Fernando et al. (2003) studied drug-sharing among injection drugs users in the South Bronx in New York City. Splitting a bag – a practice where heroin is shared between two people after dissolving it in a common cooker prior to injection– is a known high risk factor for HIV exposure.  Interviews with drug users in a residential treatment facility provided the following information.

Â

 Gender

Have you split drugs?

Total

Yes (Y)

No (N)

Male (M)

390

321

711

Female (F)

179

131

310

Total

569

452

1021

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What is the probability that a person chosen at random will have split drugs?

What is the probability that a person chosen at random is not female?

What is the probablity that a person has split drugs given they are a male?

WHat is the probablility that someone chosen at random will be a female or will not have split drugs?

Given taht someone has split drugs, what is the probablity that they are female?Â

1. In a sample of 25 adolescents who served as the subjects in an immunologic study, one variable of interest was the diameter of skin test reaction to an antigen. The sample mean and standard deviation were 24 and 15 mm, respectively.

a) Conduct an appropriate test (state H0, HA and critical value) to see if we can conclude that the population mean of the diameter of skin test reaction to this antigen is different from 30, use a=0.05. Report the p-value of this test and conclude. (Use SAS or use the tables)
b) Construct a 95% confidence interval for the population mean (µ) of the diameter of skin test reaction to this antigen in this population. Based on the 95% confidence interval, would you have expected to reject or not to reject the null hypothesis in part (a)? Why? (Use SAS or use the tables)
c) In another sample of size 100, the 95% confidence interval of diameter for skin test reaction is found to be (18.25, 25.75). What are the sample mean and sample standard deviation of the diameter for skin test reaction to this antigen of this sample? (Hand calculation – show your working)
d) The true population mean for the diameter of skin test reaction to this antigen is 25 mm, and the population variance for this variable is 100. If you want to have a 90% power to detect this difference, using a two-sided test conducted at the 0.01 level of significance, what sample size will be needed? (Use SAS software)

2. A researcher was interested in knowing if preterm infants with late metabolic acidosis and preterm infants without the condition differ with respect to urine levels of a certain chemical.
The mean levels, standard deviations, and sample sizes for the two samples studied were as follows:

Sample n ??¯ S
With condition 5 5.5 3.3
Without condition 9 8.7 1.5

a) Assume normality and unequal variances for the two groups, ??12 ? ??22, find the 90% confidence interval for the difference of average chemical levels, ??1- ??2, between the preterm infants with and without the condition. (Use SAS Software)

b) Test the null hypothesis (state H0, HA, and critical value) that the preterm infants with and without the condition have the same population mean urine levels for this chemical, using a two-sided test at a=0.10 for this test.

c) If we don’t want to assume the urine levels of this chemical are normally distributed but want to test if the medians of the urine level of this chemical for the two populations are equal, what test shall we use? The urine levels of this chemical in the 2 samples are shown below. Test this hypothesis at a=0.10 level (state H0, HA, and p-value). (Hand calculation)

With condition: 9.7, 1.5, 7.9, 4.9, 3.4
Without condition: 11.1, 9.6, 7.4, 8.5, 9.8, 6.5, 8.1, 7.3, 10.2
3. In a study of dyslipidemia, a sample of 18 male participants was drawn from male adult population and a sample of 36 female participants was drawn from female adult population to measure their low-density lipoprotein (LDL) cholesterol level in their blood. The study results found 4 participants in the male sample had high LDL level (i.e., LDL level was = 100 mg/dL); and 9 participants in the female sample had high LDL.

a) It is postulated that 30% of male adults have high LDL in the male adult population. If this hypothesis is true, what is the probability of observing 4 or less male participants with a high LDL in a male adult sample of size 18? (SAS Software)
b) At the 0.05 level of significance, test the null hypothesis (state H0, HA, critical value, and p-value) that the proportions of high LDL are equal among male and female populations. (Hand calculation)
c) Construct a 95% confidence interval for the true difference in population proportions of high LDL (p1-p2) between female and male populations. (hand calculation)

4. A study was conducted to evaluate the relative efficacy of flu shot in preventing flu each year in college students. In 2009, a group of college students who took flu shot were asked whether they had flu that year. In the same year, a different group of college students who did not take flu short were asked the same question. The results are show in the following table:
a) Test the null hypothesis (state H0, HA, and critical value) that there is no association between flu shot and flu occurrence at the 0.05 level of significance. (SAS Software)

b) Estimate the relative odds (odds ratio) of flu occurrence for those who took flu shot versus those did not take flu shot in 2009.Calculate a 99%confidence interval for the population odds ratio. Based on the 99% confidence interval, would you have expected to reject or not to reject the null hypothesis in part (a) at the 0.01 level of significance? Why?

c) In 2010, the first group of college students (those who took flu shot in 2009) were asked to NOT take flu shot. They were then asked to report whether they had flu in 2010. The results of flu occurrence in 2009 and 2010 are shown below for each student in that group, with a Y denoting “Yes” for flu occurrence and a N denoting “No” for flu occurrence:

