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Time allowed: Three hours Total marks: 40

Question Booklet

There are FIVE questions over TWO sections in this booklet. Section A contains Questions One and Two, Section B contains Questions Three, Four, and Five.

You should attempt ALL the questions in this booklet.

Use the Formulae and Tables Booklet provided.

Write your answers in the Answer Booklet.

Show ALL working. Start your answer to each question on a new page. Carefully number each question.

Check that this booklet has pages 2–11 in the correct order and that none of these pages is blank.

You mAY keep this bookLet At the end of the exAminAtion.

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new Zealand scholarshipstatistics

sAmpLe Question bookLet

You have three hours to complete this examination.

SECTION A: EMPLOYMENT IN NEW ZEALAND

Questions One and Two refer to the Household Labour Survey carried out by Statistics New Zealand. In each quarter, a sample of 15 000 private households (about 30 000 New Zealand residents aged 15 years or more) is taken. Each person in the survey is classified as employed, unemployed, or not in the workforce.

Question one

Figure 1 below shows the actual and seasonally adjusted numbers of people employed in New Zealand from the December 1995 quarter to the December 2010 quarter. The scale on the vertical axis is in thousands (000).

Figure 1

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Figures 2 and 3 below show the seasonally adjusted numbers of people employed full-time and part-time respectively, from the December 2006 quarter to the December 2010 quarter. The scale on the vertical axis is in thousands (000). A smoothed series is also shown on each graph (labelled ‘Trend’).

Figure 2 Figure 3

(a) Using the graphs, give SIX distinct observations that can be made about employment in New Zealand. Some of your observations should relate to the components of a time series.

(b) The employment data in Figure 1 were smoothed and a line fitted to the smoothed series. The equation of the line is y = 1670 + 40t, where y represents the number of people employed in a quarter and t represents the number of years after 1995 (for 1995, t = 0).

(i) Given that estimates of the seasonal effects for the March, June, September, and December quarters (in 000) are -18.6, -90.3, +42.1, and +77.4 respectively, obtain forecasts for the last two quarters in 2012.

(ii) Discuss the validity of the forecasts in (i).

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(c) Figure 4 below shows the seasonally adjusted employment rate from the December 2006 quarter to the December 2010 quarter. Figure 5 below shows the seasonally adjusted unemployment rate by sex for the same period.

Figure 4 Figure 5

Discuss what the graphs show about the unemployment rate over the period from the December 2006 quarter to the December 2010 quarter.

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Question tWo

The diagram in Figure 6 below summarises the labour market in the December 2010 quarter. ‘Not in the labour force’ refers to people who are retired, attending educational institutions, or who are home-makers.

The results shown in Figure 6 are based on a sample of 29 456 people selected from the New Zealand working-age population in the December 2010 quarter.

The diagram was presented in an official report.

Figure 6 Labour market in the December 2010 quarter

Working-age population3 448 000

In theLabour force

2 341 000

Not in thelabour force1 107 000

Employed2 182 000

Unemployed159 000

Source: Statistics New Zealand

(a) (i) Find the survey percentage (point estimate) of the labour force in the December 2010 quarter who were employed.

Hence obtain a 95% confidence interval for the percentage of the labour force in the December 2010 quarter who were employed.

(ii) Explain the meaning of the confidence interval in (i).

(b) In the March 2011 quarter, the number in the labour force was 2 369 000. In this quarter, a large sample of those in the labour force was taken, and the percentage of those in the sample who were employed was found. Repeated sampling with replacement was carried out on this sample, and the percentage of those who were employed in each resampling was recorded.

Table 1 below shows the cumulative percentage distribution of the resampling data. For example, 30% of the resampling data was less than 93.32%.

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Table 1

Cumulative % Resampling data0% 92.10%10% 92.12%20% 92.19%30% 92.34%40% 92.65%50% 93.20%60% 93.74%70% 94.03%80% 94.16%90% 95.19%

100% 95.21%

(i) Using the resampling data, obtain a 95% confidence interval for the true (population) percentage of those in the labour force who were employed in the March 2011 quarter, and hence give 95% confidence limits for the number in the labour force who were employed in this quarter.

(ii) In a previous survey, the survey percentage of the labour force who were employed was 93.04%. The claim is made that the percentage of those in the labour force who were employed in the March 2011 quarter has increased since the previous survey. Discuss this claim.

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SECTION B: TELLER EMPLOYMENT AND PERfORMANCE AT XYZ BANk

Questions Three, Four, and Five relate to tellers at the XYZ Bank.

Question thRee

In a study of the effect of work experience on the efficiency of tellers, a random sample of 36 tellers was selected. Data were collected for the following variables:

E: Experience (the time the teller has been employed, in months)

S: Service rate (the mean number of customers served per hour)

The data are shown in Table 2 below.

