Question 1 – 3 marks
The data are given in the file households.mtw. In this file there are the following variables.
• id: a serial number that identifies each area.
• area: the name of the area.
• type: whether the area is a county council or unitary authority: ‘1’ – county council, ‘2’ – unitary authority.
• size: number of households in the area: ‘1’ – less than 125 000, ‘2’ – at least 125000.
• older: the percentage of households that consist only of people aged 65 or over (‘older’ households).
• family: the percentage of households that include dependent children (‘family’ households).
By looking at the data file households.mtw, answer the following.
(a) Which area has the highest percentage of ‘older’ households?
(b) Is the area that you identified in part (a) a unitary authority or a county? For this area, what is the percentage of ‘family’ households?

Question 2 – 13 marks
A contingency table of the size and type of area is given below.
                                                         County council    Unitary authority    Total
‘Small’: less than 125 000 households          0                         20                20
‘Big’: at least 125 000 households                14                         6                   20
Total                                                                14                      26                   40
Note that this table is given in the file areas.mtw.
(a) Explain carefully why this table is correctly described as a contingency table. [3]
(b) What is the probability that a randomly selected area had less than 125 000 households? Give your answer to three decimal places. [1]
(c) What is the probability that a randomly selected unitary authority has at least 125000 households? Give your answer to three decimal places. [1]
(d) Suppose that a researcher is interested in the following question: Are unitary authorities more likely to be big compared to county councils?
(Here take ‘big’ to mean at least 125000 households.)
(i) Write down suitable null and alternative hypotheses. [2]
(ii) Using the χ2 test for contingency tables, test the hypotheses that you wrote down in part (d)(i). Make sure to include in your answer:
• a table of the expected values (with the expected values given as whole numbers)
• the value of the test statistic
• the degrees of freedom
• the p-value or the values of CV5 and CV1
• your conclusion from the test. [6]

Question 3 – 5marks
(a) Using Minitab, produce a stemplot of the percentage of ‘family’ households in which outliers are displayed separately. Include this stemplot in your answer. (In doing this, select the Trim outliers option but leave the Increment field blank and make sure that the stemplot includes all the areas.) [1]
(b) Use your stemplot to describe the shape of the distribution of the percentages of family households. Justify your answers.

Question 4 – 18 marks
(a) Using Minitab, or otherwise, obtain boxplots of the percentages of ‘older’ households in areas that are unitary authorities and areas that are county councils. (These percentages are given in separate columns in the file older households.mtw.)You should ensure that the boxplots are horizontal and drawn on
the same axes. You should prepare these boxplots ready for inclusion in a report. That is, ensure that the title and the label for the horizontal axis are clear and informative. Include the finished boxplots in your answer. [4]
(b) Complete the following table of summary statistics for the percentage of ‘older’ households. (You should give the number of areas to the nearest whole number, and all the other values rounded to two decimal places.)

               County councils Unitary authorities
Number
Mean
Median
Variance
Interquartile range
Range
(c) A sociologist claims that county councils and unitary authorities have the same mean percentage of ‘older’ households. Write down suitable null and alternative hypotheses to test this claim, stating clearly the meaning of any symbols that you use. [2]
(d) You are going to use a two-sample t-test (for populations with a common variance) to test the hypotheses that you wrote down in part (c). The appropriateness of this test depends on two assumptions – state these assumptions. Using your answers to parts (a) and (b), explain why these assumptions do not seem unreasonable. [5]
(e) Use the two-sample t-test (for populations with a common variance) and the data in olderhouseholds.mtw to test the hypotheses that you wrote down in part (c). Make sure to include in your answer:
• the pooled estimate of the common population variance (you should give your answer to two decimal places)
• the value of the test statistic
• the p-value or the value of CV5 (if you give the value of CV5, you are advised to state how many degrees of freedom this is based on)
• your conclusion from the test. [5]

Question 5 – 13 marks
The sociologist is also interested in the relationship between the percentage of ‘older’ households and the percentage of ‘family’ households.
(a) Suppose that a scatterplot is to be drawn of the percentage of ‘older’ households and the percentage of ‘family’ households. Why is it not clear which variable should be drawn on the x-axis and which on the y-axis? [2]
(b) Obtain the scatterplot described in part (a), and include it in your answer. Interpret this scatterplot, giving reasons for your interpretation. [9]
(c) The correlation coefficient for these two variables is −0.788. With reference to the scatterplot given in part (b), state whether this correlation coefficient provides a good representation of the strength of the relationship between the percentages of ‘older’ households and the percentages of ‘family’ households. Justify your answer. [2]

Question 6 – 4marks
The sociologist is also interested in doing some in-depth examinations of the areas, but can afford to investigate only eight areas in this way. Using stratified sampling, the sociologist selects the following four small areas and four large areas. (Here small areas are taken to be those with less than 125 000 households, and large areas are those with at least 125000 households.)
4 16 28 38
13 19 24 34
By comparing the numbers of county councils and unitary authorities in the sample with the numbers for all the areas represented in the file households.mtw, comment on the representativeness of the selected sample. Why must the sample be representative with respect to the size of the areas? Justify your opinion. [4]

Question 7 – 4marks
In one of the areas, a couple of social workers are interested in whether a new scheme would improve the quality of life for older people who live in single-person households. They intend to investigate this by carrying out a trial comparing the new scheme with the status quo.
Through local advertising, the social workers hope to generate a list of older people who live in single-person households and are willing to take part in the trial. Using this list, the social workers intend to split the older people into pairs in such a way that the pairs have backgrounds and circumstances that are as similar as possible. They will then offer the new scheme to one person in each pair, the person being chosen at random.
(a) What design of trial is this? [1]
(b) Why is it important that within each pair the older person who is offered the new scheme is chosen at random? [1]
(c) One of the outcomes of the trial will be the number of people that each participant knows in the local area two months after the trial has started. Explain why such data arising out of the trial could be considered to be objective and measured on an interval scale. [2]

Question 8 – 10 marks
Suppose that it is known that for all areas in England, the population mean percentage of ‘family’ households is 28.83% and the population standard deviation is 3.13%.
(a) For a sample of 32 areas, what is the approximate distribution of the sample mean? [4]
(b) For a sample of 32 areas, the sample mean percentage of ‘older’ households turns out to be 21.50%, and the corresponding estimated standard error is 0.69%. For this sample, calculate by hand a 95% confidence interval for the mean. Give the interval correct to one decimal place, and show your working. [4]
(c) What does the interval that you calculated in part (b) imply about the result of a hypothesis test that tests the hypothesis that the population mean percentage of ‘older’ households is 18%?

Question 1 – 3 marks The data are given in the file households.mtw...