e-book? Use a one-tailed test with α = .05.
Final answer:
By conducting a One-Sample t-test, with the data provided, it was found that there is a significant decrease in test scores for students who studied with an e-book compared to the overall class average. The calculated t-score exceeded the critical value in a one-tailed test at alpha level of .05, leading to the rejection of the null hypothesis.
Explanation:
To determine if there is a significant decrease in test scores for students who studied using an e-book compared to the overall class average, we need to conduct a hypothesis test. First, let's define our null hypothesis (H0) as there being no difference in means, implying that the e-book does not affect scores. The alternative hypothesis (H1) suggests that the e-book usage leads to lower scores. Given that the overall class average (μ) is 85 and the mean (M) for the 10 students who used the e-book is 77 with a standard deviation (s) of 6, we can use the One-Sample t-test to analyze the data.
To calculate the t-score, we use the formula t = (M - μ) / (s / √n), where n is the sample size, which is 10 in this case. Plugging in our values gives us t = (77 - 85) / (6 / √10), resulting in a t-score of approximately -4.216.
Consulting a t-table or using statistical software with a one-tailed test at α = .05 and df = 9 (n-1), we find that our critical t-value is around -1.833. Since our calculated t-score of -4.216 is more extreme than -1.833, we reject the null hypothesis. This indicates a significant decrease in test scores for students who studied with the e-book, as compared to the class average.
Josh rents a kayak at a nearby state park. He pays a flat rate of $12.99 plus $3.75 for each hour that he spends in the water. How much did Josh spend if he was on the river for 4 1 2 hours?
Answer:
$29.87
Step-by-step explanation:
Josh's fee is $12.99 plus $3.75 for each of 4.5 hours, so is ...
$12.99 + 3.75×4.5 = $29.865 ≈ $29.87
Josh spent $29.87.
Suppose the interval [negative 2,0] is partitioned into nequals4 subintervals. What is the subinterval length Upper Delta x? List the grid points x0, x1, x2, x3, x4. Which points are used for the left, right, and midpoint Riemann sums?
Answer:
We find the length of each subinterval dividing the distance between the endpoints of the interval by the quantity of subintervals that we want.
Then
Δx= [tex]\frac{0-(-2)}{4}=\frac{2}{4}=\frac{1}{2}[/tex]
Now, each [tex]x_i[/tex] is found by adding Δx iteratively from the left end of the interval.
So
[tex]x_0=-2\\x_1=-2+\frac{1}{2}=\frac{-3}{2}\\x_2=\frac{-3}{2}+\frac{1}{2}=-1\\x_3=-1+\frac{1}{2}=-\frac{1}{2}\\x_4=\frac{-1}{2}+\frac{1}{2}=0[/tex]
Each subinterval is
[tex]s_1=[-2,-3/2]\\s_2=[-3/2,-1]\\s_3=[-1,-1/2]\\s_4=[-1/2,0][/tex]
The midpoints of the subintervals are
[tex]m_1=\frac{-2-3/2}{2}=\frac{-7/2}{2}=\frac{-7}{4}\\m_2=\frac{-1-3/2}{2}=\frac{-5/2}{2}=\frac{-5}{4}\\m_3=\frac{-1/2-1}{2}=\frac{-3/2}{2}=\frac{-3}{4}\\m_4=\frac{0-1/2}{2}=\frac{-1}{4}[/tex]
The points used for the
1. left Riemann sums are the left endpoints of the subintervals, that is
[tex]x_0=-2, x_1=\frac{-3}{2}, x_2=-1, x_3= \frac{-1}{2}[/tex]
2. right Riemann sums are the right endpoints of the subinterval,
[tex]x_1=-\frac{3}{2}, x_2=-1, x_3=-\frac{1}{2}, x_4=0[/tex]
3. midpoint Riemann sums are the midpoints of each subinterval
[tex]m_1,m_2,m_3,m_4.[/tex]
The subinterval length in the given range is 0.5. The grid points are -2, -1.5, -1, -0.5, and 0. The points used for the left, right, and midpoint Riemann sums vary based on the position of the subinterval.
