Name:

  1. A filling operation is designed to fill Gatorade bottles to a mean weight of 32 ounces. A sample of bottles is periodically selected and weighed to determine whether the bottles are close to the specified weight. If a hypothesis test suggests the bottles are different from 32 ounces, then the production line will be shut down and adjusted to obtain the proper weight.
  1. Specify the appropriate null and alternative hypotheses:

\[H_{o}: \mu = 32 \\ H_{a}: \mu ≠32 \] b. What is the conclusion of the test when the null is not rejected?

There is not enough statistical evidence to suggest that the average fill weight is different from 32 ounces.

  1. What is the conclusion of the test when the null is rejected?

There is enough information to suggest that the true average fill weight is different from 32 ounces.

  1. A study is conducted to determine if a new manufacturing method reduces costs. The current production method operates with a mean cost of $200 per hour. The study will estimate the costs of the new method over a sample production period.
  1. Specify the appropriate null and alternative hypotheses:

\[ H_{o}: \mu = 200 \\ H_{a}: \mu < 200 \] a. What is the conclusion of the test when null is not rejected?

There is not enough statistical information to suggest that average costs were reduced from the new manufacturing method.

  1. What is the conclusion of the test when the null is rejected?

There is enough statistical informatino to suggest that average costs were reduced as a result of the new manufacturing method.

  1. Sales at an electronics store average $25,000 per week. The company is considering implementing a new incentive program to sales associates to improve sales. Use this information to answer the following questions.
  1. Specify the appropriate null and alternative hypotheses:

\[H_{o}:\mu = 25,000 \\ H_{a}: \mu >25,000 \] b. Conceptually, what is the Type I error for this particular problem? What are the consequences of making this error?

The Type I error is a false positive. In this case it would refer to the case where we suggest that the new incentive program increases average sales when it really doesn’t. The consequence would be that the electronics store would likely adopt the new incentive program when it would likely have no effect on increasing sales.

  1. Conceptually, what is the Type II error for this particular problem? What are the consequences of making this error? Would suggest that sales remained the same when it really increased.

The Type II error is a false negative. In this case it would represent the case where we suggest the new incentive program doesn’t increase sales when in reality the program does. The consequence of making this error is that the electronics store would likely not adopt the new program when it would likely increase sales.

  1. A new production method will be implemented if a hypothesis test suggests that the new method reduces the mean operating cost at the facility.
  1. Conceptually, what is the Type I error for this particular problem? What are the consequences of making this error?

The Type I error for this particular problem would be suggesting that the new method reduces costs when in reality they don’t. The consequence is that the facility would likely adopt the new method to reduce costs while the new method actually won’t reduce costs at all.

  1. Conceptually, what is the Type II error for this particular problem? What are the consequences of making this error? Suggest that the operating cost stayed the same when it really declined.

The Type II error for this particular problem is to fail to suggest the new method reduces costs when in reality it does. The facility would likely not adopt the new method when it would likely reduce costs.

  1. Individuals who file income tax returns two weeks early receive an average refund of $1100. Consider people who file their taxes at the last minute, in the five days prior to the deadline. A sample of 400 people who submitted their taxes yielded a sample mean of $970. The population standard deviation for returns is assumed to be $2,000. Let \(\alpha = 0.05\).
  1. Specify the null and alternative hypotheses to test if people who waited until the last minute received a lower tax refund than people who filed their taxes early.

\[H_{o}: \mu = 1100 \\ H_{a} : \mu < 1100 \]

  1. Report the test statistic and the p-value associated with this test.

The test statistic is -1.3 and the p-value associated with a lower-tail test is 0.1.

  1. What is the conclusion of the test? Clearly explain this result.

Since the p-value of 0.097 is greater than 0.05 - the specified level of \(\alpha\) - we fail to reject the null in favor of the alternative. We do not have enough statistical information to suggest that last minute filers receive a lower average tax refund.

  1. A random sample of 25 cell phone bills indicate that the sample average price is $89 per month. The sample standard deviation is $33. Use this information to answer the following questions.
  1. Construct a 95% confidence interval around the population mean. Report the margin of error, along with the upper and lower bounds. Interpret the confidence interval in the context of the question.

\[ CI = 75.38, 102.62\]

  1. What is the effect on the width of the confidence interval if we decrease the sample size?

The width of the confidence interval will increase if the sample size decreases since there is less information available in the sample.

  1. A study wants to determine if cell phone bills are higher than $100. Specify the null and alternative hypothesis for this particular test. \[H_{o}: \mu=100 \\ H_{a} : \mu >100 \]

  2. Report the test statistic and p-value from this test and interpret these values. Let α = .01. What is the result from the test? Explain.

The test statistic is -1.67 and the p-value is 0.95. The p-value is much grater than the specified level of alpha. We do not reject the null in favor of the alternative. There is not enough information to suggest that the average cell phone bill is more than $100 per month.

  1. In 2015 the average rental price for an apartment in Chicago was $1200 per month. Previous studies suggest a population standard deviation rental price of σ = $800. A recent sample of rental prices for 180 apartments in Chicago is provided in the excel file “chicagorent”. Perform a test to determine if mean apartment rental prices currently exceed $1200?
  1. Specify the appropriate null and alternative hypotheses for this question.

\[Ho: \mu = 1200 \\ Ha: \mu >1200 \]

  1. Report the test statistic and p-value. Interpret these values. The test statistic is 2.04 and the p-value for an upper tail test is 0.02. The p-value is approximately 0.

  2. Let α = .05. What is the result of the test? Explain this result.

The p-value is approximately 0. So, we reject the null in favor of the alternative. This suggests that average rental rates in Chicago do exceed $1200 per month.

  1. Nationally, the average price for a used car is $9,500. A sample of 25 used car prices in the Midwest is provided in the excel file “midwestcars”. Perform a hypothesis test to determine if used car prices are higher than $9,500 in the Midwest.

  1. Specify the appropriate null and alternative hypotheses for this question. \[H_{o}: \mu=9500 \\ H_{a}: \mu >9500 \]

  2. Report the test statistic and p-value. Interpret these values.

The test statistic is 1.37 and the p-value for an upper tail test is 0.09.

  1. Let α = .05. What is the conclusion of the test? Explain this result.

Since the p-value is greater than the specified level of \(\alpha\) then we fail to reject the null in favor of the alternative. This suggests that there is no indication that average used car prices are more expensive in the midwest.