Student
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
2009 Y N N Y N N N N Y N N N Y N N N N Y N N N
2010 N Y N Y N Y Y Y N Y Y N Y N N Y N Y N Y N
Based on this data, complete the following table for flu occurrence and test the null hypothesis (state Ho, HA,) that there is no association between flu shot and the flu occurrence at the 0.01 level of significance.(Hand calculation)
2009
(with flu shot) 2010 (without flu shot)
Yes No
Yes
No

1. In a sample of 25 adolescents who served as the subjects in an immunologic study, one variable of interest was the diameter of skin test reaction to an antigen. The sample mean and standard deviation were 24 and 15 mm, respectively.

a) Conduct an appropriate test (state H0, HA and critical value) to see if we can conclude that the population mean of the diameter of skin test reaction to this antigen is different from 30, use a=0.05. Report the p-value of this test and conclude. (Use SAS or use the tables)
b) Construct a 95% confidence interval for the population mean (µ) of the diameter of skin test reaction to this antigen in this population. Based on the 95% confidence interval, would you have expected to reject or not to reject the null hypothesis in part (a)? Why? (Use SAS or use the tables)
c) In another sample of size 100, the 95% confidence interval of diameter for skin test reaction is found to be (18.25, 25.75). What are the sample mean and sample standard deviation of the diameter for skin test reaction to this antigen of this sample? (Hand calculation – show your working)
d) The true population mean for the diameter of skin test reaction to this antigen is 25 mm, and the population variance for this variable is 100. If you want to have a 90% power to detect this difference, using a two-sided test conducted at the 0.01 level of significance, what sample size will be needed? (Use SAS software)

2. A researcher was interested in knowing if preterm infants with late metabolic acidosis and preterm infants without the condition differ with respect to urine levels of a certain chemical.
The mean levels, standard deviations, and sample sizes for the two samples studied were as follows:

Sample n ??¯ S
With condition 5 5.5 3.3
Without condition 9 8.7 1.5

a) Assume normality and unequal variances for the two groups, ??12 ? ??22, find the 90% confidence interval for the difference of average chemical levels, ??1- ??2, between the preterm infants with and without the condition. (Use SAS Software)

b) Test the null hypothesis (state H0, HA, and critical value) that the preterm infants with and without the condition have the same population mean urine levels for this chemical, using a two-sided test at a=0.10 for this test.

c) If we don’t want to assume the urine levels of this chemical are normally distributed but want to test if the medians of the urine level of this chemical for the two populations are equal, what test shall we use? The urine levels of this chemical in the 2 samples are shown below. Test this hypothesis at a=0.10 level (state H0, HA, and p-value). (Hand calculation)

With condition: 9.7, 1.5, 7.9, 4.9, 3.4
Without condition: 11.1, 9.6, 7.4, 8.5, 9.8, 6.5, 8.1, 7.3, 10.2
3. In a study of dyslipidemia, a sample of 18 male participants was drawn from male adult population and a sample of 36 female participants was drawn from female adult population to measure their low-density lipoprotein (LDL) cholesterol level in their blood. The study results found 4 participants in the male sample had high LDL level (i.e., LDL level was = 100 mg/dL); and 9 participants in the female sample had high LDL.

a) It is postulated that 30% of male adults have high LDL in the male adult population. If this hypothesis is true, what is the probability of observing 4 or less male participants with a high LDL in a male adult sample of size 18? (SAS Software)
b) At the 0.05 level of significance, test the null hypothesis (state H0, HA, critical value, and p-value) that the proportions of high LDL are equal among male and female populations. (Hand calculation)
c) Construct a 95% confidence interval for the true difference in population proportions of high LDL (p1-p2) between female and male populations. (hand calculation)

4. A study was conducted to evaluate the relative efficacy of flu shot in preventing flu each year in college students. In 2009, a group of college students who took flu shot were asked whether they had flu that year. In the same year, a different group of college students who did not take flu short were asked the same question. The results are show in the following table:
a) Test the null hypothesis (state H0, HA, and critical value) that there is no association between flu shot and flu occurrence at the 0.05 level of significance. (SAS Software)

b) Estimate the relative odds (odds ratio) of flu occurrence for those who took flu shot versus those did not take flu shot in 2009.Calculate a 99%confidence interval for the population odds ratio. Based on the 99% confidence interval, would you have expected to reject or not to reject the null hypothesis in part (a) at the 0.01 level of significance? Why?

c) In 2010, the first group of college students (those who took flu shot in 2009) were asked to NOT take flu shot. They were then asked to report whether they had flu in 2010. The results of flu occurrence in 2009 and 2010 are shown below for each student in that group, with a Y denoting “Yes” for flu occurrence and a N denoting “No” for flu occurrence:

Student
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
2009 Y N N Y N N N N Y N N N Y N N N N Y N N N
2010 N Y N Y N Y Y Y N Y Y N Y N N Y N Y N Y N
Based on this data, complete the following table for flu occurrence and test the null hypothesis (state Ho, HA,) that there is no association between flu shot and the flu occurrence at the 0.01 level of significance.(Hand calculation)
2009
(with flu shot) 2010 (without flu shot)
Yes No
Yes
No