Explanation:The interval specified is [negative 2,0] and is partitioned into n=4 subintervals. The length of each subinterval, Upper Delta x, is determined by subtracting the lower boundary from the upper boundary and dividing by the number of subintervals, in this case (0 - (-2)) / 4 = 0.5. Therefore, the subinterval length is 0.5.
The grid points are calculated by adding the subinterval length to each previous point, starting from the lower boundary: x0 is -2, for x1, add the subinterval length 0.5 to x0, resulting in -1.5. Repeat this process to find x2, x3 and x4 which are -1, -0.5 and 0 respectively.
Now, for the left Riemann sum, you use the left-hand side points of each subinterval, which are x0, x1, x2, x3. The right Riemann sum uses the right-hand side points, x1, x2, x3, x4. Lastly, the midpoint Riemann sum uses points in the middle of each subinterval. As the subintervals here each have a length of 0.5, their midpoints fall on x0.5, x1.5, x2.5 and x3.5 respectively.
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Hypothetical. A box is full of thousands of tickets labeled either O or 1 It is believed that the ave age of all he tickets in the box that ·the ro tion of S S 020 or 2 To test the null hypothesis that 20% of the tickets are labeled 1, we draw 400 tickets at random. Of these, 102 are labeled 1. Round your values to three decimal places. You can choose to work with the sum or the average (that is, the number of 1's or the proportion of 1's); the test statistic will be the same. Use this problem to practice this idea. If the null hypothesis is right, then the expected value of the number of 1's in 400 random draws from this box is If the null hypothesis is right, then the expected value of the proportion of 1's in 400 random draws from this box is with a standard error of The observed number of 1's is The observed proportion of 1's is 0.255 with a standard error of The test statistic is 2.75 The P-value (using a 2-test) is approximately 0.006
Answer: 28 * 1
Step-by-step explanation:
A billboard designer has decided that a sign advertising the movie Fight Club II - Rage of the Mathematicians should have 1-ft. margins at the top and bottom, and 2-ft. margins on the left and right sides. Furthermore, the billboard should have a total area of 200 ft2 , including the margins. Find the dimensions that would maximize printed area.
Answer:
Length = 20 ft
Height = 10 ft
Step-by-step explanation:
Let 'X' be the total width of the billboard and 'Y' the total height of the billboard. The total area and printed area (excluding margins) are, respectively:
[tex]200 = x*y\\A_p = (x-4)*(y-2)[/tex]
Replacing the total area equation into the printed area equation, gives as an expression for the printed area as a function of 'X':
[tex]y=\frac{200}{x} \\A_p = (x-4)*(\frac{200}{x} -2)\\A_p=208 -2x -\frac{800}{x}[/tex]
Finding the point at which the derivate for this expression is zero gives us the value of 'x' that maximizes the printed area:
[tex]\frac{dA_p(x)}{dx} =\frac{d(208 -2x -\frac{800}{x})}{dx}=0\\0=-2 +\frac{800}{x^2} \\x=\sqrt{400}\\x=20\ ft[/tex]
If x = 20 ft, then y=200/20. Y= 10 ft.
The dimensions that maximize the printed area are:
Length = 20 ft
Height = 10 ft
The dimensions that maximize the printed area of the billboard are 12 ft by 12 ft (excluding the margins) which gives a printed area of 144 square feet.
Explanation:To solve this problem, we use the relationship between area, length, and width. The total area, including marginal spaces is given as 200 square feet. The top and bottom margins sum to 2 feet, and the side margins sum to 4 feet.
Let's denote the width of the printed area (excluding the margins) as x and the length as y. Therefore, the width of the whole billboard is x + 4 ft and the length is y + 2 ft.
We know that Area = length * width, so we can establish the equation (x + 4)(y + 2) = 200.
To maximize the printed area, it's best when x and y are equal (a principle in mathematics), making the printed area a square. Also, x + y being constant, the square gives the maximum product.
Solving equation (x + 4) = (y + 2), we find that x = y = 12 feet, which maximizes the printed area giving us 144 square feet.
