A physical therapist wants to determine the difference in the proportion of men and women who participate in regular sustained physical activity. What sample size should be obtained if he wishes the estimate to be within two percentage points with 90​% ​confidence, assuming that
​(a) he uses the estimates of 21.6​% male and 19.2​% female from a previous​ year?
​(b) he does not use any prior​ estimates?

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:

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:

The 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.

Answer: 28 * 1

Step-by-step explanation:

Answer: The part of the stock owned by neither Rob nor​ Talia is

13/16

Step-by-step explanation:

Let the total stock in the local bread company be represented by x

Rob owns 1/8 of the stock in the local bread company. This means that the amount of stock owned by Rob is 1/8 × x = x/8

His sister Talia owns half as much stock as Rob. This means that the amount of stock owned by Talia would be 1/2 × x/8 = x/16

Total part of the stock owned by Talia and Rob would be

x/8 + x/16 = (2x+x)/16 = 3x/16

The part of the stock owned by neither Rob nor​ Talia will be the total stock minus the part of the stock owned by Rob and Talia.

It becomes

x - 3x/16 = (16x - 3x)/16 = 13x/16

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.

Answer: Our required probability is 0.351.

Step-by-step explanation:

Since we have given that

Probability of getting classroom 1 = [tex]\dfrac{1}{2}[/tex]

Probability of getting classroom 2 = [tex]\dfrac{1}{2}[/tex]

Probability of getting girl from classroom 1 = [tex]\dfrac{13}{18}[/tex]

Probability of getting girl from classroom 2 = [tex]\dfrac{9}{23}[/tex]

Using "Bayes theorem".

so, the probability that classroom 2 was chosen given that a girl was chosen would be

[tex]\dfrac{\dfrac{1}{2}\times \dfrac{9}{23}}{\dfrac{1}{2}\times \dfrac{9}{23}+\dfrac{1}{2}\times \dfrac{13}{18}}\\\\=\dfrac{0.195}{0.195+0.361}\\\\=0.351[/tex]

Hence, our required probability is 0.351.

Answer:

The 99% confidence interval would be given by (12.10;12.20)

Step-by-step explanation:

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

[tex]\bar X=12.15[/tex] represent the sample mean  

[tex]\mu[/tex] population mean (variable of interest)

[tex]\sigma=0.15[/tex] represent the population standard deviation

n=60 represent the sample size  

Assuming the X follows a normal distribution

[tex]X \sim N(\mu, \sigma=0.15[/tex]

The sample mean [tex]\bar X[/tex] is distributed on this way:

[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]

The confidence interval on this case is given by:

[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex]   (1)

The next step would be find the value of [tex]\z_{\alpha/2}[/tex], [tex]\alpha=1-0.99=0.01[/tex] and [tex]\alpha/2=0.005[/tex]

Using the normal standard table, excel or a calculator we see that:

[tex]z_{\alpha/2}=2.58[/tex]

Since we have all the values we can replace:

[tex]12.15 - 2.58\frac{0.15}{\sqrt{60}}=12.10[/tex]  

[tex]12.15 + 2.58\frac{0.15}{\sqrt{60}}=12.20[/tex]  

So on this case the 99% confidence interval would be given by (12.10;12.20)  

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]

Learn More:https://brainly.com/question/6077878

(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

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.

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.

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.

Answer:

Hence the value of damping coefficient = 1.49071.

Step-by-step explanation:

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.

Answer:

41 is correct

Step-by-step explanation:

90 + 49 = 139

180 - 139 = 41

I'm going to assume the joint density function is

[tex]f_{X,Y}(x,y)=\begin{cases}\frac67(x^2+\frac{xy}2\right)&\text{for }0<x<1,0<y<2\\0&\text{otherwise}\end{cases}[/tex]

a. In order for [tex]f_{X,Y}[/tex] to be a proper probability density function, the integral over its support must be 1.

[tex]\displaystyle\int_0^2\int_0^1\frac67\left(x^2+\frac{xy}2\right)\,\mathrm dx\,\mathrm dy=\frac67\int_0^2\left(\frac13+\frac y4\right)\,\mathrm dy=1[/tex]



b. You get the marginal density [tex]f_X[/tex] by integrating the joint density over all possible values of [tex]Y[/tex]:

[tex]f_X(x)=\displaystyle\int_0^2f_{X,Y}(x,y)\,\mathrm dy=\boxed{\begin{cases}\frac67(2x^2+x)&\text{for }0<x<1\\0&\text{otherwise}\end{cases}}[/tex]

c. We have

[tex]P(X>Y)=\displaystyle\int_0^1\int_0^xf_{X,Y}(x,y)\,\mathrm dy\,\mathrm dx=\int_0^1\frac{15}{14}x^3\,\mathrm dx=\boxed{\frac{15}{56}}[/tex]

d. We have

[tex]\displaystyle P\left(X<\frac12\right)=\int_0^2\int_0^{1/2}f_{X,Y}(x,y)\,\mathrm dx\,\mathrm dy=\frac5{28}[/tex]

and by definition of conditional probability,

[tex]P\left(Y>\dfrac12\mid X<\dfrac12\right)=\dfrac{P\left(Y>\frac12\text{ and }X<\frac12\right)}{P\left(X<\frac12\right)}[/tex]

[tex]\displaystyle=\dfrac{28}5\int_{1/2}^2\int_0^{1/2}f_{X,Y}(x,y)\,\mathrm dx\,\mathrm dy=\boxed{\frac{69}{80}}[/tex]

e. We can find the expectation of [tex]X[/tex] using the marginal distribution found earlier.

[tex]E[X]=\displaystyle\int_0^1xf_X(x)\,\mathrm dx=\frac67\int_0^1(2x^2+x)\,\mathrm dx=\boxed{\frac57}[/tex]

f. This part is cut off, but if you're supposed to find the expectation of [tex]Y[/tex], there are several ways to do so.

[tex]f_Y(y)=\displaystyle\int_0^1f_{X,Y}(x,y)\,\mathrm dx=\begin{cases}\frac1{14}(4+3y)&\text{for }0<y<2\\0&\text{otherwise}\end{cases}[/tex]

[tex]\implies E[Y]=\displaystyle\int_0^2yf_Y(y)\,\mathrm dy=\frac87[/tex]

[tex]f_{Y\mid X}(y\mid x)=\dfrac{f_{X,Y}(x,y)}{f_X(x)}=\begin{cases}\frac{2x+y}{4x+2}&\text{for }0<x<1,0<y,2\\0&\text{otherwise}\end{cases}[/tex]

The law of total expectation says

[tex]E[Y]=E[E[Y\mid X]][/tex]

We have

[tex]E[Y\mid X=x]=\displaystyle\int_0^2yf_{Y\mid X}(y\mid x)\,\mathrm dy=\frac{6x+4}{6x+3}=1+\frac1{6x+3}[/tex]

[tex]\implies E[Y\mid X]=1+\dfrac1{6X+3}[/tex]

This random variable is undefined only when [tex]X=-\frac12[/tex] which is outside the support of [tex]f_X[/tex], so we have

[tex]E[Y]=E\left[1+\dfrac1{6X+3}\right]=\displaystyle\int_0^1\left(1+\frac1{6x+3}\right)f_X(x)\,\mathrm dx=\frac87[/tex]

The question deals with the joint probability density function fX,Y (x, y). It covers verification of the function, calculation of density function of X, probabilities of certain conditions, and finding expectation values of X and Y.

The question revolves around the concept of continuous probability density functions (pdf). Here, the function fX,Y (x, y) is the joint pdf of X and Y under given interval constraints.

(a) To verify if fX,Y (x, y) serves as a joint density function, one would need to confirm that it complies with two conditions: The function should be nonnegative, and the integral over its entire domain should equal to one. You would need to compute a double integral over the range of fX,Y and ascertain if it equals one.

(b)The density function of X can be obtained by integrating fX,Y (x, y) over the range of y i.e., integrate from 0 to 2 for given x.

(c) To compute P(X > Y), the region where this condition holds true needs to be calculated. The marginal density of X must be integrated over this region.

(d) P(Y > 1/2 | X < 1/2) is computed by integrating the joint density function over the region defined by Y > 1/2 and X < 1/2, normalised by the probability of the event X < 1/2.

(e) E(X) is the expectation of X and can be calculated by integrating x*fX(x), where fX(x) is the marginal density function of X.

(f) Similarly, E(Y) can be computed by integrating y*fY(y), with fY(y) being the marginal density of Y.

https://brainly.com/question/35625239

#SPJ3

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]

Answer:

Step-by-step explanation:

To compute the tSTAT, plug the given values into the formula tSTAT = (X1 - X2) / sqrt(Sp2 * (1/n1 + 1/n2)). For the degrees of freedom, calculate df = n1 + n2 - 2. Whether to reject or not the null hypothesis H0 depends on whether the absolute value of tSTAT is greater than the critical value tCritical.

