1.
The probability that a randomly chosen inhabitant (of any age) of a particular town is in paid employment is 0.4. A random sample of seven people is drawn from the population of the town. You may assume that the number of people in paid employment in this sample has an appropriate binomial distribution.
For the questions given, select the correct answer (to three decimal places) from the drop-down list.
What is the probability that exactly four of these people are in paid employment?
Answer 1
What is the probability that two or fewer of these people are in paid employment?
Answer 2
2. A horticulturalist is interested in which of two different new varieties of broad bean produces the highest average yield. She grows forty plants of each of the two varieties in her company’s research garden and measures the yield of beans (in grams) from each plant.Answer 1A craft worker makes wool felt objects for sale. Because felt shrinks when it is made, it is difficult to predict exactly the finished size of the articles. In the past, on average, they have been 13.5cm long (and the lengths have, approximately, a normal distribution).The worker develops a quicker way of making the felt, but is concerned that the average length may be different from what it previously was. She makes up 12 articles using the new method and measures their length.Answer 2
3. The two-sample t-test with unequal variances has, as its null hypothesis, that the variances of the two populations involved are the same.
A researcher has collected two samples of data and the sample variances are 0.34 and 2.11. It would be appropriate to use the two-sample t-test with a common variance.
The same formula is used for the ESE (estimated standard error) in the two-sample t-test with unequal variances and the two-sample z-test.
A researcher has collected two samples of data and the sample variances are 2.16² and 4.82². It would not be appropriate to use the two-sample t-test with a common variance.
The critical values for the two-sample t-test with unequal variances and the two-sample z-test, for given sample sizes, are always the same.
4. A boxplot of some data is as follows.
Using this boxplot, answer the following questions.
Does the boxplot show that the data are left-skew, symmetric or right-skew?
Answer 1
Approximately what percentage of the data lies between -40 and 0?
Answer 2
Approximately what percentage of the data lies between 0 and 20?
Answer 3
5. In cluster sampling, each member of the population should be in exactly one cluster.
If a survey uses clustering, it must not use stratification as well.
In cluster sampling, each cluster must differ as far as possible from the other clusters with respect to the subject under investigation.
Stratified sampling generally decreases the sampling error compared to a simple random sample of the same size.
In a stratified sample, the sizes of the subsamples from each stratum must always be equal to one another.
6. The following table gives data from the 2011 UK Census, on a simple random sample of 12 parliamentary constituencies drawn from all the constituencies in England and Wales. The data for each constituency are x, the percentage of households in the constituency where all the residents are of age 65 and over, and y, the percentage of the dwellings in the constituency that are detached houses.
Constituency
x
y
Aberavon
22.4
14.4
Croydon South
18.7
26.0
Doncaster North
21.5
19.4
Dudley North
23.5
20.4
Hastings and Rye
22.3
23.1
Morley and Outwood
19.3
25.0
New Forest West
35.3
47.0
Peterborough
18.9
24.2
Reigate
20.5
26.4
Torfaen
21.2
16.9
West Bromwich East
21.7
12.0
Workington
24.5
25.0
For these data ∑x² = 6279.86, ∑y² = 7380.7 and ∑xy = 6591.75.
Choose the option that is closest to the slope of the least squares regression line for these data, rounded to 3 significant figures, treating x as the explanatory variable and y as the response variable.
Select one:
1.41
857
301
-8.32
0.351
214
0.711
7. Data were collected, using a sample survey, on the inhabitants of a British town. Among other things, the respondents were asked their age and their satisfaction with the shopping opportunities in the local high street. These data were used to construct a contingency table, with 3 age categories for its rows, and 5 categories of satisfaction for its columns. The researchers wanted to use this table to test the null hypothesis that satisfaction with the high street opportunities and age are independent. The χ² test statistic turned out to be 22.312. (You may assume that all the expected values are greater than 5.) Choose the one correct option that describes what the researchers should conclude.
Select one:
The null hypothesis is rejected at the 5% significance level, but cannot be rejected at the 1% significance level. Therefore there is moderate evidence that satisfaction with high street shopping is related to age.
The null hypothesis cannot be rejected at the 5% significance level. Therefore there is strong evidence that satisfaction with high street shopping is related to age.
The null hypothesis cannot be rejected at the 5% significance level. Therefore there is little evidence that satisfaction with high street shopping is related to age.
The null hypothesis is rejected at the 5% significance level, but cannot be rejected at the 1% significance level. Therefore there is little evidence that satisfaction with high street shopping is related to age.
The null hypothesis is rejected at the 1% significance level. Therefore there is strong evidence that satisfaction with high street shopping is related to age.
The null hypothesis is rejected at the 1% significance level. Therefore there is little evidence that satisfaction with high street shopping is related to age.
8. The following table gives data from the 2011 UK Census, on a simple random sample of 10 parliamentary constituencies drawn from all the constituencies in England and Wales. The data for each constituency are x, the total number of full-time students living in the constituency (during term time, in thousands) and y, the total numbers of cars and vans available to all households in the constituency (in thousands).
Constituency
x
y
Ashford
5.0
64.3
Carlisle
4.5
41.1
Central Devon
3.2
58.6
Dewsbury
6.9
51.8
Meon Valley
4.0
60.4
North East Cambridgeshire
4.2
65.0
North Swindon
4.2
57.6
Portsmouth South
20.3
37.1
Redcar
4.3
39.6
Romford
4.8
47.6
For these data ∑x = 61.4, ∑y = 523.1, ∑x² = 608.0, ∑y² = 28342.15 and ∑xy = 2959.8.
Calculate the correlation coefficient for these two variables, and type the answer, rounded to two decimal places, in the box below.
9. An experiment was carried out to compare the wear resistance of two different materials for making the soles of children’s shoes. Special pairs of shoes were made up with one shoe sole in Material A and the other in Material B. (Either the right or the left shoe was chosen at random to have the Material A sole.) Children were then given the shoes and asked to wear them normally for two months. At the end of that time, the amount of wear (in mm) of each sole was measured, at the centre of the sole.
The differences in wear amounts (Material A – Material B) were calculated. In all there were 15 pairs of shoes. The sample mean of the differences was 1.608 mm, and the standard deviation of the differences was 0.449 mm. Using a method based on the matched-pairs t-test, calculate a 95% confidence interval for the difference between the mean wear amounts for the two materials. Round the end points of your interval to three decimal places and choose the appropriate option.
Select one:
(1.381, 1.835)
(1.351, 1.865)
(0.645, 2.571)
(1.359, 1.857)
(1.488, 1.728)
(1.492, 1.724)
(0.728, 2.488)
(1.361, 1.855)
10. Select the TWO options that are TRUE statements about clinical trials.
Select one or more:
Randomisation is used in clinical trials mainly to avoid selection bias, but can also help to facilitate blinding.
In a placebo-controlled clinical trial with a crossover design, some patients receive the placebo first and then change over to the drug under test, while others receive the drug first and change over to the placebo.
In a double-blind clinical trial, the researchers who assess the outcomes of the treatments do not actually see the patients, but instead make their assessment solely on the basis of the information in the patients’ records (which, of course, include details of which treatments they are receiving). The patients are not allowed to see the records till the trial is over.
In a clinical trial with a matched-pairs design, the available patients must be divided up into pairs entirely at random in order to ensure that the sample is representative.
In a group-comparative trial with two treatment groups, it is good practice to allocate alternate patients to the two treatments, while using a random method to decide which treatment is given to the very first patient to be enrolled.
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