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Find the t-coordinates of all critical points of the given function. Determine whether each critical point is a relative maximum, minimum, or neither by first applying the second derivative test, and, if the test fails, by some other method.
f(t)= -3t^3+2t
a.) f has (a relative maximum, relative minimum, or no relative extrema) at the critical point t=________ * (smaller t-value)
b.) f has (a relative maximum, relative minimum, or no relative extrema) at the crtical point t=__________* (larger t-value)
Answer:
f has relative maximum at t = [tex]+\frac{\sqrt2}{3}[/tex]
and
f has relative minimum at t = [tex]-\frac{\sqrt2}{3}[/tex]
Step-by-step explanation:
Data provided in the question:
f(t) = -3t³ + 2t
Now,
To find the points of maxima or minima, differentiating with respect to t and putting it equals to zero
thus,
f'(t) = (3)(-3t²) + 2 = 0
or
-9t² + 2 = 0
or
t² = [tex]\frac{2}{9}[/tex]
or
t = [tex]\pm\frac{\sqrt2}{3}[/tex]
to check for maxima or minima, again differentiating with respect to t
f''(t) = 2(-9t) + 0 = -18t
substituting the value of t
at t = [tex]+\frac{\sqrt2}{3}[/tex]
f''(t) = [tex](-18)\times\frac{\sqrt2}{3}[/tex]
= - 6√2 < 0 i.e maxima
and at t = [tex]-\frac{\sqrt2}{3}[/tex]
f''(t) = [tex](-18)\times(-\frac{\sqrt2}{3})[/tex]
= 6√2 > 0 i.e minima
Hence,
f has relative maximum at t = [tex]+\frac{\sqrt2}{3}[/tex]
and
f has relative minimum at t = [tex]-\frac{\sqrt2}{3}[/tex]
Which data set has the highest standard deviation (without doing calculations)?
A) 1,2,3,4
B) 1,1,1,4
C) 1,2,2,4
D) 4,4,4,4
Answer:
B)1,1,1,4
Step-by-step explanation:
A.1,2,3,4
Mean=[tex]\bar x=\frac{1+2+3+4}{4}=2.5[/tex]
x [tex]x-\bar x[/tex] [tex](x-\bar x)^2[/tex]
1 -1.5 2.25
2 -0.5 0.25
3 0.5 0.25
4 1.5 2.25
[tex]\sum(x-\bar x)^2=2.25+0.25+0.25+2.25=5[/tex]
B.
Mean=[tex]\bar x=\frac{1+1+1+4}{4}=1.75[/tex]
[tex](x-\bar x)^2[/tex]
0.5625
0.5625
0.5625
5.0625
[tex]\sum(x-\bar x)^2=0.5626+0.5625+0.5625+5.0625=6.75[/tex]
C.
Mean=[tex]\bar x=\frac{1+2+2+4}{4}=2.25[/tex]
[tex](x-\bar x)^2[/tex]
1.5625
0.0625
0.0625
3.0625
[tex]\sum (x-\bar x)=1.5625+0.0625+0.0625+3.0625=4.75[/tex]
D.[tex]Mean=\bar x=\frac{4+4+4+4}{4}=4[/tex]
[tex](x-\bar x)^2[/tex]
0
0
0
0
[tex]\sum (x-\bar x)^2=0+0+0+0=0[/tex]
We know that
S.D is directly proportional to [tex]\sum (x-\bar x)^2[/tex].
When [tex]\sum (x-\bar x)^2[/tex] is highest then the S.D is also highest.
We can see that the value of [tex]\sum (x-\bar x)^2[/tex] is highest in option B.
Therefore, S.D of the date set of option B is highest.
Final answer:
The data set with the numbers 1 through 4 (option A) likely has the highest standard deviation due to having a greater spread compared to others.
