Question

Suppose you perform a study about the hours of sleep that college students get. You know that for all people, the average is about 7 hours. You randomly select 35 college students and survey them on their sleep habits. From this sample, the mean number of hours of sleep is found to be 6.2 hours with a standard deviation of 0.97 hours. You want to construct a 99% confidence interval for the mean nightly hours of sleep for all college students.

(a) What is the point estimate for the mean nightly hours of sleep for all college students?

(b) Construct the 99% confidence interval for the mean nightly hours of sleep for all college students. Round your answers to 1 decimal place.

(c) Are you 99% confident that the mean nightly hours of sleep for all college students is below the average for all people of 7 hours per night? Why or why not?

a. Yes, because 7 is above the upper limit of the confidence interval for college students.

b. No, because 7 is below the upper limit of the confidence interval for college students.

c. Yes, because 7 is below the upper limit of the confidence interval for college students.

d. No, because 7 is above the upper limit of the confidence interval for college students.

(d) We are never told whether or not the parent population is normally distributed. Why could we use the above method to find the confidence interval?

a. Because the margin of error is positive.

b. Because the sample size is greater than 30.

c. Because the margin of error is less than 30.

d. Because the sample size is less than 100.

Answer 1

Answer)

A)

Point estimate = 6.2

B)

As the population s.d is not mentioned in the question and we are given with the sample s.d as the best estimate

We will use the t distribution to estimate the interval

N = 35

Mean = 6.2

S.d = 0.97

Degrees of freedom is = n-1 = 34

For 34 dof and 99% confidence level critical value t from t distribution is = 2.73

MOE = t*s.d/√n = 2.73*0.97/√35 = 0.447

Interval is given by

(Mean - MOE, Mean + MOE)

[5.753, 6.647].

We are 99% confident that the population mean (μ) falls between 5.753 and 6.647.

C)

Yes, because 7 is above the upper limit of the confidence interval for college students.

D)

b. Because the sample size is greater than 30.

As according to the central limit theorem

If sample size is greater than 30

Data is approximately normal