Question 1. For each question below, choose one answer from the bulleted list.
a. The meaning of a p-value is...
- the power of the statistical test
- the probability that the null hypothesis is true
- the probability of making a type II error
- the probability of getting a result as extreme or more extreme than the one observed if the null hypothesis is false
- the probability of getting a result as extreme or more extreme than the one observed if the null hypothesis is true
b. An unbiased estimate...
- is equal to the true parameter
- has the smallest variance of all possible estimates
- is never an efficient estimate
- is always a maximum likelihood estimate
- none of the above
c. A 99% confidence interval for a mean...
- is wider than a 95% confidence interval
- is narrower than a 95% confidence interval
- excludes the mean with 99% probability
- is obtained as the sample average plus two standard deviations
- none of the above
d. A scientist reports that based on his sample of size 10 individuals, the 95% confidence interval for mean height is 66-74 inches. This result is to be interpreted as meaning...
- we have 95% confidence that a person's height lies between 66 and 74 inches
- we have 95% confidence that the population mean height lies between 66 and 74 inches
- 95% of the population has a height between 66 and 74 inches
- all of the above
- none of the above
e. A 95% confidence interval implies that...
- the t-test gives correct intervals 95% of the time
- the hypothesis will be false in 95 out of 100 such intervals
- the probability that the interval is false is 95%
- there is a 95% probability that the underlying distribution is normal
- if we repeatedly select representative samples and construct such interval estimate, 95 out of 100 of the intervals would be expected to bracket the true mean
Question 2. A study was conducted to examine whether the use of an intrauterine device (IUD) was associated with increased systolic blood pressure (SYSBP) in women. The following data were obtained:
|
ID
|
SYSBP when not using IUD
|
SYSBP when using IUD
|
|
1
|
115
|
124
|
|
2
|
106
|
115
|
|
3
|
109
|
99
|
|
4
|
102
|
109
|
|
5
|
122
|
132
|
|
6
|
126
|
120
|
|
7
|
114
|
121
|
a. What statistical procedure would you use to construct a 90% confidence interval for the mean difference in systolic blood pressure when using an IUD versus not. State any assumptions you need for the statistical procedure to be valid.
b. Construct the 90% confidence interval for the mean difference in systolic blood pressure when using an IUD versus not.
c. Interpret the confidence interval you obtained above.
Question 3. A randomized controlled trial was conducted to compare men receiving a treatment regimen of Omega-X vs. a treatment regimen of Delta-M for treatment of hepatitis C. Overall, 400 men participated in this study; 200 were randomized to each treatment. Twelve weeks after treatment, the participants were tested to see if their infection has improved or gotten worse (an outcome called virologic failure). The results of this study are below.
|
Omega-X
|
Delta-M
|
Total
|
|
Virologic failure
|
56
|
42
|
98
|
|
No virologic failure
|
144
|
158
|
302
|
|
Total
|
200
|
200
|
400
|
a. Calculate by hand the expected cell counts in each of the four shaded cells in the table below.
|
Omega-X
|
Delta-M
|
Total
|
|
Virologic failure
|
|
|
98
|
|
No virologic failure
|
|
|
302
|
|
Total
|
200
|
200
|
400
|
b. You wish to determine whether there was a treatment effect. Specify your null and alternative hypotheses, indicate your test statistic, perform your calculations, determine whether you reject or fail to reject the null hypothesis, and provide a conclusion/interpretation regarding the results of this study.
Question 4. Ms. Schuman's 4th grade class had 17 students with a mean statewide test score of 81 and a sample standard deviation of 7. Mr. Hart's 4th grade class had 20 students with a mean statewide test score of 87 and a sample standard deviation of 9.
Carry out the appropriate statistical hypothesis test to determine whether there is evidence that Mr. Hart's class performed statistically significantly better on the statewide test than Ms. Schuman's class. You may assume that the samples are simple random samples from normal distributions and that the two population variances are equal.
In your answer, please state the null and alternative hypotheses, indicate the name and result of the correct test statistic, perform your calculations, and interpret your findings with regard to your hypotheses. Please put your final conclusion in the context of the question regarding the test performance in Ms. Schuman and Mr. Hart's classes.
Question 5. An investigator is interested in examining whether the prescribed treatment regimen for diabetes differs by the patient's gender. The table below displays the number of male and female patients according to their prescribed treatment regimen.
|
Gender
|
Diet and exercise
|
Insulin
|
Oral medication
|
Total
|
|
Female
|
103
|
156
|
204
|
463
|
|
Male
|
119
|
151
|
193
|
463
|
|
Total
|
222
|
307
|
397
|
926
|
Based on the data above, is there evidence of a significant relationship between gender and prescribed treatment regimen? In your answer, please indicate the null and alternative hypotheses, the name and value of the correct test statistic, perform your calculations, and interpret your findings with regard to your hypotheses. Please put your final conclusion in the context of the question.
Question 6. For part a-e below, indicate whether the statement is "true" or "false" and explain why.
a. The data come from a normal distribution when the histogram is bell shaped, bimodal, and the median is less than the mean.
b. If the assumptions are satisfied and the degrees of freedom are sufficiently large, a chi square distribution can be used in place of the t-distribution.
c. The power of a statistical test increases with the variance of the population.
d. For a given sample variance and a given sample mean, a 90% confidence interval is narrower than a 99% confidence interval.
e. Consider the construction of a 95% confidence interval. I repeat the sampling process over and over and a new 95% confidence interval is obtained for each sample drawn. If for each sample, I indicate that the population parameter is contained in the interval, this statement would be accurate 95% of the time.