Explanation:
The student has asked which data set has the highest standard deviation without doing calculations. In looking at the provided options, we can determine that the higher the variability in the data set, the higher the standard deviation will be. Data set A has numbers 1 through 4, which are more spread out compared to the other sets. Therefore, option A) 1,2,3,4 is likely to have the highest standard deviation. Option D) 4,4,4,4 will have a standard deviation of zero since all the data points are the same. Standard deviation is a measure of how spread out numbers are and a zero standard deviation occurs when all values in a data set are identical.
Factor the polynomial completely. 8x^4y – 16x^2y^2
8x^4y – 16x^2y^2=
=8x^4y-16x^4y
=-8x^4y
=-8*(x^4y)
Answer:
8\,x^2\,y\,(x-\sqrt{2} )(x+\sqrt{2} )
Step-by-step explanation:
Let's start by extracting all common factors from the two terms of this binomial. These common factors are: 8, [tex]x^2[/tex], and [tex]y[/tex].
The extraction renders:
[tex]8\,x^2\,y\,(x^2-2)[/tex]
In the real number system, the binomial in parenthesis can still be factored out considering that 2 is the perfect square of [tex]\sqrt{2}[/tex], that is:
[tex]2=(\sqrt{2} )^2[/tex]
We can then forwards with the factoring of this binomial using the factorization of a difference of squares as:
[tex](x^2-2) = (x^2-(\sqrt{2} )^2)=(x-\sqrt{2} )(x+\sqrt{2} )[/tex]
Thus giving the complete factorization as:
[tex]8\,x^2\,y\,(x-\sqrt{2} )(x+\sqrt{2} )[/tex]
Two points are selected randomly on a line of length 40 so as to be on opposite sides of the midpoint of the line. In other words, the two points X and Y are independent random variables such that X is uniformly distributed over [0,20) and Y is uniformly distributed over(20,40]. Find the probability that the distance between the two points is greater than 12 .
Answer:
Probability that we have an ordered pair (x,y) representing two points that satisfy the conditions is 328/400.
Step-by-step explanation:
This is a geometric probability.
set of all possible x in [0,20) and y in (20,40] we obtain a square in the x-y plane (in the first quadrant).
Area of this square is 20x20 = 400.
Set of all (x,y), that satisfy the distance being greater than 12
y > x, this means that (y - x) > 12 which is the same as the inequality y > x + 12.
For inequality, we obtain the region in the plane above the line y = x + 12. The area of this region is
(1/2)bh = (1/2)(12)(12) = 72.
Thus the area of the set of all points in our square that satisfy the condition (y - x) > 12 is 400 - 72 = 328.
Probability that we have an ordered pair (x,y) representing two points that satisfy the conditions is 328/400.
Use the definition of the derivative to find an expression for the instantaneous velocity of an object moving with rectilinear motion according to the given function relating s (in ft) and t (in s). Then calculate the instantaneous velocity for the given value of t. s = 8t + 19; t=4 (Simplify your answer)
Answer:
The instantaneous velocity for [tex]s(t) = 8t + 19[/tex] when t = 4 is [tex]v(t)=8 \:\frac{ft}{s}[/tex].
Step-by-step explanation:
The average rate of change of function f over the interval is given by this expression:
[tex]\frac{f(b)-f(a)}{b-a}[/tex]
The average velocity is the average rate of change of distance with respect to time
[tex]average \:velocity=\frac{distance \:traveled}{time \:elapsed} =\frac{\Delta s}{\Delta t}[/tex]
The instantaneous rate of change is defined to be the result of computing the average rate of change over smaller and smaller intervals.
The derivative of f with respect to x, is the instantaneous rate of change of f with respect to x and is thus given by the formula
[tex]f'(x)= \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}[/tex]
For any equation of motion s(t), we define what we call the instantaneous velocity at time t to be the limit of the average velocity, between t and t + Δt, as Δt approaches 0.
[tex]v(t)=\lim_{\Delta t \to 0} \frac{s(t+\Delta t)-s(t)}{\Delta t}[/tex]
We know the equation of motion [tex]s(t) = 8t + 19[/tex] and we want to find the instantaneous velocity for the given value of t = 4.
