Posted: February 26th, 2023

Statistics

Question 1
Motor vehicles have a recommended tire pressure of 32 psi (pounds per square inch). The
road safety authority launches a roadside checkpoint to measure the actual tire pressure in the
front left tire of a random sample of 50 passing cars. The authority hopes to use this
information to decide whether an additional investment in air pumps is required at filling
stations.

(a) Define the population characteristic of interest in the above example. (2 marks)

(b) The following SPSS output has been produced from a one sample t-test. Using the 5-step
approach to hypothesis testing, determine whether the road safety authority should invest in
additional air pumps at filling stations.

(10 marks)

One-Sample Statistics

N Mean
Std.

Deviation Std. Error Mean
Tire pressure 50 32.8200 1.98659 .28095

One-Sample Test

Test Value = 32

t df
Sig.

(2-tailed)
Mean

Difference

95% Confidence Interval of
the Difference

Lower Upper
Tire pressure 2.919 49 .005 .82000 .2554 1.3846

(c) The following 95% confidence interval has been computed for the mean tire pressure of
all motor vehicles. Does the confidence interval support the conclusion you made in part (b)?
Explain your answer in detail. (4 marks)

(d) Assume that the median tire pressure of all motor vehicles is 32.82. Produce an
appropriate graph that gives an approximation of the likely distribution of the tire pressure
variable. (3 marks)

(Note: You may use insert: shapes: lines: freeform scribble or manually sketch the graph and
insert a photographic image of the graph in your word solution document.)

(e) Redraw the graph assuming that the median tire pressure of all motor vehicles is actually
34. (3 marks)

(f) Comment on the graph in part (e) and provide an example of a dataset that would likely
follow a similar distribution. (3 marks)

Total 25 marks

95% Confidence
Interval for Mean

Lower Bound 32.2554
Upper Bound 33.3846

Question 2
Political science students at a university carry out a survey of 1,669 randomly selected
registered voters. They wish to determine if there is an association between the respondents’
political party (Republican or Democrat) and their opinion about a newly proposed gun
control law (support or oppose). The following two-way table summarizes the count data.

Gun control law

Total Support Oppose
Political

party
Democrat 438 247 685
Republican 374 610 984

Total 812 857 1669

(a) Among those who are democrats, what percentage supports the newly proposed law?

(1 mark)
(b) Among those who are republicans, what percentage supports the newly proposed law?

(1 mark)
(c) Do your calculations in parts (a) and (b) indicate that there is an observed relationship
between political party and opinion about the gun control law? Explain your answer.

(3 marks)
(d) Assuming that a relationship between the two variables is observed in the sample, can the
political science students infer from this that there is a relationship in the population?
If not, how would they go about making this inference?

(5 marks)

The economic and social research institute has gathered data on the potential factors that
influence current salary levels in the market. The following statistical output has been
produced from a linear regression, with current salary as the response variable and education
(no. of years) and previous experience (no. of months) as the explanatory variables.

Coefficientsa

Model

Unstandardized
Coefficients

Standardized
Coefficients

t Sig. B Std. Error Beta
1 (Constant) -20978.304 3087.258 -6.795 .000

Education 4020.343 210.650 .679 19.085 .000
Previous experience 12.071 5.810 .074 2.078 .038

a. Dependent Variable: Current salary

(e) Does an individual’s education predict their current salary well? Explain your answer with
specific reference to both the p-value of 0.000 and the beta coefficient of 4020.343.

(5 marks)

(f) Provide an example of a possible confounding variable that might change your
interpretation in part (e). Explain your answer. (4 marks)

(g) What would influence your conclusion on whether an individual’s previous experience is
significantly related to their current salary? Explain your answer in detail.

(6 marks)

Total 25 marks

Question 3
The department of health is currently drafting a new health initiative to promote weekly
exercise. The department wants to identify which groups in society most need additional
supports to increase their amount of weekly exercise. The department issues a survey to a
random sample of 100 employed individuals asking them “How many minutes do you
typically exercise in a week?” The summary statistics from the sample are:

Society group N Mean Standard
deviation

Employed in low income job

55 180.63 33.65

Employed in high income job

45 201.08 44.23

(a) Suggest an appropriate statistical test that would provide useful information to allow the
department to identify where supports are most needed in the population.
Explain your answer. (3 marks)

(b) Assuming a p-value of 0.073, write out the approach to the statistical test you suggested in
part (a). Clearly state your conclusion.

(6 marks)

(c) Based on your conclusion in part (b), what recommendation would you make to the
department of health regarding which group most needs additional supports to increase their
amount of weekly exercise. Explain your answer. (4 marks)

(d)
Assume that the department of health gathers additional data on the minutes of weekly
exercise from a third group in society:

Society group N Mean Standard
deviation

Unemployed

47 111.35 18.79

Suggest an appropriate statistical test that would allow for this additional data to be
incorporated into the previous analysis. (2 marks)

(e) Assume that the distribution of weekly exercise in minutes is bell-shaped. Using the
Empirical Rule, explain whether or not it would be unusual for an unemployed individual to
exercise for the same amount of time as an average individual employed in a high income
job.

(5 marks)

Total 20 marks

Question 4
Bailey Office Ltd. sells office supplies including a popular printer refill ink. Weekly demand
for the ink is normally distributed with a mean of 60 litres and a standard deviation of 24
litres. When the stock of this ink drops to 80 litres, Bailey Office Ltd places a replenishment
order. The delivery from the supplier takes place within one week of the order being placed.
The store manager is concerned that sales are being lost due to stockouts while waiting for
orders to be delivered.

(a) What is the probability of a stockout occurring? Explain in detail your approach to
calculating the probability. (5 marks)

(b) If the store manager wants the probability of a stockout to be no more than 5%, at what
level of stock should he place the replenishment order? (5 marks)

The following two boxplots show the weekly sales (€’000) of a retailer for the summer
season and the winter season:

(c) Discuss the ways in which the retailer’s sales activity differs between summer and winter
by comparing the two samples of data.

(10 marks)

(d) Provide a personal reflection on how this module has helped you to both analyse
quantitative data and to interpret and present results from such analyses. (10 marks)

Total 30 marks

END

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Summer sales Winter sales