A study of 20 worldwide financial institutions showed the correlation between their assets and pretax profit to be 0.81.State the decision rule for 0.025 significance level: H0: rho ≤ 0; H1: rho > 0. (Round your answer to 3 decimal places.)Compute the value of the test statistic. (Round your answer to 2 decimal places.)Can we conclude that the correlation in the population is greater than zero? Use the 0.025 significance level.

Answer:

  c = 22

Step-by-step explanation:

The given equation can be written as ...

  x^2 = 103 -c

In order for -9 and +9 to be solutions, the equation needs to be ...

  x^2 = 81

So, we must have ...

  103 -c = 81

  c = 103 -81 = 22

The value of c must be 22.

Answer: height of kite is 147.042 feets

Step-by-step explanation:

The diagram of the kite is shown in the attached photo

Triangle ABC is formed and it is a right angle triangle.

The kite string made an angle of 33 degrees with the ground. The string used was 90 yards We will convert the 90 yards to feets.

I yard = 3 feets

90 yards would become

90×3 = 270 feets

This 270 feets form the hypotenuse of the triangle.

To determine the height of the kite h, we will use trigonometric ratio

Sin# = opposite / hypotenuse

Where

# = 33 degrees

Hypotenuse = 270 feets

Opposite = h feets

Sin 33 = h/270

h = 270sin33

h = 270 × 0.5446 = 147.042 feets

Answer:

The hypotheses used in this situation

[tex]H_0:\mu = 14.6[/tex]

[tex]H_a:\mu \neq 14.6[/tex]

Step-by-step explanation:

We are given that  Business Week reported that at the top 50 business schools, students studied an average of 14.6 hours.

Mean = [tex]\mu = 14.6[/tex]

Claim : The amount UMSL students study is different from this 14.6 hour benchmark.

The hypotheses used in this situation

[tex]H_0:\mu = 14.6[/tex]

[tex]H_a:\mu \neq 14.6[/tex]

Answer:

a. Sample proportion ^p= 0.12

b. It is appropiate.

c. [0.0447;0.1953]

d. [^p ± d]

Step-by-step explanation:

Hello!

Given the information I'll assume that the variable of study has a binomial distribution:

X~Bi(n;ρ)

The sample data:

n= 50

"Success" x= 6

Sample proportion ^p= x/n = 6/50 = 0.12

Now, your study variable has a binomial distribution, but remember that the Central Limit Theorem states that given a big enough sample size (usually n≥ 30) you can approximate the sample proportion distribution to normal.

Since the sample is 50 you can apply the approximation, your sample proportion will have the following distribution:

^p≈ N( p; [p(1 - p)]/n)

With E(^p)= p and V(^p)= [p(1 - p)]/n.

This allows you to estimate the population proportion per Confidence Interval using the Z-distribution:

[^p±[tex]Z_{1-\alpha /2}[/tex]*√(^p(1 - ^p)/n)]

Since you are estimating the value of p, you'll use the estimated standard deviation (i.e. with the sample proportion instead of the population proportion)

to calculate the interval.

At level 90% the interval is:

[0.12±1.64*√([0.12(1 - 0.12)]/50)]

[0.0447;0.1953]

The margin of error (d) of an interval is half its amplitude (a)

if a= Upper bond - Low bond

then d= (Upper bond - Low bond)/2

d= (0.1953-0.0447)/2

d= 0.0753

And since the interval structure is "estimator" -/+ "margin of error" you can write it as:

[^p ± d]

I hope you have a SUPER day!

Answer:

(0.6430, 0.8170)

Step-by-step explanation:

Given that during a recent drought a water utility in a certain town sampled 100 residential water bills and found out that 73 of the residences had reduced their water consumption over that of the previous year.

Sample size n = 100

Sample proportion p = [tex]\frac{73}{100} =0.73[/tex]

q = 1-p = 0.23

Std error of proportion = [tex]\sqrt{\frac{pq}{n} } \\=\sqrt{\frac{0.73*0.27}{100} } \\=0.04440[/tex]

95% Z critical value = 1.96

Margin of error = [tex]1.96*0.0444 = 0.0870[/tex]

Confidence interval = sample proportion ±margin of error

0.642983946

0.817016054

(0.6430, 0.8170)

The 95% confidence interval for the proportion of residences that reduced their water is (0.6430, 0.8170).

This is used to determine the measure of how much uncertainty there is with any particular statistic.

Sample size (n) = 100

Sample proportion(p) = 73/100 = 0.73

q = 1-p = 0.23

√pq/n = √0.73×0.27/100 = 0.04440

95% Z critical value = 1.96

Margin of error = 1.96×0.04440 = 0.0870

Confidence interval = sample proportion ± margin of error

= (0.6430, 0.8170)

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By Green's theorem,

[tex]\displaystyle\int_{x^2+y^2=9}\vec F(x,y)\cdot\mathrm d\vec r=\iint_D\left(\frac{\partial(xy)}{\partial x}-\frac{\partial(x^2)}{\partial y}\right)\,\mathrm dx\,\mathrm dy=\iint_Dy\,\mathrm dx\,\mathrm dy[/tex]

where [tex]C[/tex] is the circle [tex]x^2+y^2=9[/tex] and [tex]D[/tex] is the interior of [tex]C[/tex], or the disk [tex]x^2+y^2\le1[/tex].

