An article in the National Geographic News reports that Americans are sleeping l

Question

An article in the National Geographic News reports that Americans are sleeping less. A researcher wants to determine if Americans are sleeping less than the recommended 7 or more hours on weekdays. Assume the population is normally distributed with a known standard deviation of 2.1 hours.

a) Create Null and Alternative Hypotheses:

Set up the null and alternative hypotheses that would allow you to test if Americans are sleeping less than 7 hours on weekdays. Is this a one-tail, or two-tailed test? Explain why.

b) Figure out the Decision Rule

If the required level of significance () is 0.01, calculate the z critical value(s). What is the decision rule (i.e. when do you reject the null and when do you accept it?), Draw a picture showing sampling distribution and the rejection region(s).

c) Conduct the Hypothesis Test (Calculate the test statistic)

Suppose a random sample of 150 Americans yields the following sample statistic:

            Sample mean = 6.7 hours on weekdays.

What is the value of the z-test statistic?

Do you accept or reject the null hypothesis using the critical value method? Explain your decision

d) Using the p-value method, is your decision the same as in (c)? Explain your answer. Be sure to include the calculation of the p-value.

Explanation / Answer

Solution:-

a) State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: > 7

Alternative hypothesis: < 7

Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected if the sample mean is too small.

b) Formulate an analysis plan. For this analysis, the significance level is 0.01. The test method is a one-sample z-test.

Analyze sample data. Using sample data, we compute the standard error (SE), and the z statistic test statistic (z).

SE = s / sqrt(n)

S.E = 0.1715

z = (x - ) / SE

c) z = - 1.75

zcritical = - 2.33

Reject the null hypothesis if z score is less than - 2.33.

where s is the standard deviation of the sample, x is the sample mean, is the hypothesized population mean, and n is the sample size.

d) Thus the P-value in this analysis is 0.04

Interpret results. Since the P-value (0.04) is greater than the significance level (0.01), we cannot reject the null hypothesis.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.