Table 2 E S E S

15 5.1 11 19.32 11.3 2 7.94 10.6 18 22.13 12.0 14 20.06 13.4 19 26.45 14.9 16 22.77 15.3 1 23.88 17.2 17 21.26 10.1 4 23.0

10 15.6 12 19.113 20.5 10 19.214 21.3 3 25.616 20.6 13 28.313 19.4 6 24.620 20.8 8 25.116 21.6 10 26.218 19.5 14 28.313 21.8 16 28.5

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A scatter plot of the data is shown in Figure 7 below.

Figure 7

(a) Discuss the relationship between the variables E and S.

(b) The regression lines y = 0.411x + 15.1, y = 0.646x + 9.84, and y = 0.255x + 23.6 were obtained in relation to the data.

(i) Select one of the above regression lines that will enable you to make predictions of the service rate. Write the equation of the line you have selected and justify your selection.

(ii) Predict the service rate for tellers with 9 months experience and tellers with 27 months experience.

(iii) Discuss your predictions. In your discussion you should refer to:• the target population for your predictions• the validity of the predictions.

(c) Other variables that may be related to the service rate are:A: age of tellerH: mean height of teller’s customersT: mean time taken to serve customers.

Discuss the nature of the relationship that might be expected between each of these variables and S. In your answer, you should:

• describe the strength and direction of each relationship, and justify your descriptions• use appropriate statistical terminology.

You may find it helpful to use diagrams.

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Question fouR

(a) Chris is one of three people who work ‘on call’ as replacements when regular staff call in sick for a day. The ‘on call’ workers are used randomly and are always available. On any given day, the number of staff who call in sick for that day can be modelled by a Poisson distribution having a mean of 1.4.

If Chris is working on a particular day, find the probability that exactly one other ‘on call’ worker is being used that day.

(b) The number of hours that a part-time worker works varies from week to week. Over a long period, the worker finds that their working time exceeds 20 hours in a given week in 32.5% of weeks.

(i) Find the probability that in any period of eight weeks their working time exceeds 20 hours in more than five of the weeks.

(ii) State what assumption(s) it is necessary to make in (i), and comment on the reasonableness of the assumption(s).

(c) The weekly earnings of each of two part-time employees A and B vary from week to week. The weekly earnings for A and B are independent. They are paid every four weeks.

The weekly earnings (in dollars) for A and B can be modelled by a normal distribution with parameters as shown in Table 3 below.

Table 3 Employee Mean Standard deviation

A x 21.7B 620.4 29.4

A and B compare their weekly earnings over a large number of pay periods (four weeks) and find that in 24% of the pay periods A’s weekly earnings exceed B’s weekly earnings in exactly two of the weeks in the pay period.

Let the probability that A’s weekly earnings exceed B’s weekly earnings in a randomly chosen week be p. Find the value(s) of p and hence find A’s mean weekly earnings (shown as x in Table 3).

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Question fiVe

(a) The following is an extract from a media report concerning XYZ bank.

New Zealand men tend to save more money than women, according to a recent survey carried out by XYZ Bank on its customers. Men are also more likely than women to invest money in shares that are high risk.

The findings come from an online survey of 1 200 XYZ Bank customers. Customers were randomly selected to complete the survey when they were logged into their accounts online.

The survey found that men had an average of nearly $5 568 in their savings account, whereas women had an average of just over $3 827 in their savings account. Nearly 36% of the men surveyed had invested in shares that were high risk, but only about 23% of women had done the same.

The survey also included questions about age, income, and education.

From the extract, evaluate the report. In your discussion, you should: • evaluate the statistical evidence and processes• evaluate the claims made in the extract• suggest potential sources of bias, and for each explain why it may affect the survey findings• use appropriate statistical terminology.

(b) An experiment to test the effectiveness of a bank teller training course was carried out. Forty bank tellers were assigned randomly to two equally sized groups: A and B. Group B tellers attended a training course but group A tellers did not. After the training course had been completed, the time (in seconds) taken to complete a particular transaction was found for each teller in both groups.

Figure 8 below shows:• the actual results for each teller• a dot plot of the actual results for each group• a box-and-whisker plot for the results for each group• a bar graph of the results of 1 000 re-randomised differences in the medians for the two

groups.

Figure 9 below shows:• the actual results for each teller (as for Figure 8)• a dot plot of the actual results for each group (as for Figure 8)• a histogram of the results of 1 000 re-randomised differences in the means for the two groups.

Table 4 below gives the median and mean for each group.

Table 4 Group Median Mean

A 27 27.2B 26 25.0

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Figure 8 Median difference

Source: iNZight

Figure 9 Mean difference

Source: iNZight

(i) What can be concluded from this output? Justify your answer.

(ii) Discuss what factors need to be considered before a claim of a causal relationship can be made for this situation.

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