Convert to polar coordinates, taking

[tex]\begin{cases}x=r\cos\theta\\y=r\sin\theta\end{cases}\implies\mathrm dx\,\mathrm dy=r\,\mathrm dr\,\mathrm d\theta[/tex]

Then the work done by [tex]\vec F[/tex] on the particle is

[tex]\displaystyle\iint_Dy\,\mathrm dx\,\mathrm dy=\int_0^{2\pi}\int_0^3(r\sin\theta)r\,\mathrm dr\,\mathrm d\theta=\left(\int_0^{2\pi}\sin\theta\,\mathrm d\theta\right)\left(\int_0^3r^2\,\mathrm dr\right)=\boxed0[/tex]

The work done in a force field on a particle moving around a circle is found by calculating and integrating the line integral of the force field over the path defined by the circle. It involves substituting the parametric representation of the circle into the force field equation and incorporating the directional aspect of the line integral.

This problem is related to calculating work done in a force field and involves concepts from vector calculus. The work done is calculated based on the line integral of the force field F over a path C, defined by the parametric representation of the circle. More specifically, we'll need to find the line integral of F along the path C, and then integrate that from 0 to 2π (since the particle moves around the circle once).

The parametric representation of the circle x² + y² = 9 is x = 3cosθ, y = 3sinθ, where -π ≤ θ ≤ π. Substitute these into the force field equation you'll get F(3cosθ, 3sinθ) = 9cos²θ i + 9cosθsinθ j.

To find the work done, we'll compute the line integral of F over the path C, which in this case is the circle. Since the movement is in the clockwise direction and when it comes to line integrals, the direction matters, we'll need to use -θ instead of θ to represent our parameter. So we're integrating F along C from 0 to 2π.

The exact calculation of the integral might require a bit of time and effort, but you should end up with the work done by the force field on the particle that moves once around the circle oriented in the clockwise direction.

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Answer:

1. Distance from Katie's House to the beach = 75 km

2. Hours that it takes to get to the beach = 6 hrs

3. -12.5 which tells us the rate at which katie traveled which was 12.5 km per hour

Steps to 3.

Rise/Run = -75/6 = -12.5 = slope

Answer:

Step-by-step explanation:

The graph is a plot of distance in kilometers on the vertical axis against time, in hours on the horizontal axis.

1) The x-intercept is the point on the horizontal axis at which y = 0. This means that at this point, the distance from the beach to her house is zero. From the graph, the point is 6 hours.

2) The y-intercept is the point on the vertical axis at which x = 0. This means that this point is the point at which she started to ride. At this point,time = 0 hours and the distance from the beach to her house is 75 kilometers.

3) slope is expressed as

Change in y / change in x

We will pick 2 points on the y axis and two corresponding points on the x axis.

Change in y = y2 - y1

Change in x = x2 - x1

Slope = (0-75) / (6 - 0) = -75/6

Slope = - 12.5 kilometers per hour

The slope represents the rate at which her distance from the beach changes with time. It represents her velocity. It is negative because her distance from the beach is decreasing with time

To estimate the required sample size to estimate the population proportion, we can use the formula n = (Z^2 * p * (1 - p)) / E^2. In part (a), assume p=0.5, while in part (b), use a known value of p=0.4 from a prior study. Compare the results in part (c).

To find the minimum sample size needed to estimate the population proportion of adults who think the president of their country can control the price of gasoline with a 95% confidence level and an accuracy within 5%, we can use the formula:

n = (Z^2 * p * (1 - p)) / E^2

Where:

n = required sample size

Z = Z-value for the desired confidence level (for 95%, Z-value is approximately 1.96)

p = estimated proportion from a prior study or 0.5 if no prior study is available

E = desired margin of error (5% or 0.05)

Using this formula, we can plug in the values and calculate the minimum sample size. In part (a), p is assumed to be 0.5, while in part (b), the p-value is given as 0.4 from a prior study. Finally, in part (c), you can compare the results obtained from the different approaches.

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To estimate a population proportion with 95% confidence and an error of 5%, a sample size of approximately 385 is needed without a preliminary estimate. If a prior estimate of 40% is used, the sample size required is about 370. Using an estimate closer to 0.5 generally results in a larger, more conservative sample size.

A researcher wishes to estimate the population proportion of adults who think the president of their country can control the price of gasoline with 95% confidence. The estimate must be accurate within 5% of the true proportion.

No preliminary estimate available: When you do not have a preliminary estimate, the most conservative approach is to use 0.5 as the estimated population proportion (p). The formula for the minimum sample size (n) is:

Therefore, the minimum sample size needed is approximately 385.

Using a prior study where 40% of respondents think the president can control gasoline prices: Here, p is 0.4.

Thus, the minimum sample size needed is approximately 370.

