Examples of Confidence Intervals and Hypothesis Testing

Problem 1: Confidence Interval for a Sample Mean

1)A simple random sample of 60 items resulted in a sample mean of 80. The population standard deviation is σ = 15.

a.What is the 95% confidence interval for the population mean?

b.Assume that the sample size was 120, rather than 60. Recalculate the 95% confidence interval for the population mean.

c.What does the increased sample size due to the confidence interval?

2)A simple random sample with n=54 provided a sample mean of 22.5 and sample standard deviation of 4.4.

a.What is the 90% confidence interval for the population mean?

b.What is the 95% confidence interval for the population mean?

c.What is the 99% confidence interval for the population mean?

d.What happens to the margin of error and the confidence interval as the confidence level is increased?

3)The average cost per night of a hotel room in New York City is $273. Assume that this estimate is based on a sample of 45 hotels and that the sample standard deviation is $65. 

a.What is the 95% confidence interval for the population mean?

b.Two years ago the average cost of a hotel room in NYC was $229. Discuss the change in cost over the two-year period. (If the price two years ago is not in the current confidence interval, then we can be confident that the actual price has gone up in that time period)

4)A researcher is interested in whether or not high school students are passing a new statewide exit exam.  A random sample of 100 students has a mean of 76 and a standard deviation of 15.  Calculate and draw the 95% confidence interval around the mean.  Can we be confident that the population mean is above a passing grade of 70?

5)A university wants to know more about the knowledge of students regarding international events. They are concerned that their students are uninformed in regards to news from other countries. A standardized test is used to assess student’s knowledge of world events (national reported mean=65, standard deviation=5). A sample of 30 students are tested (sample mean=58, standard deviation =3.2). Compute a 99 percent confidence interval based on this sample's data. How do these students compare to the national sample?

6)Carpetland salespersons average $8,000 per week in sales. Steve Contois, the firm’s vice president, proposes a compensation plan with new selling incentive. Steve hopes that the results of a trial selling period will enable him to conclude that the compensation plan increase the average sales per salesperson. 

a.Develop the appropriate null and alternative hypotheses.

b.What is the Type I error in this situation? What are the consequences of making this error?

c.What is the Type II error in this situation? What are the consequences of making this error?

7)Consider the following hypothesis test:

H0 :  ≤ 25

Ha :  > 25

A sample of 40 observations provided a sample mean of 26.4. The standard deviation is 6.

a.What is the test statistic for this sample

b.At α = .01 what is your critical value?

c.What is your conclusion?

8)Individuals filing federal income tax returns prior to March 31 received an average refund of $1,056. Consider the population of “last minute” filers who mail their tax return during the last five days of the income tax period (typically April 10 to April 15). 

a.A researcher suggests that a reason individuals wait until the last five days is that on average, these individuals receive lower refunds than early filers. Develop an appropriate hypothesis test such that rejection of H0 will support the researcher’s contention.

For a sample of 400 individuals who filed a tax return between April 10 and April 15, the sample mean refund was $910. Based on prior experience the standard deviation of σ = $1600 may be assumed. 

b.What is the test statistic for this sample?

c.With an α = .05, what is the critical value?

d.Can you support the researcher’s contention?

9) Consider the following hypothesis test:

H0 :  ≥ 15

Ha :  <15

A sample of 50 observations provided a sample mean of 14.15. The sample standard deviation is 3. The p-value for the test statistic is .028.

At α = .05 what is your conclusion for this test? 

10)A manufacturer of electric lamps is testing a new production method that will be considered acceptable if the lamps produced by this method result in a normal population with an average life of 2,400 hours and a standard deviation equal to 300. A sample of 100 lamps produced by this method has an average life of 2,320 hours. Can the hypothesis of validity for the new manufacturing process be accepted with a risk equal to or less than 5%?