Question

An article in the National Geographic News (February 24, 2005) reports that Americans are increasingly skimping on their 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. Use Table 1.

a.

Select the relevant null and the alternative hypotheses.

	H0: μ = 7; HA: μ ≠ 7
	H0: μ ≤ 7; HA: μ ≥ 7
	H0: μ ≥ 7; HA: μ < 7

b.	Calculate the value of the test statistic. (Negative value should be indicated by a minus sign. Round your answer to 2 decimal places.)

Test statistic

c.

The p-value is: (Round p-value to 4 decimal places.)

	p-value < 0.01
	0.01 ≤ p-value < 0.025
	0.025 ≤ p-value < 0.05
	0.05 ≤ p-value < 0.10
	p-value ≥ 0.10

d.

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 do not sleep less than the recommended 7 hours of sleep.
	There is insufficient evidence to suggest that Americans sleep less than the recommended 7 hours of sleep.

Answer 1

Solution:

a)

H₀: μ ≥ 7; H_A: μ < 7

b)

The test statistic t is

z = ( - )/[/n]

= [6.7 - 7]/[2.1/150]

= -1.75

Test statistic : -1.75

c)

Left tailed test

So ,

p value = P(Z < z) = P(Z < -1.75) = 0.0401

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.