Answer:

Hence increasing in  (-\infty,0) U (1,5)

c) Decreasing in (0,1)

Step-by-step explanation:

Given that y(t) satisfies the differential equation

[tex]\frac{dy}{dt} =y^4-6y^2+5y^2\\=y^2(y^2-6y+5)\\=y^2(y-1)(y-5)[/tex]

Separate the variables to have

[tex]\frac{dy}{y^2(y-1)(y-5)} =dt[/tex]

Left side we can resolve into partial fractions

Let [tex]\frac{1}{y^2(y-1)(y-5)} =\frac{A}{y} +\frac{B}{y^2}+\frac{C}{y-1} \frac{D}{y-5}[/tex]

Taking LCD we get

[tex]1= Ay(y-1)(Y-5) +B(y-1)(y-5)+Cy^2 (y-5)+Dy^2 (y-1)\\Put y =1\\1 =  -4C\\Put y =5\\ 1 = 25(4)D\\Put y =0\\1=5B\\[/tex]

By equating coeff of y^3 we have

A+C+D=0

[tex]C=\frac{-1}{4} \\D=\frac{1}{100} \\B =\frac{1}{5} \\A = -C-D = \frac{6}{25}[/tex]

Hence left side =

[tex]\frac{6}{25y} +\frac{1}{5y^2}+\frac{-1}{4(y-1)}+ \frac{1}{100(y-5)}=dt\\\frac{6}{25}ln y -\frac{1}{5y}-\frac{1}{4}ln|(y-1)| +\frac{1}{100}ln|y-5| = t+C[/tex]

b) y is increasing whenever dy/dt>0

dy/dt =0 at points y =0, 1 and 5

dy/dt >0 in (-\infty,0) U (1,5)

Hence increasing in  (-\infty,0) U (1,5)

c) Decreasing in (0,1)

Answer:

a) y = 0 , 5,1

b) y ⊂ (- ∞,0) ∪ (0,1)∪(5,∞)

Step-by-step explanation:

Given data:

differential equation is given as

[tex]\frac{dy}[dt} = y^4 -6y^3+ 5y^2[/tex]

a) constant solution

[tex] y^4 -6y^3+ 5y^2 = 0 [/tex]

taking y^2 from all part

[tex]y^2(y^2 - 6y -5) = 0[/tex]

solution of above equation is

y = 0 , 5,1

b) for which value y is increasing

[tex]\frac{dy}{dt}  > 0[/tex]

y^2(y - 5) (y -1) > 0

y ⊂ (- ∞,0) ∪ (0,1)∪(5,∞)

Answer:

d) Squared differences between actual and predicted Y values.

Step-by-step explanation:

Regression is called "least squares" regression line. The line takes the form = a + b*X where a and b are both constants. Value of Y and X is specific value of independent variable.Such formula could be used to generate values of given value X.

For example,

suppose a = 10 and b = 7. If X is 10, then  predicted value for Y of 45 (from 10 + 5*7). It turns out that with any two variables X and Y. In other words, there exists one formula that will produce the best, or most accurate predictions for Y given X. Any other equation would not fit as well and would predict Y with more error. That equation is called the least squares regression equation.

It minimize the squared difference between actual and predicted value.

The least squares regression line minimizes the sum of the squared differences between the actual and predicted Y values, making option (D) the correct answer.

The least squares regression line is a method used in statistics to find the line that best fits a set of data points. This method minimizes the sum of squared differences between the actual Y values and the predicted Y values on the regression line. In other words, the objective of the least squares regression line is to find the coefficients that minimize the SSE (sum of squared errors). So, the correct answer is:

Using this method allows for the most accurate prediction within the given set of data, but it is important to note that it may not be suitable for predicting values outside that set.

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Answer:

Parameter of interest= proportion of parts that present serious problems with porosity (defective)

z= 5.08  

Based on the p value obtained and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of parts defective (problems with porosity) NOT exceeds 0.15 or 15% .  

Step-by-step explanation:

1) Data given and notation n  

n=300 represent the random sample taken

X=58 represent the parts that present serious problems with porosity in the sample

[tex]\hat p=\frac{58}{300}=0.193[/tex] estimated proportion of parts that present serious problems with porosity in the sample

[tex]p_o=0.15[/tex] is the value that we want to test

[tex]\alpha=0.05[/tex] represent the significance level

Confidence=95% or 0.95

z would represent the statistic (variable of interest)

[tex]p_v[/tex] represent the p value (variable of interest)  

2) Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that defective rate (proportion defective) exceeds 15%. :  

Null hypothesis:[tex]p\leq 0.15[/tex]  

Alternative hypothesis:[tex]p>0.15[/tex]

We assume that the proportion follows a normal distribution.  

When we conduct a proportion test we need to use the z statistic, and the is given by:  

[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)  

The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly (different,higher or less) from a hypothesized value [tex]p_o[/tex].

3) Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

[tex]z=\frac{0.193 -0.15}{\sqrt{\frac{0.15(1-0.15)}{300}}}=2.086[/tex]  

4) Statistical decision  

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

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 test the p value would be:  

[tex]p_v =P(z>2.086)=0.0185[/tex]  

Based on the p value obtained and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the proportion of parts defective (problems with porosity) NOT exceeds 0.15 or 15% .  

