Question

According to the Centers for Disease Control and Prevention (February 18, 2016), 1 in 3 American adults don’t get enough sleep. A researcher wants to determine if Americans are sleeping less than the recommended 7 hours of sleep on weekdays. He takes a random sample of 150 Americans and computes the average sleep time of 6.7 hours on weekdays. Assume that the population is normally distributed with a known standard deviation of 2.1 hours. Test the researcher’s claim at α = 0.01

a. Select the relevant null and the alternative hypotheses.
A. H0: μ = 7; HA: μ ≠ 7
B. H0: μ ≤ 7; HA: μ ≥ 7
C. H0: μ ≥ 7; HA: μ < 7
b. Calculate the value of the test statistic. (Negative value should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)
Test statistic = ?
c. Find the p-value
A. p-value < 0.01
B. 0.01 ≤ p-value < 0.025
C. 0.025 ≤ p-value < 0.05
D. 0.05 ≤ p-value < 0.10
E. p-value ≥ 0.10
d. What is the conclusion at α = 0.01?
A. Reject H0 since the p-value is greater than α.
B. Reject H0 since the p-value is smaller than α.
C. Do not reject H0 since the p-value is greater than α.
D. Do not reject H0 since the p-value is smaller than α.
e. Make an inference.
A. There is insufficient evidence to suggest that Americans sleep less than the recommended 7 hours of sleep.
B. There is sufficient evidence to suggest that Americans sleep less than the recommended 7 hours of sleep.

Answer 1

A) Option C: H0: μ ≥ 7; HA: μ < 7

B) test statistic = -1.75

C) Option C: 0.025 ≤ p-value < 0.05

D) Option C: Do not reject H0 since the p-value is greater than α.

E) Option A: There is insufficient evidence to suggest that Americans sleep less than the recommended 7 hours of sleep.

Explanation:

We are given;

Population mean; μ = 7

Sample mean; x = 6.7

Standard deviation; s = 2.1

Sample size; n = 150

A) Defining the hypothesis, we have;

Null Hypothesis; H0: μ ≥ 7

Alternative hypothesis; HA: μ < 7

B) Formula for the test statistic is;

z = (x - μ)/(s/√n)

Plugging in the relevant values, we have;

z = (6.7 - 7)/(2.1/√150)

z = -0.3/0.1715

z = -1.75

C) Using online p-value from z-score calculator attached with; z = - 1.75 ; α = 0.01 and one tailed distribution, we have;

The P-Value = 0.040059

Thus,the most appropriate answer is;

Option C: 0.025 ≤ p-value < 0.05

D) P-value is greater than the significance level, thus we fail to reject the null hypothesis.

E) We will conclude that there is insufficient evidence to suggest that Americans sleep less than the recommended 7 hours of sleep.