Question

Exercise 3. In a class survey, students are asked how many hours they sleep per night. In the sample of 22 students, the (sample) mean is 6.77 hours with a (sample) standard deviation of 1.472 hours. We wish to know if this shows that the class’s student population on average has not gotten at least the recommended 8 hours of sleep at the α = .1 level. The distribution of sleep for this population follows a normal distribution.

a. Write appropriate hypotheses and α level. Use both the symbols and the words.Please write your answer outside your R code!

b. What do you need to check to use your test? Please verify the conditions are met. Please write your answer outside your R code!

c. Please use R to compute the test statistic and obtain a p-value.

d. Make conclusions using the test statistic and p-value. Please write your answer outside your R code!

e. Also, find the 90% upper confidence bound of average sleep for this class.

Answer 1

Part A:

Let X denote the number of hours of sleep for a student per night.

Given

Here our null hypothesis is

And our alternate hypothesis is

The level of significance is given as,

Part B:

The test we are going to use is a single sample t-test. A single sample t-test is used when we want to compare the population mean of a sample drawn from a normal population, to a hypothesized value and the population variance is unknown to us.

The main assumption for t-test are

1. The sample is drawn from a normal population.
2. The test variable is continuous.
3. The sample observations are independent of each other.
4. The sample drawn is a random sample of the population.

Checking of the assumptions:

1. Here it is given that the distribution of sleep for this population follows a normal distribution. So assumption 1 is satisfied.
2. Since the number of hours of sleep is a continuous variable, assumption 2 is satisfied.
3. Since student's sleep pattern does not depend on one another, assumption 3 is satisfied.
4. The class survey was conducted on a random school. So we can safely assume this is a random sample of the population. So assumption 4 is satisfied.

So all of our assumptions are satisfied and we can proceed to the test.

Part C:

Since raw data is not provided, we can not use the R function t.test here.

The test statistic is

,

where population mean under H₀, , sample mean, , sample variance, and sample size, .

Under H₀, test statistic, , which is a t-distribution with (n-1) = 21 degrees of freedom.

Using R, we get

[ R-code: (6.77-8)/(1.472/sqrt(22)) ]

For a lower tailed test p-value is

Using R, we get

[ R-code: pt(-3.919301,21)]

Part D:

We have level of significance, and .

So p-value < .

Thus we reject our null hypothesis and conclude that the data suggest that the population mean is less than 8.

Part E:

The 90% Upper confidence bound is

where sample mean, , sample variance, , sample size, and is 0.9 percentile point of a t-distribution with 21 degrees of freedom.

Using R we get,

[ R-code: qt(0.9,21)]

Thus we get, 90% Upper confidence bound of average sleep for this class is 7.185258.

[ R-code: 6.77+(1.323188*(1.472/sqrt(22)))]

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