Answer: time it will take him to 17.5 minutes to complete the test from start to finish

Step-by-step explanation:

During the standardized test, Paul answered the first 22 questions in 5 minutes.

If he answers 22 questions 5 minutes

He would answer one question in 5/22 = 0.227 minutes

He continues to answer questions at the same rate. This means that his unit rate of answering 1 question in 0.227 minutes is constant throughout the test.

Total number of questions on the test is 77. The time it will take him to complete the test from start to finish will be

Unit rate of answering questions × total number of questions

= 0.227 × 77 = 17.5 minutes

Answer:

a) Null hypothesis:[tex]\mu \leq 40[/tex]    

Alternative hypothesis:[tex]\mu > 40[/tex]  

b) [tex]t=\frac{45-40}{\frac{5}{\sqrt{25}}}=5[/tex]    

[tex]p_v =P(t_{(24)}>5)=2.08x10^{-5}[/tex]    

If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that mean age of all employees is significantly more than 40 years at 5% of signficance.

Step-by-step explanation:

1) Data given and notation    

[tex]\bar X=45[/tex] represent the sample mean    

[tex]s=15[/tex] represent the sample standard deviation    

[tex]n=25[/tex] sample size    

[tex]\mu_o =40[/tex] represent the value that we want to test    

[tex]\alpha=0.05[/tex] represent the significance level for the hypothesis test.    

t would represent the statistic (variable of interest)    

[tex]p_v[/tex] represent the p value for the test (variable of interest)    

2) State the null and alternative hypotheses.    

We need to conduct a hypothesis in order to check if the mean age of all the employees is significantly more than 40 years, the system of hypothesis are :    

Null hypothesis:[tex]\mu \leq 40[/tex]    

Alternative hypothesis:[tex]\mu > 40[/tex]    

Since we don't know the population deviation, is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:    

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (1)    

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".    

3) Calculate the statistic    

We can replace in formula (1) the info given like this:    

[tex]t=\frac{45-40}{\frac{5}{\sqrt{25}}}=5[/tex]    

4) P-value    

First we need to calculate the degrees of freedom given by:  

[tex]df=n-1=25-1=24[/tex]  

Since is a one-side upper test the p value would be:    

[tex]p_v =P(t_{(24)}>5)=2.08x10^{-5}[/tex]    

5) Conclusion    

If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the mean age of all employees it's significantly more than 40 years at 5% of signficance.

To test whether the mean age of employees has increased, use a one-sample t-test with a significance level of 0.05.

To test whether the mean age of all the employees is significantly more than 40 years, we can use a hypothesis test.

The null hypothesis (H0) states that the mean age is equal to or less than 40 years, while the alternative hypothesis (Ha) states that the mean age is greater than 40 years.

We can conduct a one-sample t-test to compare the sample mean of 45 years to the population mean of 40 years. We will use a significance level (α) of 0.05. If the p-value of the test is less than α, we reject the null hypothesis and conclude that there is evidence of a significant increase in the average age of all the employees.

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To answer this question, we use the binomial probability formula to calculate the probability of picking 0, 1, and 2 green jelly beans out of 13. Summing up these probabilities gives us our answer. The closest option to our calculated value is (b) 0.3100.

The subject of this question is the calculation of probabilities that is handled in Mathematics, specifically in Statistics. To solve this, we can use the binomial probability formula P(X=k) = C(n, k) * (p^k) * (1 - p)^(n - k), where k is the number of successful trials, n is the total number of trials, and p is the probability of success in one trial.

In this case, we want to find the probability that at most 2 jelly beans will be green if Sally chooses 13. We need to calculate the probability of getting 0, 1, and 2 green jelly beans, then sum those probabilities. Therefore, P(X≤2) = P(X=0) + P(X=1) + P(X=2).

Using the above binomial formula, we will not get an exact value from the options provided. Only option (b) 0.3100, is closest to our calculated value and would be correct if we were to pick from these options.

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Answer:

a) quotient = 2,  remainder = 5

b) quotient = -10,  remainder = -1

c) quotient = 34,  remainder = 7

d) quotient = 77,  remainder = 0

e) quotient = 0,  remainder = 0

f) quotient = 0,  remainder = 3

g) quotient = 0,  remainder = -1

h) quotient = 4,  remainder = 0

Step-by-step explanation:

The relationship between the quotient, remainder and divisor is that the product of the quotient and divisor added to the reminder gives the dividend.

Based on this, when

a)19 is divided by 7 = 19/7

= 2[tex]\frac{5}{7}[/tex]

Hence the quotient is 2, the remainder is 5

Applying the same principle, when

b) −111 is divided by 11

= -10[tex]\frac{1}{11}[/tex]

Hence the quotient is -10, the remainder is -1

c) 789 is divided by 23

=34[tex]\frac{7}{23}[/tex]

Hence the quotient is 34, the remainder is 7

d) 1001 is divided by 13

= 77

Hence the quotient = 77, the remainder is 0

e) 0/19 = 0

Hence the quotient is 0, the remainder is 0

f) 3/5

Hence the quotient is 0, the remainder is 3

g) -1/3

Hence the quotient is 0, the remainder is -1

h) 4/1

Hence the quotient is 4, the remainder is 0

The division operations result in the following quotients: 2, -10, 34, 77, 0, 0, 0, 4, and remainders: 5, 0, 3, 0, 0, 3, -1, 0 respectively.

