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 6 to 6.9 Age Fewer than 7 to 7.9 8 or more Total 38 74 49 or younger 50 or older a. Conduct a test of independence to determine whether the hours of sleep on weeknights are independent of age. use。= .05. Use Table 12.4 73 95 240 260 31 59 75 Compute the value of the x2test statistic (to 2 decimals). The p value is Select What is Select b Using the total sample of s00, 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:

a. Conduct a test of independence to determine whether the hours of sleep on weeknights are independent of age

Compute the value of the $\dpi{100} \small \chi^2$ test statistic.

Answer: $\dpi{100} \small \boldsymbol{\chi^2= 2.94}$

The p-value is $\dpi{100} \small \boldsymbol{0.4005}$

What is your conclusion?

Since the p-value = 0.4005 > 0.05, we conclude there is strong evidence that hours of sleep on weeknights
are independent of age.

Explanation:

$\dpi{100} \small \chi^2=\sum \frac{(O-E)^2}{E}$

Where:

O are the observed frequencies

E are the expected frequencies

The observed frequencies are:

The expected frequencies are calculated as:

$\dpi{100} \small \chi^2= \frac{(38-33.12)^2}{33.12}+\frac{(55-54.72)^2}{54.72}+\frac{(74-71.52)^2}{71.52}+\frac{(73-80.64)^2}{80.64}+\frac{(31-35.88)^2}{35.88}+\frac{(59-59.28)^2}{59.28}+\frac{(75-77.48)^2}{77.48}+\frac{(95-87.36)^2}{87.36}$

$\dpi{100} \small = 2.94$

b. Using the total sample of 500, estimate the percentage of people who sleep 6, 6 to 6.9, 7 to 7.9, and 8 or more hours on weeknights.

Answer: