Question

Mall security estimates that the average daily per-store theft is less than $250, but wants to determine the accuracy of this statistic. The company researcher takes a sample of 81 clerks and finds that =$252 and s = $10.

a) Test at α = .1

b) Construct a 90% CIE of μ

2) A company that manufactures batteries is interested in constructing a two-sided 95% confidence interval for the population mean.

They sampled 121 batteries and found that the sample mean is 97.0 hours and the sample standard deviation is 3 hours.

Answer 1

Mall security estimates that the average daily per-store theft is less than $250, but wants to determine the accuracy of this statistic. The company researcher takes a sample of 81 clerks and finds that =$252 and s = $10.

a) Test at α = .1

b) Construct a 90% CIE of μ

Here claim statement is "the average daily per-store theft is less than $250"

So it is one sided t test ( because we don't know population standard deviation).

let's write the given information

= sample mean = 252

sample standard deviation = 10

sample size = n =81

level of significance = 0.1

so confidence level = c = 1 - 0.1 = 0.9

Using minitab commands:

The command is Stat>>>Basic Statistics >>1 sample t...

Click on "Summarized data"

Sample size : 81

Mean: 252

Standard deviation: 10

Hypothesized mean: 250

then click on Option select level of confidence = c = 0.9*100 = 90

Alternative " less than"

Click on Ok

Again "click on OK"

We get the following output

Decision rule:

1) If p-value < level of significance (alpha) then we reject null hypothesis

2) If p-value > level of significance (alpha) then we fail to reject null hypothesis.

Here p value = 0.5 > 0.1 so we used 2nd rule.

That is we fail to reject null hypothesis

Conclusion: At 5% level of significance there are not sufficient evidence to say that the average daily per-store theft is less than $250

b) Since the test is one sided so we need to find one sided confidence interval.

From the above output the upper bound of the 90% confidence interval is  251.44

2) A company that manufactures batteries is interested in constructing a two-sided 95% confidence interval for the population mean.

They sampled 121 batteries and found that the sample mean is 97.0 hours and the sample standard deviation is 3 hours.

We want to find 95% confidence interval for the population mean µ using the t-distribution

let's write the given information

= sample mean = 97

sample standard deviation = 3

sample size = n =121

Using minitab commands:

The command is Stat>>>Basic Statistics >>1 sample t...

Click on "Summarized data"

Sample size : 121

Mean: 97

Standard deviation: 3.0

then click on Option select level of confidence = c = 95

Alternative " not equal"

Click on Ok

Again "click on OK"

We get the following output

From the above output the 95% confidence interval for population mean is as (96.460, 97.540)