Here are the results of the given divisions: a) When 19 is divided by 7, the quotient is 2 and the remainder is 5. b) When −111 is divided by 11, the quotient is −10 and there is no remainder. c) When 789 is divided by 23, the quotient is 34 and the remainder is 3. d) When 1001 is divided by 13, the quotient is 77 and there is no remainder. e) When 0 is divided by 19, the quotient and the remainder are both 0. f) When 3 is divided by 5, the quotient is 0 and the remainder is 3. g) When −1 is divided by 3, the quotient is 0 and the remainder is -1. h) When 4 is divided by 1, the quotient is 4 and there is no remainder.

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We divide [0, 3] into [tex]n[/tex] subintervals,

[tex]\left[0,\dfrac3n\right]\cup\left[\dfrac3n,\dfrac6n\right]\cup\left[\dfrac6n,\dfrac9n\right]\cup\cdots\cup\left[\dfrac{3(n-1)}n,3\right][/tex]

so that the right endpoint of each subinterval is given according to the arithmetic sequence,

[tex]r_k=\dfrac{3k}n[/tex]

for [tex]1\le k\le n[/tex].

The Riemann sum is then

[tex]\displaystyle\sum_{k=1}^nf(r_k)\Delta x_k[/tex]

where

[tex]\Delta x_k=r_k-r_{k-1}=\dfrac{3k}n-\dfrac{3(k-1)}n=\dfrac3n[/tex]

With [tex]f(x)=2x^2[/tex], we have

[tex]\displaystyle\frac3n\sum_{k=1}^n2\left(\frac{3k}n\right)^2=\frac{54}{n^3}\sum_{k=1}^nk^2[/tex]

Recall that

[tex]\displaystyle\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}6[/tex]

The area under the curve [tex]f(x)[/tex] over the interval [0, 3] is then

[tex]\displaystyle\int_0^32x^2\,\mathrm dx=\lim_{n\to\infty}\frac{54n(n+1)(2n+1)}{6n^3}=\lim_{n\to\infty}9\left(2+\frac3n+\frac1{n^2}\right)=\boxed{18}[/tex]

Answer:

(a) 0.28347

(b) 0.36909

(c) 0.0039

(d) 0.9806

Step-by-step explanation:

Given information:

n=12

p = 20% = 0.2

q = 1-p = 1-0.2 = 0.8

Binomial formula:

[tex]P(x=r)=^nC_rp^rq^{n-r}[/tex]

(a) Exactly two will be drunken drivers.

[tex]P(x=2)=^{12}C_{2}(0.2)^{2}(0.8)^{12-2}[/tex]

[tex]P(x=2)=66(0.2)^{2}(0.8)^{10}[/tex]

[tex]P(x=2)=\approx 0.28347[/tex]

Therefore, the probability that exactly two will be drunken drivers is 0.28347.

(b)Three or four will be drunken drivers.

[tex]P(x=3\text{ or }x=4)=P(x=3)\cup P(x=4)[/tex]

[tex]P(x=3\text{ or }x=4)=P(x=3)+P(x=4)[/tex]

Using binomial we get

[tex]P(x=3\text{ or }x=4)=^{12}C_{3}(0.2)^{3}(0.8)^{12-3}+^{12}C_{4}(0.2)^{4}(0.8)^{12-4}[/tex]

[tex]P(x=3\text{ or }x=4)=0.236223+0.132876[/tex]

[tex]P(x=3\text{ or }x=4)\approx 0.369099[/tex]

Therefore, the probability that three or four will be drunken drivers is 0.3691.

(c)

At least 7 will be drunken drivers.

[tex]P(x\geq 7)=1-P(x<7)[/tex]

[tex]P(x\leq 7)=1-[P(x=0)+P(x=1)+P(x=2)+P(x=3)+P(x=4)+P(x=5)+P(x=6)][/tex]

[tex]P(x\leq 7)=1-[0.06872+0.20616+0.28347+0.23622+0.13288+0.05315+0.0155][/tex]

[tex]P(x\leq 7)=1-[0.9961][/tex]

[tex]P(x\leq 7)=0.0039[/tex]

Therefore, the probability of at least 7 will be drunken drivers is 0.0039.

(d) At most 5 will be drunken drivers.

[tex]P(x\leq 5)=P(x=0)+P(x=1)+P(x=2)+P(x=3)+P(x=4)+P(x=5)[/tex]

[tex]P(x\leq 5)=0.06872+0.20616+0.28347+0.23622+0.13288+0.05315[/tex]

[tex]P(x\leq 5)=0.9806[/tex]

Therefore, the probability of at most 5 will be drunken drivers is 0.9806.

a) Exactly two will be drivers: 0.2835. b) Three or four will be drivers: 1.5622; c) At least 7 will be drivers: 32.5669 (rounded to four decimal places). d) At most 5 will be drivers: 0.8749

a) Exactly two will be drivers:

For this case, we'll use the binomial probability formula:

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

Where:

n is the number of trials (in this case, the number of drivers in the sample), which is 12.

k is the number of successful trials (in this case, the number of drivers), which is 2.

p is the probability of success in a single trial (in this case, the probability of a driver being), which is 0.20.

