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The Centers for Disease Control and Prevention (CDC) defines a healthy sleep duration
to be at least seven hours per day. The CDC reports that the percentage of people who
report a healthy sleep duration varies by marital status. The CDC also reports that in
2018, 67% of those who are married report a healthy sleep duration; 62% of those who
have never been married report a healthy sleep duration; and 56% of those who are
divorced, widowed, or separated report a healthy sleep duration. The file Sleep Habits
contains sample data on the sleeping habits of people who have never been married
that are consistent with the CDC's findings. Use these data to answer the following
questions. Click on the datafile logo to reference the data,
DATA file
a. Develop a point estimate and a 95% confidence interval for the proportion of those
who have never been married who report a healthy sleep duration,
Point estimate
(to 4 decimals)
to
Confidence interval
(to 4 decimals)
4
b. Develop a point estimate and a 95% confidence interval for the mean number of
hours of sleep for those who have never been married.
(to 4 decimals)
Point estimate
to
Confidence interval
(to 4 decimals)
c. For those who have never been married, estimate the number of hours of sleep per
day for those who report a healthy sleep duration (to 4 decimals).

Sagot :

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a. The point estimate of the married people that sleep well is = 0.6299. The confidence interval for the mean hours of sleep for those that have never been married but report good sleep is 0.6186, 0.6412

b. The point estimate of the number of hours for people that have never been married = 7.2257

95% confidence interval for the mean number of  hours of sleep for those who have never been married = 7.2065, 7.2447

c. the estimated number of sleep for those that have never been married = 7.729 hours

a. The point estimate

[tex]P=\frac{Xp}{n}[/tex]

[tex]Xp = successes\\n = sample size\\[/tex]

Those who have never been married n = 7044

Xp = those who sleep for 7 hours at least

To get Xp, the excel function

Countif(range, criteria)

Countif(A:A, >=7)

This gives Xp = 4437

Point estimate =

[tex]\frac{4477}{7044}[/tex]

= 0.6299

The point estimate of the married people that sleep well is = 0.6299

At 95% interval of confidence,

α = 1-0.95 = 0.05

α/2 = 0.05/2 = 0.025

Z0.025 = 1.96 ( Use the standard probability normal table)

Margin of Error

=[tex]M.E = 1.96\sqrt \frac{0.6299(1-0.6299)}{7044} \\\\= 1.96\sqrt{0.000033} \\\\1.96*0.0057[/tex]

= 0.0113

The 95% confidence interval

= 0.6299 - 0.0113 , 0.6299 + 0.0113

= 0.6186, 0.6412

The confidence interval for the mean hours of sleep for those that have never been married is 0.6186, 0.6412

b. Sample mean

x = ∑x/n

x = number of sleep of the never been married

Mean is calculated using excel function

Average(range)

Average(a:a)

This gives the sample mean = 7.2257

The point estimate of the number of hours for people that have never been married = 7.2257

Standard deviation using excel =

Stdev(A:A) = 0.8211

[tex]Margin of error =1.96*\frac{0.8211}{\sqrt{7044} }[/tex]

= 0.0192

The confidence interval =

= 7.2257-0.0192, 7.2257+0.0192

= 7.2065, 7.2447

95% confidence interval for the mean number of  hours of sleep for those who have never been married = 7.2065, 7.2447

c. From the available data, the mean number of sleep per day that is gotten from the 4437 people that are unmarried and sleep healthily for at least 7 hours = 34295.1

Mean = 34295.1/4437

= 7.729

the estimated number of sleep for those that have never been married = 7.729 hours

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