The question pertains to statistical hypothesis testing, specifically the t-test for comparing two independent samples. To solve this:

a. First, calculate the tSTAT as follows: tSTAT = (X1 - X2) / sqrt(Sp2 * (1/n1 + 1/n2)). Using your given values, this becomes tSTAT = (42 - 34) / sqrt(776 * (1/8 + 1/15)). Evaluating the right side will give you the tSTAT.

b. For this t-test, the degree of freedom (df) is calculated as df = n1 + n2 - 2, which gives df = 8+15-2 = 21.

c. The decision whether to reject or fail to reject the null hypothesis H0 depends on the comparison of the calculated tSTAT with the critical value (from the t-distribution table for df = 21). If |tSTAT| > tCritical, reject H0; otherwise, fail to reject H0.

https://brainly.com/question/31829815

#SPJ3

Final answer:

The demand for tie-dyed T-shirts will drop by approximately 23 T-shirts when the price is increased from $15 to $17. This calculation is based on the function q = 510 − 90[tex]p^{0.5}[/tex], which represents the demand in relation to the price.

Explanation:

The student is asking how fast the demand for tie-dyed T-shirts will drop if the price increases by $2 when the demand is represented by the function q = 510 − 90[tex]p^{0.5}[/tex]. Since the current price is $15, we need to calculate the demand at both $15 and $17 to find the rate of change in demand when the price is raised by $2.

To find the demand at the current price of $15, substitute p = 15 into the demand function:
q = 510 − 90[tex](15)^{0.5}[/tex]
q = 510 − 90(3.87)
q = 510 − 348.3
q = 161.7

To find the demand when the price is $17, substitute p = 17 into the demand function:
q = 510 − 90[tex](17)^{0.5}[/tex]
q = 510 − 90(4.12)
q = 510 − 371
q = 139

The change in demand is the difference between the two quantities:
Δq = 161.7 - 139 = 22.7
The demand drops by approximately 23 T-shirts when the price increases by $2.

The demand will drop by approximately [tex]\( \boed{23} \)[/tex] T-shirts per month.

Step 1

To determine how fast the demand for tie-dyed T-shirts will drop when the price increases by $2, we need to calculate the rate of change of the demand q with respect to the price p. The given demand function is:

[tex]\[ q = 510 - 90p^{0.5} \][/tex]

We need to find the derivative of q with respect to p, which gives us the rate of change of demand with respect to price.

Step 2

First, let's find the derivative [tex]\( \frac{dq}{dp} \):[/tex]

[tex]\[ q = 510 - 90p^{0.5} \]\[ \frac{dq}{dp} = -90 \cdot \frac{1}{2} p^{-0.5} \]\[ \frac{dq}{dp} = -45 p^{-0.5} \]\[ \frac{dq}{dp} = -45 \cdot \frac{1}{\sqrt{p}} \][/tex]

Step 3

Now, we need to evaluate this derivative at the current price [tex]\( p = 15 \)[/tex]:

[tex]\[ \frac{dq}{dp} \bigg|_{p=15} = -45 \cdot \frac{1}{\sqrt{15}} \][/tex]

Calculate \[tex]( \sqrt{15} \)[/tex]:

[tex]\[ \sqrt{15} \approx 3.872 \][/tex]

So,

[tex]\[ \frac{dq}{dp} \bigg|_{p=15} = -45 \cdot \frac{1}{3.872} \approx -11.62 \][/tex]

This means the rate of change of demand with respect to price at [tex]\( p = 15 \)[/tex] is approximately [tex]\(-11.62\)[/tex] T-shirts per dollar.

Given that the price is increased by $2, we need to find how much the demand drops for this price increase:

[tex]\[ \frac{dq}{dp} \times 2 \approx -11.62 \times 2 \approx |-23.24| \][/tex]

Rounded to the nearest whole number, the demand drops by approximately 23 T-shirts per month when the price is raised by $2.

Answer:

We conclude that the telecommuting reduced the number of sick days.

Step-by-step explanation:

We are given the following in the question:

Population mean, μ = 5.4

Sample mean, [tex]\bar{x}[/tex] = 4.4

Sample size, n = 80

Alpha, α = 0.05

Sample standard deviation, s = 2.8

First, we design the null and the alternate hypothesis

[tex]H_{0}: \mu = 5.4\\H_A: \mu < 5.4[/tex]

We use One-tailed z test to perform this hypothesis.

Formula:

[tex]z_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}} }[/tex]

Putting all the values, we have

[tex]z_{stat} = \displaystyle\frac{4.4 - 5.4}{\frac{2.8}{\sqrt{80}} } = -3.1943[/tex]

Now,

We calculate the p-value from the z-table

P-value = 0.000702, which is very low.

b) Since we have a very low p-value that is it is less than the significance level, we fail to accept the null hypothesis and reject it at significance level of 5%. We would require a very small value of alpha for us to accept the null hypothesis thus we would conclude that there is statistically significant evidence to reject the null hypothesis that the mean was equal to 5.4.

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.

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.