Applying the definition of instantaneous velocity, we get
[tex]v(t)=\lim_{\Delta t \to 0} \frac{s(t+\Delta t)-s(t)}{\Delta t}\\\\v(4)=\lim_{\Delta t \to 0} \frac{8(4+\Delta t)+19-(8(4)+19)}{\Delta t}\\\\v(t)=\lim_{\Delta t \to 0} \frac{32+8\Delta t+19-32-19}{\Delta t}\\\\v(t)=\lim_{\Delta t \to 0} \frac{8\Delta t}{\Delta t}\\\\v(t)=8 \:\frac{ft}{s}[/tex]
The instantaneous velocity for [tex]s(t) = 8t + 19[/tex] when t = 4 is [tex]v(t)=8 \:\frac{ft}{s}[/tex].
A report describes the results of a large survey involving approximately 3500 people that was conducted for the Center for Disease Control. The sample was selected in a way that the Center for Disease Control believed would result in a sample that was representative of adult Americans. One question on the survey asked respondents if they had learned something new about a health issue or disease from a TV show in the previous 6 months. Data from the survey was used to estimate the following probabilities, where L-event that a randomly selected adult American reports learning somethlng new about a health Issue or disease from a TV show in the previous 6 months and F-event that a randomly selected adult American is female Assume that P(F) 0.5. Are the events L and F independent events? Use probabilities to justify your answer. Land Flare notindependent events, because P(L) P(F)- 0.3 , which is not equal to P(L n F)
Some manufacturers claim that non-hybrid sedan cars have a lower mean miles-per-gallon (mpg) than hybrid ones. Suppose that consumers test 21 hybrid sedans and get a mean of 32 mpg with a standard deviation of 8 mpg. Thirty-one non-hybrid sedans get a mean of 21 mpg with a standard deviation of three mpg. Suppose that the population standard deviations are known to be six and three, respectively. Conduct a hypothesis test at the 5% level to evaluate the manufacturers claim.
A polling organization took a random sample of 2,500 single parents to determine how many single parents have two jobs. Suppose 38% of all single parents have two jobs. If the total population of single parents is 12 million, is the 10% condition met? Justify your answer.
Answer:
Yes it is below 10% of the population
Step-by-step explanation:
10% of 12 million is 1,200,00
One of the conditions is as 2,500
The population is at least 10 times as large as the sample.
here 10*n = 10* 2500 = 25000 < 12000000
So the correct option is "Yes, 10(2,500) = 25,000, which is less than the total population."
5. The Lakeland Post polled 1,231 adults in the city to determine whether they wash their hair before their body while in the shower. Of the respondents, 63% said they washed their hair first. Suppose 51% of all adults actually wash their hair first. What are the mean and standard deviation of the sampling distribution?
Sampling distribution of sample proportion ( \hat p ) is approximately normal for large n with mean and standard deviations are as
mean = p = 0.51
for finding the standard deviation we need to use p = 0.51
б = √p·(1 - p )/ n = √0.51 . 0.49/ 1234 = 0.014248
So the correct choice is "The mean is 0.51 and the standard deviation is 0.0142"
Learn more about polling organizations at
https://brainly.com/question/971724
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Trainees must complete a specific task in less than 2 minutes. Consider the probability density function below for the time it takes a trainee to complete the task.f(x) = 0.85 - 0.35x0 < x < 2a) What is the probability a trainee will complete the task in less than 1.31 minutes? Give your answer to four decimal places.b) What is the probability that a trainee will complete the task in more than 1.31 minutes? Give your answer to four decimal places.c) What is the probability it will take a trainee between 0.25 minutes and 1.31 minutes to complete the task? Give your answer to four decimal places.d) What is the expected time it will take a trainee to complete the task? Give your answer to four decimal places.e) If X represents the time it takes to complete the task, what is E(X2)? Give your answer to four decimal places.f) If X represents the time it takes to complete the task, what is Var(X)? Give your answer to four decimal places.
