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X7037711
©
National
Qualications
AH 2016
Total marks — 100
Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions which contain appropriate working. State the units for your answer where appropriate. Write your answers clearly in the answer booklet provided. In the answer booklet you must clearly identify the question number you are attempting. Use blue or black ink. Before leaving the examination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper. A booklet of Statistical Formulae and Tables is supplied for all candidates.
X703/77/11 Statistics
TUESDAY, 10 MAY
1:00 PM – 4:00 PM
A/HTP
Total marks — 100 Attempt ALL questions
1. A biologist in Africa is investigating grazing patterns. The heights in a random sample of vegetation were measured to the nearest centimetre at various points on a line transect and are listed below.
26 15 28 23 25 15 16 20 22 27 17 30 48
(a) Draw a suitable statistical diagram to represent this sample.
(b) Calculate the upper fence and comment on the measurement of 48 cm.
2. A large number of knee operations is undertaken by 10 different surgeons.
Amongst these surgeons there is considerable variation in the number of operations they undertake. Some surgeons perform only a few and others perform hundreds. A sample of 100 patients is to be taken to evaluate patient satisfaction using a quality of life questionnaire.
(a) If a simple random sample of 100 patients is taken, explain why this might be an unrepresentative sample.
(b) Describe how a more representative sample might be taken, stating the type of sampling used.
(c) Suggest another possible cause of variation in patient satisfaction other than the skill of the surgeon.
3. The content weights of a random sample of 12 jars of honey yielded a mean of 147·8 g with standard deviation 2·379 g.
(a) Assuming that the weights are normally distributed, assess the evidence, at the 1% level of significance, that the population mean content weight is less than 150 g.
(b) Explain why a z -test would be inappropriate in this situation.
6. Working at Trondheim University Hospital in 2003, researchers Sigurd Fasting and Sven E. Gisvold used control charts to monitor the quality of anaesthesia processes. An adapted version of their work is considered below. They investigated the percentage of surgical operations in which patients had a difficult recovery from general anaesthetic every month in the period from January 1997 (Month 1) to June 1999 (Month 30). The p-chart for the period of study, with 3-sigma limits, is shown below.
p-chart showing rate of difficult recovery from general anaesthesia
p^ ˆ
Month
UCL
LCL
Proportion difficult recovery
(a) Without further calculation, describe a feature of the control chart that indicates that the process is possibly influenced by special causes.
Another possible indication of special cause variation is illustrated in the data from months 15 to 28 where 12 out of 14 consecutive data points lie below the centre line.
(b) Calculate the probability that exactly 12 out of 14 consecutive points lie either above or below the centre line, when a process is in control.
Before the study was undertaken, changes had been made to anaesthetic practice in early 1998 and the data can be more sensibly viewed as two separate processes, before and after the change.
(c) State the numbers of the two months between which the change in practice was most likely to have been made.
The estimated proportion of difficult recoveries, p ˆ^ , illustrated on the centre line of the control chart, is 0·025 and the Upper Control Limit is 0·04.
(d) Show that the number of operations performed each month was 975.
(e) Explain why it was appropriate to use the normal approximation to the binomial distribution in this study and calculate the approximate percentage of sample proportions which would be greater than 0·03.
7. During a viral epidemic a doctor examines 150 people suffering from symptoms commonly associated with the virus. Of the 150 people examined, 90 are male of whom 40 actually have the virus. 10 of the examined females have the virus, the rest do not.
(a) Calculate the probability that an individual selected at random from this group is infected with the virus.
(b) If 3 different people are selected at random without replacement from this group, what is the probability that all 3 have the disease?
Of the people in this group with the virus 94% react positively to a clinical test to confirm the viral infection, as do 7% of the people without the virus.
(c) (i) Calculate the probability that a person selected at random reacts positively.
(ii) Calculate the probability that a person selected at random has the virus given that he or she reacted positively.
8. A new drug was trialled on 100 randomly chosen patients who had a particular disease and yielded a 75% recovery rate. Another 100 randomly chosen patients with the disease were used as a control group, treated with an existing drug, giving a 65% recovery rate.
(a) Perform a z -test to determine whether there is any evidence that the new drug
had a higher recovery rate.
(b) For this particular type of drug trial explain why a two-tailed test might be more appropriate.
9. In a particular school, the number of sporting injuries X occurring in a week can be modelled by the Poisson distribution with mean 4.
(a) State two assumptions that underlie the valid use of a Poisson model in this situation.
(b) Calculate the probability that the number of sporting injuries in a week is more than two standard deviations above the mean.
(c) Assuming that a school year is 38 weeks, use a suitable approximation to find the probability that there are fewer than 140 sporting injuries in a school year.
(d) Considering the circumstance in which sporting injuries might occur, give a reason why the Poisson distribution might not be an appropriate model.
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12. A study entitled “A relationship between attractiveness and performance in professional cyclists” was published in Biology Letters in January 2014 and a slightly modified version of the results is introduced below. Participants in the study were shown head and shoulders portraits of a sample of 70 of the leading riders in the 2012 Tour de France cycle race. They were asked to then rate each rider, on a scale of 1 to 5 (with 5 the highest), in terms of attractiveness.
Scores from all participants were combined to give a single value of attractiveness ( y )
for each rider.
A value for performance for each rider ( x ) was obtained by considering outcomes
from different parts of the race, where higher values of x signify better
performance. A scatterplot of all 70 results is shown below.
Performance
Attractiveness
Tour de France riders 2012
y
x
(a) From the scatterplot, comment on the relationship between attractiveness and performance for the population of professional cyclists sampled.
For the 70 results, the following statistics were obtained.
∑ 12 ^0345 ∑^179 ^9185
Sxx 58 3111 S (^) yy 26 1816 Sxy 15 6348
x y
(b) Calculate the product moment correlation coefficient of x and y.
State what this measures and comment on its value.
[Turn over for next question
12. (continued)
(c) Show that there is evidence at the 5% level that this correlation coefficient is significantly different from zero.
(d) With reference to the test performed in (c) explain why relatively low correlations can be statistically significant. What would be your final conclusion to this study?
(e) Give a reason why it might not be useful to fit a least squares regression line to this data.
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