Question

A survey was conducted to estimate the mean number of books (denoted by µ) each university student read in the last year. Among a random sample of 51 students, the number of books each student read in the last year was recorded. The sample mean was 8.039 and the sample standard deviation was 2.545.

a. Write down the point estimate of µ.

b. Calculate the standard error of the point estimate in a.

c. To construct a 95% two-sided confidence interval for µ, what’s the multiplier to use?

d. Calculate the lower endpoint of the 95% two-sided confidence interval for µ.

e. Calculate the upper endpoint of the 95% two-sided confidence interval for µ.

f. Consider a hypothesis testing problem where the null hypothesis is H0 : µ = 10 and the alternative hypothesis is Ha : µ 6= 10. Use a significance level of 0.05. Based on the 95% two-sided confidence interval for µ above, what’s your decision about whether to reject H0? Give a brief explanation.

g. Suppose you’re asked to conduct a hypothesis testing to show that on average a student read more than 5 books last year. Which pair of hypotheses should be used? (5 points) A. H0 : µ = 5 against Ha : µ 6= 5 B. H0 : µ < 5 against Ha : µ ≥ 5 C. H0 : µ ≤ 5 against Ha : µ > 5 D. H0 : µ ≥ 5 against Ha : µ < 5

Answer 1

a) The point estimate of mean of a population is equal to the sample mean.

Hence

Point Estimate of  

b) The standard error of point estimate of i

Since the standard deviation of the population is not known, hence instead of the standard deviation of the population, we would use sample standard deviation to calculate the standard error.

Hence

The standard error of point estimate of i where is the sample standard deviation and is the sample size.

The required standard error of point estimate  

c) The two sided confidence interval is given by the expression

So in order to calculate the confidence interval the multiplier that we would use with the standard error is

Now, for a 95% confidence interval,

Hence the required multiplier to be multiplied with the standard error is 1.96 in order to calculate the 95% confidence interval.

d) The low end point of the 95% confidence interval for the mean is given by

e)

The upper end point of the 95% confidence interval for the mean is given by

f) Our 95% confidence interval for the mean gives the value of to be in the interval (7.339, 8.739)

The null hypothesis is

  

Since the value of 10 for doesn't lie in between the calculated confidence interval, hence we are going to reject the null hypothesis.

g) Determination of null and alternate hypothesis is always tricky. In the above question, the hypothesis testing is done to show that on average a student read more than 5 books last year. Now, what we want to show to prove is generally considered as the alternate hypothesis whereas what we want to reject, that we will choose as a null hypothesis.

In the above question, we want to show that on average a student read more than 5 books last year. So, this is the alternate hypothesis. A complementary of this hypothesis would be our null hypothesis,

Hence,

This is our null hypothesis.

and

This is our alternate hypothesis.

Hence, Option C is correct.

Thank you!!

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