# Statistics Practice Problems

1) The health of employees is monitored by periodically weighing them in. A sample of 54 employees has a mean weight of 183.9 lb. Assuming that σ is known to be 121.2 lb, use a 0.10 significance level to test the claim that the population mean of all such employees weights is less than 200 lb. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim.

2) A researcher wants to check the claim that convicted burglars spend an average of 18.7 months in jail. She takes a random sample of 11 such cases from court files and finds that the sample mean = 20.5 months and s = 7.9 months. Test the null hypothesis that μ = 18.7 at the 0.05 significance level. Test the given claim using the traditional method of hypothesis testing. Assume that the sample has been randomly selected from a population with a normal distribution. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim.

3) A researcher wishes to determine whether people with high blood pressure can reduce their blood pressure by following a particular diet. Use the sample data below to test the claim that the treatment population's mean μ1 is smaller than the control population's mean μ2. Test the claim using a significance level of 0.01. Test the indicated claim about the means of two populations. Assume that the two samples are independent and that they have been randomly selected. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim.

(see attached file for data)

4) Use the sample data above to construct a 99% confidence interval for u1 - u2 where u1 and u2 represent the mean for the treatment group and the control group respectively. Interpret the confidence interval in the context of the study described in problem 3 above.

5) Five students took a math test before and after tutoring. Their scores were as follows.

Using a 0.01 level of significance, test the claim that the tutoring improves the math scores. Use the traditional method of hypothesis testing to test the given claim about the means of two populations. Assume that two dependent samples have been randomly selected from normally distributed populations. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim.

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#### Solution Summary

This solution is comprised of a detailed explanation on appropriate statistic method used for the objectives defined in the problem. Full description is given for the method along with the null and alternative hypotheses is provided in the solution.

Glucose Level, Total Cost and LOS, Waiting Time

1. You want to test the effectiveness of a new diabetes management program on lowering fasting blood sugar (glucose) levels for individuals who have diabetes.

Background:

According to information found on the internet, the following are glucose level ranges and their associated meaning:

Glucose Level (mg/dL) Test Result

70-99 Normal

100-125 Pre-diabetes (impaired glucose tolerance)

126 and above Diabetes (must confirm with an additional test)

Study:

The study includes 25 individuals who have been diagnosed with diabetes. Prior to implementing the diabetes management program, data on fasting glucose levels is collected for these 25 individuals each day for a period of 30 days. Each participant then begins the diabetes management program, which includes managing diet and engaging in a moderate amount of exercise. After participating in the program for 30 days, glucose levels on each participant are then collected again on a daily basis for another 30 days, while participants continue with the program. Therefore, the total study period is 3 months.

Once all the data is collected, average glucose levels are then calculated for each individual - one for the 30 observations collected prior to participating in the program and another for the 30 days after following the guidelines of the program. The mean glucose levels (before and after the program) for each of the 25 individuals is in the Excel file "HW5glucose.xls" and is also listed below:

Patient ID Mean Glucose Level (30 days prior to diabetes management program) Mean Glucose level (30 days post diabetes management program)

1 140 110

2 135 121

3 126 95

4 150 134

5 128 95

6 132 99

7 141 102

8 132 104

9 133 110

10 152 125

11 129 95

12 145 115

13 146 122

14 128 98

15 130 108

16 134 112

17 150 130

18 126 97

19 135 100

20 138 106

21 140 112

22 142 115

23 128 103

24 131 105

25 136 102

Analysis:

a) Describe the statistical test that can be used to determine if the diabetes management program has had a significant impact on decreasing average fasting glucose levels for these individuals with diabetes. Explain why this is the appropriate statistical test.

b) Describe any special considerations that must be taken into account when entering/modifying the data in order to conduct this test in SPSS.

c) Conduct this test using SPSS (Include output in Appendix).

d) Is this test significant? Explain how you know whether your result is significant or not. If it is significant, at what level of significance can you report the results (.01, .05, .1)?

e) Write out your conclusion based on your results (include the level of significance/confidence)

Preserving Original Data

Many times when you receive data, it will not be in the exact form you need in order to conduct your analysis. Sometimes you will need to create variables (as we did last week when we added dummy variables for each doctor), and other times you will need to modify the format in which the data is entered in order to run the test(s) you need.

Regardless of what modifications you need to make, NEVER change your original data set!

Before making a single change, resave your file under a new name and make modifications to that file. That way if you make a mistake or the data becomes corrupt, you will always have the original so you can start again with accurate data. If the study is really important (or if the data was difficult or expensive to collect), then it may also be wise to make an additional backup and keep it at another location (to protect from loss due to fire, flood, computer virus, etc).

For problem 2, you will be given data that will need to be modified. Before starting, resave the original file under a new name and then make your modifications.

1. You have hospital data ("HW5patientCostLOS.xls") on total inpatient "Cost" and "LOS" before (n=100) and after (n=105) the purchase of a new piece of diagnostic equipment.

a) Describe the statistical test that can be used to determine if the new diagnostic equipment has had a significant impact on changing total inpatient Cost. Explain why this is the appropriate statistical test.

b) Describe any special considerations that must be taken into account when entering/modifying the data in order to conduct this test in SPSS.

c) What did you rename your file? Print your modified data set (Include in Appendix)

d) Conduct the test you described above using SPSS (Include output in Appendix).

e) Is this test significant? Explain how you know whether your result is significant or not. If it is significant, at what level of significance can you report the results (.01, .05, .1)?

f) Write out your conclusion based on your results (include the level of significance/confidence)

1. Recently I had the opportunity to wait in the emergency room of Children's Hospital to have my son seen. I knew before we went that the wait time would probably be longer at this hospital than if we had gone to a hospital outside of Boston. Not unexpectedly, our wait to see a doctor was quite lengthy. Think about the factors that would influence wait time in an emergency room.

a) Make a list of 6-7 factors that you think would influence the wait to see a doctor in an emergency room. (Hint: a good way to do this is to think of different scenarios, especially extreme cases. For example, think about what conditions would have to exist in order for your wait time to be REALLY, REALLY short? Or REALLY, REALLY long? Make a list. Try to think outside the box as well - would it make a difference if I had gone to a different hospital? Why? What else could make a difference?? Brainstorm and then choose your most relevant items as your independent variables).

b) Indicate how each of these independent variables would affect your wait time - will it increase (+) or decrease (-) your wait time? You may need to provide a short explanation for each variable (or indicate how it will be measured) to illustrate why the sign you chose is appropriate.

c) Write a regression equation with "Wait time" as the dependent variable and include the independent variables and associated signs you indicated above. In writing out your equation, follow the model in the yellow handout from our discussion on regression 2 classes ago.

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