Question

The National Sleep Foundation used a survey to determine whether hours of sleeping per night are independent of age (Newsweek, January 19, 2004). The following show the hours of sleep on weeknights for a sample of individuals age 49 and younger and for a sample of individuals age 50 and older.

Hours of Sleep
	Age	Fewer than 6	6 to 6.9	7 to 7.9	8 or more	Total
	49 or younger	38	56	70	76	240
	50 or older	30	56	71	103	260

Conduct a test of independence to determine whether the hours of sleep on weeknights are independent of age. Use  = .05. Use Table 12.4.

A.) Compute the value of the  2 test statistic (to 2 decimals).

B.) Using the total sample of 500, estimate the percentage of people who sleep less than 6, 6 to 6.9, 7 to 7.9, and 8 or more hours on weeknights (to 1 decimal).

Less than 6 hours	%
6 to 6.9 hours	%
7 to 7.9 hours	%
8 or more hours	%

Answer 1

Solution:-

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

H₀: Age and hours of sleep on weeknights are independent.
H_a: Age and hours of sleep on weeknights are not independent.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a chi-square test for independence.

Analyze sample data. Applying the chi-square test for independence to sample data, we compute the degrees of freedom, the expected frequency counts, and the chi-square test statistic. Based on the chi-square statistic and the degrees of freedom, we determine the P-value.

DF = (r - 1) * (c - 1) = (2 - 1) * (4 - 1)
D.F = 3
E_r,c = (n_r * n_c) / n

where DF is the degrees of freedom.

The P-value is the probability that a chi-square statistic having 3 degrees of freedom is more extreme than 4.23.

We use the Chi-Square Distribution Calculator to find P( > 4.23) = 0.238

Interpret results. Since the P-value (0.238) is greater than the significance level (0.05), we have to accept the null hypothesis.

Thus, we conclude that there is no relationship between Age and hours of sleep on weeknights.