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  {\Large{\bf {Assignment set on Bio-statistics}}}\\
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\item Prove that sample variance is an unbiased estimator of population variance.
\item  We wish to estimate the average number of heartbeats per minute for a certain population. The average number of heartbeats per minute for a sample of 49 subjects was found to be 90. Assume that these 49 patients constitute a random sample, and that the population is normally distributed with a standard deviation of 10.
Construct 90, 95, and 99 percent confidence intervals for the population mean, and state the practical and probabilistic interpretations of each.
3. A sample of 16 ten-year-old girls had a mean weight of 71.5 and a standard deviation of 12 pounds, respectively. Assuming normality, find the 90, 95, and 99 percent confidence intervals for $\mu$.
\item  Confidence interval for the difference between two population means: Iannelo et al. (A-8) performed a study that examined free fatty acid concentrations in 18 lean subjects and 11 obese subjects. The lean subjects had a mean level of 299 mEq/L with a standard error of the mean of 30, while the obese subjects had a mean of 744 mEq/L with a standard error of the mean of 62.
Construct 90, 95, and 99 percent confidence intervals for the population mean, and state the practical and probabilistic interpretations of each.
\item The purpose of a study by Granholm et al. (A-7) was to determine the effectiveness of an integrated outpatient dual-diagnosis treatment program for mentally ill subjects. The authors were addressing the problem of substance abuse issues among people with severe mental disorders. A retrospective chart review was performed on 50 consecutive patient referrals to the Substance Abuse/Mental Illness program at the VA San Diego Healthcare System. One of the outcome variables examined was the number of inpatient treatment days for psychiatric disorder during the year following the end of the program. Among 18 subjects with schizophrenia, the mean number of treatment days was 4.7 with a standard deviation of 9.3. For 10 subjects with bipolar disorder, the mean number of psychiatric disorder treatment days was 8.8 with a standard deviation of 11.5. Construct a 95 percent confidence interval for the difference between the means of the populations represented by these two samples. (See example 6.4.3 of Daniel & Cross)
\item Escobar et al. performed a study to validate a translated version of the Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC) questionnaire used with Spanish-speaking patients with hip or knee osteoarthritis. For the 76 women classified with severe hip pain, the WOMAC mean function score (on a scale from 0 to 100 with a higher number indicating less function) was 70.7 with a standard deviation of 14.6. We wish to know if we may conclude that the mean function score for a population of similar women subjects with severe hip pain is less than 75. Let  $\alpha=.01$. 

\item Cortisol level determinations were made on two samples of women at childbirth. Group 1 subjects underwent emergency cesarean section following induced labor. Group 2 subjects delivered by either cesarean section or the vaginal route following spontaneous labor. The sample sizes, mean cortisol levels, and standard deviations were as follows:$n_1=10, n_2=12, \bar{x}=435, \bar{x}=645, S_1=65, S_1=80$. Do these data provide sufficient evidence to indicate a difference in the mean cortisol levels in the populations represented? Let  $\alpha=0.05$.

\item Two pain-relieving drugs were compared for effectiveness on the basis of length of time elapsing between administration of the drug and cessation of pain. Thirteen patients received drug 1, and 13 received drug 2.The sample variances were $s^2_1=64, s_2^2=16$.Testthenullhypothesisthatthetwo populations variances are equal. Let $\alpha=0.05$. 

\item Vital capacity values were recorded for a sample of 10 patients with severe chronic airway obstruction. The variance of the 10 observations was .75. Test the null hypothesis that the population variance is 1.00. Let $\alpha=0.05$.


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