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.

https://brainly.com/question/30214499

#SPJ3

a) The probability P(student will graduate | student is female) is 0.70.

b) The probability P(student will graduate and student is female) is 0.567.

c) P(student will graduate | student is male) is 0.80.

d) The P(student will graduate and student is male) is 0.152.

e) The value of P(student will graduate) is 0.719.

f) [tex]\(P(\text{graduate and is female})\)[/tex] considers all female graduates, while [tex]\(P(\text{graduate | female})\)[/tex] considers only the female students and then calculates the proportion of them who will graduate.

Given probabilities:

P (female) = 0.81

P(male) = 0.19

P( graduate | female) = 0.70

P( graduate | male) = 0.80

(a) [tex]\(P(\text{graduate | female})\):[/tex]

This is already given as 0.70.

(b) [tex]\(P(\text{graduate and female})\):[/tex]

This can be calculated as the product of the probability of being female and the probability of graduating given female:

[tex]\[P(\text{graduate and female}) = P(\text{female}) \times P(\text{graduate | female})[/tex]

                                      [tex]= 0.81 \times 0.70[/tex]

                                      = 0.567

(c) [tex]\(P(\text{graduate | male})\)[/tex]:

This is already given as 0.80.

(d) [tex]\(P(\text{graduate and male})\):[/tex]

This can be calculated as the product of the probability of being male and the probability of graduating given male:

[tex]\[P(\text{graduate and male}) = P(\text{male}) \times P(\text{graduate | male})[/tex]

                                    = 0.19 x 0.80

                                    = 0.152

(e) This can be calculated by considering both male and female students who will graduate:

[tex]\[P(\text{graduate}) = P(\text{graduate and female}) + P(\text{graduate and male})[/tex]

                    = 0.567 + 0.152

                    = 0.719

(f) The probabilities [tex]\(P(\text{graduate and is female})\) and \(P(\text{graduate | female})\)[/tex] are different because they refer to different situations.

[tex](P(\text{graduate and is female})\)[/tex] represents the probability that a randomly selected student is both female and will graduate.

[tex](P(\text{graduate | female})\)[/tex] represents the conditional probability that a randomly selected student will graduate given that they are female.

Learn more about Probability here:

https://brainly.com/question/32117953

#SPJ12

The probabilities of a selected incoming freshman nursing student graduating are calculated using the given percentages for females and males. The results are 0.700, 0.567, 0.800, 0.152, and 0.719 for each respective part of the question. The conditional probability is different from the joint probability since it gives the likelihood of one event under the condition that another event has occurred.

To calculate the required probabilities related to incoming freshmen nursing students at Litchfield College of Nursing, we use the given percentages. We have P(F) representing the probability a student is female, and P(M) the probability a student is male. We also have P(BSN|F) representing the probability of graduating with a BSN degree given that the student is female, and P(BSN|M) for males.

The probabilities P(will graduate and is female) and P(will graduate | female) are different because the first is the probability of two events occurring together (being female and graduating), while the second is the conditional probability of graduating given that the student is already identified as female.

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.

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.

Learn more about pythagorean theorem on:

https://brainly.com/question/654982

Final answer:

The best point estimate for the average number of credit hours per semester for all students at the local college is 9.1.

Explanation:

The best point estimate for the average number of credit hours per semester for all students at the local college can be found using the sample mean. In this case, the study found that students earned an average of 9.1 credit hours per semester based on a sample of 96 students. Therefore, the best point estimate for the average number of credit hours per semester for all students at the local college is 9.1.

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]

The volume of the original cone is multiplied by 27

We have been given that the height and radius of a cone are each multiplied by 3.

We need to find what effect it has on the volume of the cone.

The volume of the original cone is given as follows:

[tex]\text {Volume of cone}=\frac{1}{3} \pi r^{2} h[/tex]

Now, the radius and height are multiplied by 3.

Therefore, the new radius is ‘3r’ and the new height is ‘3h’.

Hence the new volume is  

[tex]\text {Volume of new cone}=\frac{1}{3} \pi(3 r)^{2}(3 h)=9 \pi r^{2} h[/tex]

original volume of cone x 27 = volume of new cone

Therefore, the original volume of cone is multiplied by 27

Final answer:

When the height and radius of a cone are each multiplied by 3, the volume of the cone is multiplied by 27.

Explanation:

When the height and radius of a cone are each multiplied by 3, the effect on the volume of the cone is that the volume is multiplied by 27.

The formula for the volume of a cone is V = (1/3)πr²h. When the height and radius are multiplied by 3, the new volume would be V' = (1/3)π(3r)²(3h) = 27(1/3)πr²h = 27V.

Therefore, the volume of the cone is multiplied by 27.

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.

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.

https://brainly.com/question/34713449

#SPJ3

Answer:  2.  H0: μ = 40 vs. H1: μ < 40

Step-by-step explanation:

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).

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.

https://brainly.com/question/34171008

#SPJ12

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

#SPJ2