(n choose k) is the number of combinations of n items taken k at a time.

Calculating:

P(X = 2) = (12 choose 2) * (0.20)^2 * (0.80)^10

= 66 * 0.04 * 0.1073741824

= 0.2834678413

b) Three or four will be drivers:

For this case, we'll find P(X = 3) and P(X = 4) and then add them together.

For P(X = 3):

P(X = 3) = (12 choose 3) * (0.20)^3 * (0.80)^9

= 220 * 0.008 * 0.134456

= 0.23757696

For P(X = 4):

P(X = 4) = (12 choose 4) * (0.20)^4 * (0.80)^8

= 495 * 0.016 * 0.16777216

= 1.3245924272

Adding them together:

P(X = 3 or 4) = 0.23757696 + 1.3245924272

= 1.5621693872

c) At least 7 will be drivers:

To find this probability, we need to calculate P(X = 7) + P(X = 8) + ... + P(X = 12).

For P(X = 7):

P(X = 7) = (12 choose 7) * (0.20)^7 * (0.80)^5

= 792 * 0.128 * 0.32768

= 32.5669376

Similarly, find P(X = 8), P(X = 9), P(X = 10), P(X = 11), and P(X = 12) using the same method.

Finally, add all these probabilities together.

d) At most 5 will be drivers:

To find this probability, we need to calculate P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5).

For P(X = 0):

P(X = 0) = (12 choose 0) * (0.20)^0 * (0.80)^12

= 1 * 1 * 0.0687194767

= 0.0687194767

Similarly, find P(X = 1), P(X = 2), P(X = 3), P(X = 4), and P(X = 5) using the same method.

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Answer:

$297.50

Step-by-step explanation:

Do 15% × $350 = $52.50

Subtract that from the original

350 - 52.50 =$297.5

Answer:

$297.50

Step-by-step explanation:

15% of 350 = 0.15 x 350

0.15 x 350 = 52.50

350 - 52.50 = 297.5

Hope This Helps! :D

Answer:

The probability the mean weight will exceed 3.1 ounces is 0.0668

Step-by-step explanation:

We have a random sample of size n = 36 measures which comes from a normal distribution with a mean of 3 ounces and a standard deviation of 0.4 ounces. Then, we know that the mean weight is also normally distributed with the same mean of 3 ounces and a standard deviation of [tex]0.4/\sqrt{36} = 0.4/6[/tex]. The z-score associated to 3.1 is (3.1-3)/(0.4/6) = 1.5. We are looking for P(Z > 1.5) = 0.0668, i.e., the probability the mean weight will exceed 3.1 ounces is 0.0668

Answer:

32.9 mpg

Step-by-step explanation:

Population mean (μ) = 25.5 mpg

Standard deviation (σ) = 4.5 mpg

Assuming a normal distribution for gasoline use, the manufacturer wants his car to be at the 95-th percentile of the distribution. The 95-th percentile has a corresponding z-score of 1.645. The expression for the z-score for a given gasoline use rate 'X' is:

[tex]z=\frac{X-\mu}{\sigma} \\1.645=\frac{X-25.5}{4.5} \\X=32.9\ mpg[/tex]

The gasoline use rate for the new car must be at least 32.9 mpg

To outperform 95% of the current compacts in fuel economy, the gasoline use rate for the new car must be at least 33.7775 miles per gallon.

To develop a compact car that outperforms 95% of the current compacts in fuel economy, the gasoline use rate for the new car must be at least as good as the top 5% of the current compacts. To find this, we use the z-score formula: z = (x - mean) / standard deviation. Since we want to find the value for x that corresponds to the top 5% (or 0.05) of the distribution, we can find the z-score by using the inverse normal distribution table. Once we have the z-score, we can use the formula z = (x - mean) / standard deviation to solve for x.

Using the inverse normal distribution table, we find that the z-score corresponding to the top 5% is approximately 1.645. Plugging this value into the formula and rearranging to solve for x, we get:

x = mean + (z * standard deviation)

Substituting in the given values, we have:

x = 25.5 + (1.645 * 4.5) = 33.7775

Therefore, the gasoline use rate for the new car must be at least 33.7775 miles per gallon to outperform 95% of the current compacts in fuel economy.

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Answer:

A. Direct, strong

Step-by-step explanation:

If the Pearson's correlation coefficient is positive the relationship coefficient is direct. If it is negative, it is inverse.

If is considerate to be strong if it is larger than 0.7

In this problem, we have that:

The Pearson's correlation coefficient was found to be .78. It is positive, and larger than 0.7. So the correct answer is:

A. Direct, strong

Answer: p(2 lesser than or equal to x lesser than or equal to 4) = 0.6526

Step-by-step explanation:

20% of drivers driving between 11 PM and 3 AM are drunken drivers.

