Question 1:
The following (artificial) data record the length of stay (in days) spent on a psychiatric ward for 28 consecutive patients who have been sectioned under the mental health act, classified according to their gender:
Male: 96 113 195 22 56 155 30 50 29 38 25 79 34
Female: 39 71 17 9 35 6 57 10 18 65 35 21 32 28 16
Estimate the median length of stay, giving an approximate 95% confidence interval, for males and females separately.
Carry out a test to determine if there is a significant difference between males and females with regards to the median time spent on a psychiatric ward after being sectioned.
Estimate the probability that a randomly chosen male patient will stay longer than a randomly chosen female patient.
Question 2:
General trace organic monitoring describes the process in which engineers analyse water samples for various organic materials (e.g. contaminants). The data below record the total organic carbon concentration, X, in 13 water samples collected from a river:
X: 5.1 5.7 7.5 8.1 5.8 6.1 3.1 7.1 9.2 3.7 4.4 6.8 9.8
Estimating the sample Cumulative Distribution Function (CDF) by S(xi) = i/(n + 1), use the Kolmogorov-Smirnov test to examine if these data are consistent with coming from a Normal distribution having mean 7.0 and standard deviation 2.0.
Obtain the Normal Equivalent Deviates corresponding to the observed values of X and use these to produce a
Q-Q plot. Does your plot indicate that these data are consistent with coming from a Normal distribution?
Question 3:
a) An article "Does it pay to plead guilty? Differential Sentencing and the Functioning of Criminal Courts" [Law and Society Review, 1981] considered the punishments metered out to 255 individuals found guilty of robbery. All individuals had previous convictions. Of the 191 who pleaded "guilty", 101 were given custodial sentences, whilst of the 64 who pleaded "not guilty" 56 were sent to prison.
Present this information in the form of a 2 by 2 contingency table. Carry out a test to determine if the proportion of individuals sent to prison differs significantly for those pleading "guilty" and those pleading "not-guilty"?
b) A promotions panel in a metropolitan police force considered applications from 15 officers who applied for senior positions during one year. The table below records the outcome, classified by gender of the applicant:
|
Gender
|
Promoted
|
Not promoted
|
Total
|
|
Male
|
7
|
1
|
8
|
|
Female
|
2
|
5
|
7
|
|
Total
|
9
|
6
|
15
|
Assuming that these fifteen applications can be considered a random sample of all applications made within the police force calculate the exact probability of this arrangement occurring by chance assuming that males and females are equally likely to be promoted. Test the hypothesis that males are significantly more likely to be promoted.
Question 4:
A pharmaceutical company manufactures large batches of analgesic caplets which are designated to contain 200 mg of aspirin. The concentration of aspirin per caplet is actually a random variable X normally distributed about a mean 200 mg with standard deviation known to be 7.5 mg when the manufacturing process is in "control". As a Quality Control exercise small random samples of n caplets are taken from each production run and tested for potency.
A random sample of n = 16 caplets is selected with a view to testing the hypotheses:
H0: μ = 200 mg H1: μ ≠ 200 mg
Assuming a type I error of 5%, determine the critical region(s) for this test in terms of the sample mean.
What is the probability of a type II error for this test if the true mean is in fact 195 mg?
What impact would reducing the sample size have on the type I and type II errors?
The observed mean for the sample of 16 observations was found to be 203 mg. Calculate the "p-value" associated with this sample statistic. Based on the stated hypotheses, what do you conclude? Would your conclusion change if a = 10%?
Assuming a significance level of 5% is to be used, what size sample should be taken if it is required that we reject H0 if the true mean deviates from 200 mg by 5 mg or more with power of at least 90%?