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. (You may find it useful to reference the appropriate table:z table or t table)a. Select the relevant null and the alternative hypotheses.H0: μ = 7; HA: μ ≠ 7H0: μ ≤ 7; HA: μ ≥ 7H0: μ ≥ 7; HA: μ < 7b. 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.p-value < 0.010.01 ≤ p-value < 0.0250.025 ≤ p-value < 0.050.05 ≤ p-value < 0.10p-value ≥ 0.10d. What is the conclusion at α = 0.01?Reject H0 since the p-value is greater than α.Reject H0 since the p-value is smaller than α.Do not reject H0 since the p-value is greater than α.Do not reject H0 since the p-value is smaller than α.e. Make an inference.There is insufficient evidence to suggest that Americans sleep less than the recommended 7 hours of sleep.There is sufficient evidence to suggest that Americans sleep less than the recommended 7 hours of sleep.

Answer 1

A) Null hypothesis;H0: μ ≥ 7

Alternative Hypothesis;Ha: μ < 7

B) t ≈ -1.75

C) 0.025 ≤ p-value < 0.05

D) Do not reject H0 since the p-value is greater than α

E) 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

Sample Standard deviation; s = 2.1

Level of significance; α = 0.01

Sample size; n = 150

A) Now, the hypotheses can be stated as;

Null hypothesis;H0: μ ≥ 7

Alternative Hypothesis;Ha: μ < 7

B) The test statistic will be calculated from the formula;

t = (x' - μ)/(s/√n)

Plugging in the relevant values, we have;

t = (6.7 - 7)/(2.1/√150)

t = -0.3/0.1715

t = -1.7493

t ≈ -1.75

C) From online p-value from t-score calculator attached using t-score of -1.75; DF = 150 - 1 = 149; significance level of 0.01 and one tail, we have;

The p-value = 0.041088

The closest answer among the options is;

0.025 ≤ p-value < 0.05

D) Since the p-value is greater than the significance level of 0.01, thus we will fail to reject the null hypothesis

E) From D above, we will conclude that there is insufficient evidence to suggest that Americans sleep less than the recommended 7 hours of sleep.