We want to use the binomial distribution to determine the probability that in a random sample of 12 drivers driving between 11 PM and 3 AM, two to four will be drunken drivers.

The formula for binomial distribution is

P( x = r) = nCr × q^n-r × p^r

x = number of drivers

p = probability that the drivers that are drunken.

q= 1-p = probability that the drivers are not drunken.

n = number of sampled drivers.

From the information given,

p = 20/100 = 0.2

q = 1 - p = 1 - 0.2 = 0.8

n = 12

We want to determine

p(2 lesser than or equal to x lesser than or equal to 4)

It is equal to p(x=2) + p(x= 3) + p(x=4)

p(x=2) = 12C2 × 0.8^10 × 0.2^2 = 0.2835

p(x=3) = 12C3 × 0.8^9 × 0.2^3 = 0.2362

p(x=4) = 12C4 × 0.8^8 × 0.2^4 = 0.1329

p(2 lesser than or equal to x lesser than or equal to 4) = 0.2835 + 0.2362 + 0.1329 = 0.6526

Answer:

a) [tex]Sd(X)=\sqrt{Var(X)}=\sqrt{1.36}=1.166[/tex]

b) [tex]Sd(X)=\sqrt{Var(X)}=\sqrt{149}=12.21[/tex]

Step-by-step explanation:

Part a

So then the random variable is given by this table

X     | 1      | 2    | 3    | 4     |

P(X) | 0.2 | 0.2 | 0.2 | 0.4 |

First we need to find the expected value (first moment) and the second moment in order to find the variance and then the standard deviation.

In order to calculate the expected value we can use the following formula:

[tex]E(X)=\sum_{i=1}^n X_i P(X_i)[/tex]

And if we use the values obtained we got:

[tex]E(X)=1*0.2 +2*0.2 +3*0.2 +4*0.4=2.8[/tex]

In order to find the standard deviation we need to find first the second moment, given by :

[tex]E(X^2)=\sum_{i=1}^n X^2_i P(X_i)[/tex]

And using the formula we got:

[tex]E(X^2)=(1^2 *0.2)+(2^2 *0.2)+(3^2 *0.2)+(4^2 *0.4)=9.2[/tex]

Then we can find the variance with the following formula:

[tex]Var(X)=E(X^2)-[E(X)]^2 =9.2-(2.8)^2 =1.36[/tex]

And then the standard deviation would be given by:

[tex]Sd(X)=\sqrt{Var(X)}=\sqrt{1.36}=1.17[/tex]

Part b

So then the random variable is given by this table

X     | -20  | -10 | 0   | 10  |20  |

P(X) | 0.1 | 0.2 | 0.4 | 0.1 | 0.2  |

First we need to find the expected value (first moment) and the second moment in order to find the variance and then the standard deviation.

In order to calculate the expected value we can use the following formula:

[tex]E(X)=\sum_{i=1}^n X_i P(X_i)[/tex]

And if we use the values obtained we got:

[tex]E(X)=(-20*0.1) +(-10*0.2) +(0*0.4) +(10*0.1)+(20*0.2)=1[/tex]

In order to find the standard deviation we need to find first the second moment, given by :

[tex]E(X^2)=\sum_{i=1}^n X^2_i P(X_i)[/tex]

And using the formula we got:

[tex]E(X^2)=((-20)^2 *0.1)+((-10)^2 *0.2)+(0^2 *0.4)+(10^2 *0.1)+(20^2 *0.2)=150[/tex]

Then we can find the variance with the following formula:

[tex]Var(X)=E(X^2)-[E(X)]^2 =150-(1)^2 =149[/tex]

And then the standard deviation would be given by:

[tex]Sd(X)=\sqrt{Var(X)}=\sqrt{149}=12.21[/tex]

The standard deviation of the first probability distribution is 1.10 and of the second distribution is 15.23.

The standard deviation σ of a probability distribution is calculated by first finding its mean μ, then using the formula:

σ = √[Σ(x-μ)^2 * P(X = x)]

For the first distribution, the mean μ = (1*0.2) + (2*0.2) + (3*0.2) + (4*0.4) = 3.2. The standard deviation σ = √[(1-3.2)^2 * 0.2 + (2-3.2)^2 * 0.2 + (3-3.2)^2 * 0.2 + (4-3.2)^2 * 0.4] = 1.10.

For the second distribution, the mean μ = (-20*0.1) + (-10*0.2) + (0*0.4) + (10*0.1) + (20*0) + (30*0.2) = -2. The standard deviation σ = √[(-20+2)^2 * 0.1 + (-10+2)^2 * 0.2 + (0+2)^2 * 0.4 + (10+2)^2 * 0.1 + (20+2)^2 * 0 + (30 + 2)^2 * 0.2] = 15.23.

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Answer:

There is an 1.39% probability that 62 or more Americans would indicate that the average person is not very considerate of others when talking on a cellphone.

Step-by-step explanation:

For each American, there are only two possible outcomes. Either they indicate that the average person is not very considerate of others when talking on a cellphone, of they indicate that the average person is considerate. This means that we use the binomial probability distribution to solve this problem.