Answer:
a) 0.8132 or 81.32%
b) 0.1868 or 18.68%
c) 0.6116 or 61.16%
d) 0.7667 minutes
e) 0.8667 minutes
f) 0.1 minutes
Step-by-step explanation:
a)
If f(x) = 0.85-0.35x (0<x<2) is the PDF and X is the random variable that measures the time it takes for a trainee to complete the task, the probability a trainee will complete the task in less than 1.31 minutes is P(X<1.31)
[tex]\large P(X<1.31)=\int_{0}^{1.31}f(x)dx=\int_{0}^{1.31}(0.85-0.35x)dx=\\\\=0.85\int_{0}^{1.31}dx-0.35\int_{0}^{1.31}xdx=\\\\=0.85*1.31-0.35*\frac{(1.31)^2}{2}=0.8132[/tex]
b)
The probability that a trainee will complete the task in more than 1.31 minutes is
P(X>1.31) = 1 - P(X<1.31) = 1 - 0.8132 = 0.1868
c)
The probability it will take a trainee between 0.25 minutes and 1.31 minutes to complete the task is P(0.25<X<1.31)
[tex]\large P(0.25<X<1.31)=\int_{0.25}^{1.31}f(x)dx=\int_{0}^{1.31}f(x)dx-\int_{0}^{0.25}f(x)dx=\\\\=0.8132-\int_{0}^{0.25}(0.85-0.35x)dx=0.8132-0.85\int_{0}^{0.25}dx+0.35\int_{0}^{0.25}xdx=\\\\=0.8132-0.85*0.25+0.35*\frac{(0.25)^2}{2}=0.6116[/tex]
d)
the expected time it will take a trainee to complete the task
is E(X)
[tex]\large E(X)=\int_{0}^{2}xf(x)dx=\int_{0}^{2}x(0.85-0.35x)dx=0.85\int_{0}^{2}xdx-0.35\int_{0}^{2}x^2dx=\\\\=0.85*\frac{2^2}{2}-0.35*\frac{2^3}{3}=0.7667\;minutes[/tex]
e)
[tex]\large E(X^2)=\int_{0}^{2}x^2f(x)dx=\int_{0}^{2}x^2(0.85-0.35x)dx=0.85\int_{0}^{2}x^2dx-0.35\int_{0}^{2}x^3dx=\\\\=0.85*\frac{2^3}{3}-0.35*\frac{2^4}{4}=0.8667[/tex]
f)
[tex]\large Var(X)=E(X^2)-(E(X))^2=0.8667-0.7667=0.1000[/tex]
Does second-hand smoke increase the risk of a low birthweight? A baby is considered having low birth weight if he/she weighs less than 5.5 pounds at birth. According to the National Center of Health Statistics, about 7.8% of all babies born in the U.S. are categorized as low birthweight.
Suspecting that the national percentage is higher than 7.8%, researchers randomly select 1200 babies whose mothers had extensive exposure to second-hand smoke during pregnancy and find that 10.4% of the sampled babies are categorized as low birth weight.
Let p be the proportion of all babies in the U.S. that are categorized as "low birth weight." Give the null and alternative hypotheses for this research question.
H 0 : p = 0.078
H 0 : p = 0.078
H a : p > 0.078
H 0 : p = 0.104
Final answer:
Researchers are testing the hypothesis that second-hand smoke increases the risk of low birthweight, with a null hypothesis of H0: p = 0.078, and an alternative hypothesis of Ha: p > 0.078.
Explanation:
The researchers are questioning whether second-hand smoke increases the risk of low birthweight in babies. They have a sample where 10.4% of babies born to mothers with extensive exposure to second-hand smoke are categorized as having low birth weight. This suggests the proportion might be higher than the national average of 7.8%. The null hypothesis (H0), which is the hypothesis to be tested, states that the proportion of low birth weight for the population is equal to the national average, thus H0: p = 0.078. The alternative hypothesis (Ha), representing the researchers' suspicion, is that the proportion is greater than the national average, hence Ha: p > 0.078. The last hypothesis option provided, H0: p = 0.104, is not relevant for the null hypothesis in this context since this value represents the sample proportion, not the population proportion they are testing against.