# Determine the p-value for the following hypothesis tests involving Student’s t-distribution with 10 degrees of freedom

1) Lunch breaks are often considered too short, and employees frequently develop a habit of “stretching” them. The manager at Giant Mart randomly identified 22 employees and observed the length of their lunch breaks (in minutes) for one randomly selected day during the week.

31 22 39 35 26 34 22 28 27 20 27

35 30 27 26 32 25 24 29 32 27 42

(a) Show evidence that the normality assumptions are satisfied. Analyze the Normal Q-Q plot given below and determine whether or not the data has an approximately normal distribution. (choose one)

The data appears to follow a straight line, indicating a non-normal distribution.

The data appears to follow a straight line, indicating an approximately normal distribution.

The data appears to not follow a straight line, indicating a non-normal distribution.

The data appears to not follow a straight line, indicating an approximately normal distribution.

(b) Find the 95% confidence interval for “mean length of lunch breaks” at Giant Mart. (Give your answers correct to two decimal places.)

___ to ___ minutes

2) Determine the p-value for the following hypothesis tests involving Student’s t-distribution with 10 degrees of freedom. (Give your answers correct to three decimal places.)

(a) H0: μ = 15.1, Ha: μ < 15.1, t = -1

__________

(b) H0: μ = 15.1, Ha: μ > 15.1, t = 1.0

__________

(c) H0: μ = 15.1, Ha: μ ≠ 15.1, t = 1.0

__________

(d) H0: μ = 15.1, Ha: μ ≠ 15.1, t = -1

_______

3) The recommended number of hours of sleep per night is 8 hours, but everybody “knows” that the average college student sleeps less than 7 hours. The number of hours slept last night by 10 randomly selected college students is listed below. Complete the following hypothesis test: Ho: μ = 7, Ha: μ < 7, α = 0.05.

6.6 7.8 7.9 6.3 6.9 5.3 7.8 5.9 6 7.4

(a) Find t. (Give your answer correct to two decimal places.)

____________

(ii) Find the p-value. (Give your answer correct to four decimal places.)

____________

(b) State the appropriate conclusion. (choose one)

Reject the null hypothesis, there is not significant evidence that the mean is less than 7.

Reject the null hypothesis, there is significant evidence that the mean is less than 7.

Fail to reject the null hypothesis, there is not significant evidence that the mean is less than 7.

Fail to reject the null hypothesis, there is significant evidence that the mean is less than 7.

4)Acetaminophen is an active ingredient found in more than 600 over-the-counter and prescription medicines, such as pain relievers, cough suppressants, and cold medications. It is safe and effective when used correctly, but taking too much can lead to liver damage.†

A researcher believes the amount of acetaminophen in a particular brand of cold tablets contains a mean amount of acetaminophen per tablet different from the 600 mg claimed by the manufacturer. A random sample of 37 tablets had a mean acetaminophen content of 596.7 mg with a standard deviation of 5.2 mg.

(a) Is the assumption of normality reasonable? (choose one)

a) Yes, because sample size is large.

b) Yes, because sample size is small.

c) No, because sample size is large.

d) No, because sample size is small.

(b) Construct a 99% confidence interval for the estimate of the mean acetaminophen content. (Give your answers correct to two decimal places.)

______ to ______ mg

(c) What does the confidence interval found in part (b) suggest about the mean acetaminophen content of one pill? (choose 1)

The mean amount of acetaminophen in one pill is 600 mg exactly.

The mean amount of acetaminophen in one pill is less than 600 mg.

The mean amount of acetaminophen in one pill may be 600 mg.

The mean amount of acetaminophen in one pill is more than 600 mg.

5)Find these critical values by using Table 8 of Appendix B.

(a) χ2(11, 0.01)

answer: ________

(b) χ2(20, 0.025)

answer: _______

(c) χ2(2, 0.10)

answer: ________

(d) χ2(28, 0.01)

answer: ________

(e) χ2(26, 0.95)

answer: ________

(f) χ2(17, 0.975)

answer: ________

(g) χ2(60, 0.90)

answer: _______

(h) χ2(22, 0.99)

answer: _______

6) A random sample of 61 observations was selected from a normally distributed population. The sample mean wasx = 96.1 (x bar),and the sample variance was s2 = 33.4Does this sample show sufficient reason to conclude that the population standard deviation is not equal to 8.4 at the 0.01 level of significance?

(a) State the appropriate null and alternative hypotheses.

H0: σ _____ 8.4

Ha: σ _____ 8.4

(ii) Find χ2. (Give your answer correct to two decimal places.)

___________

(iii) Find the p-value. (Give your answer bounds exactly.)

______< p ≤______

(b) State the appropriate conclusion. (choose 1)

Reject H0. There is not sufficient reason to conclude that the population standard deviation is not equal to 8.4 at the 0.01 level of significance.

Reject H0. There is sufficient reason to conclude that the population standard deviation is not equal to 8.4 at the 0.01 level of significance.

Fail to reject H0. There is not sufficient reason to conclude that the population standard deviation is not equal to 8.4 at the 0.01 level of significance.

Fail to reject H0. There is sufficient reason to conclude that the population standard deviation is not equal to 8.4 at the 0.01 level of significance.

7) Determine the critical region and critical value(s) that would be used to test the following hypotheses using the classical approach when F is used as the test statistic.

(a) H0: σ12 = σ22 vs. Ha: σ12 > σ22, with n1 = 7, n2 = 20, and α = 0.05

critical region F > _______

(b) H0:

σ12

σ22

= 1 vs.

σ12

σ22

Ha:

≠ 1,

with n1 = 21, n2 = 41, and α = 0.05

critical region F <_____ or F > _______

c)

H0:

σ12

σ22

= 1 vs.

σ12

σ22

Ha:

>1,

with n1 = 7, n2 = 6, and α = 0.01

critical region F > ________

(d)

H0: σ1 = σ2 vs. Ha: σ1 < σ2, with n1 = 41, n2 = 12, and α = 0.01

critical region F > ________

8) A study in Pediatric Emergency Care compared the injury severity between younger and older children. One measure reported was the Injury Severity Score (ISS). The standard deviation of ISS scores for 28 children 8 years or younger was 22.9, and the standard deviation for 48 children older than 8 years was 6.2. Assume that ISS scores are normally distributed for both age groups. At the 0.01 level of significance, is there sufficient reason to conclude that the standard deviation of ISS scores for younger children is larger than the standard deviation of ISS scores for older children?

(a) Find F. (Give your answer correct to two decimal places.)

_______

(ii) Find the p-value. (Give your answer correct to four decimal places.)

_________

(b) State the appropriate conclusion. (choose 1)

Reject the null hypothesis, there is significant evidence to show that the standard deviation is greater for children 8 years or younger.

Reject the null hypothesis, there is not significant evidence to show that the standard deviation is greater for children 8 years or younger.

Fail to reject the null hypothesis, there is not significant evidence to show that the standard deviation is greater for children 8 years or younger.

Fail to reject the null hypothesis, there is significant evidence to show that the standard deviation is greater for children 8 years or younger.

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