This distribution can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

[tex]E(X) = np[/tex]

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X), \sigma = \sqrt{V(X)}[/tex]

In this problem, we have that:

There are 100 americans, so [tex]n = 100[/tex]

51% of Americans say the average person is not very considerate of others when talking on a cellphone. This means that [tex]p = 0.51[/tex].

Find the approximate probability that 62 or more Americans would indicate that the average person is not very considerate of others when talking on a cellphone.

This is 1 subtracted by the pvalue of Z when [tex]X = 62[/tex].

We have that

[tex]\mu = E(X) = np = 100*0.51 = 51[/tex]

[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{100*0.51*0.49} = 5[/tex]

So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{62 - 51}{5}[/tex]

[tex]Z = 2.2[/tex]

[tex]Z = 2.2[/tex] has a pvalue of 0.9861.

This means that there is a 1-0.9861 = 0.0139 = 1.39% probability that 62 or more Americans would indicate that the average person is not very considerate of others when talking on a cellphone.

Answer:

$-2.875 in 1 game

$-287.5 in 100 games

Step-by-step explanation:

As the probability of getting head in tossing a fair coin is 0.5, the probability of getting 3 heads with 3 coins is 0.5*0.5*0.5 = 0.125

So you have a 0.125 chance of winning and 0.875 chance of losing a game

The expected value for 1 game would be

0.125*5 - 0.875*4 = -2.875

On average you would be losing 2.875  per game. In 100 games you should expect to lose 287.5 dollar

Answer:

  7 ft 1 1/4 in

Step-by-step explanation:

The total width of 3 paintings and 2 spaces is ...

  (10.5 in) + (3 in) + (3 ft 6 in) + (3 in) + (2 ft 2 3/4 in)

  = (10.5 +3 +6 +3 +2.75) in + (3 +2) ft

  = 25.25 in + 5 ft = 1.25 in + 24 in + 5 ft

  = 7 ft 1 1/4 in

The total width of wall space needed for the paintings is 7 feet 1 1/4 inches.

_____

If two more 3-inch spaces are added, one on each end, then the total width is 7 feet 7 1/4 inches. The problem isn't clear about that, saying only that there are spaces between the paintings.

Answer:

test statistic is 0.176

Step-by-step explanation:

Given Data

p>33

a=0.05

n=50

x=33.3

d(population deviation)=12

Test statistics=?

Solution

Test statistic z=(p-x)\(d/sqrt(50))

z=(33.3-30)\(12\sqrt(50))

z=0.176

In probability, the sampling distribution of the sample proportions can be approximated by a normal probability distribution when D. Both a and b are true.

The sampling distribution of the sample proportions can be approximated by a normal probability distribution whenever np is greater than or equal to 5 and when n(1-p) is greater than or equal to 5.

When the continuous random variable is uniformly distributed between a and b, the probability density function between a and b is 1/(b - a).

The probability that the sample mean will be between 80.54 and 88.9 will be:

= P[Z = 88.9 - 84)/(12/✓36)] - P(Z = 80.54 - 84)/(12/✓36)]

= P(Z = 2.45) - P(Z = -1.73)

= 0.9929 - 0.0418

= 0.9511

Therefore, the probability is 0.9511.

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Answer:

P(The grade of the boy is A| He has played video games)

is, [tex]\simeq 0.54[/tex]

P(The grade of the boy is B| He has played video games)

is [tex]\simeq 0.33[/tex]

P(The grade of the boy is C| He has played video games)

is [tex]\simeq 0.14[/tex]

Step-by-step explanation:

The total no. of boys who have played video games,

= (730 + 444 + 190)

=1360

Now, from the given data,

P(The grade of the boy is A| He has played video games)

= [tex]\frac {730}{1360}[/tex]

[tex]\simeq 0.54[/tex]

P(The grade of the boy is B| He has played video games)

= [tex]\frac {444}{1360}[/tex]

[tex]\simeq 0.33[/tex]

P(The grade of the boy is C| He has played video games)

= [tex]\frac {190}{1360}[/tex]

[tex]\simeq 0.14[/tex]

The conditional distribution of grades for boys who have played games shows that 53% received A's and B's, 33% received C's, and 14% received D's and F's. This is calculated by dividing each grade category by the total number of boys who played games and rounding to two decimal places.

To find the conditional distribution of grades for boys who have played games, we need to calculate the proportion of each grade category relative to the total number who played.

Steps to Calculate Conditional Distribution

Therefore, the conditional distribution of grades for boys who have played games is 0.53 for A's and B's, 0.33 for C's, and 0.14 for D's and F's.

Answer:3

Step-by-step explanation:plug in the numbers in the algebraic expression. Should look like this: 6(9+3)/(3+9)2. After you set it up like this you have to get rid of the parentheses by multiplying the outside number to the numbers in the parentheses. So do: 6×9=54, then 6×3=18 so now your problem should look like this 54+18/ (3+9)2 do the same with the other side so: 2×3=6, then 9×2=18 now your problem should look like this: 54+18/6+18 now you add each side up 54+18=72 and 6+18=24 . Then divide those 2 answers which looks like this 72/24 = 3.