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

05

1015

Summer sales Winter sales

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.

The population characteristic of interest is the probability that a car has an air pressure less than 32 psi. The mean and standard deviation of this distribution are

p = .05 = .05, s = .10 = .10.

The expected number of cars that have an air pressure less than 32 psi is the average (mean) of the data. The mean can be used to estimate the probability of a low air pressure.

- On your graph, plot the observed value for each car vs. its probability of having a low air pressure. (a) How do you know that the data are approximately normal?
- How would you interpret this graph?

Graphical methods are useful for understanding the shape of a distribution. They can help you understand how data are distributed and make comparisons between different distributions easier. For example, a histogram is useful for visualizing the distribution of a variable in your sample or population. A bar graph displays values on horizontal axis, while each bar represents one observation (or sample). The x-axis is used to label categories such as “categories” or “values.” This type of chart is often used when trying to identify what proportion each category represents in your data set:

- Histograms show whether there’s any deviation from normality in your data set; if so, it may indicate that something needs investigating further before drawing conclusions about its meaning (e.g., whether this difference could be due to chance).

We’ve all seen it: 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. In this case study, we’ll use some simple graphical methods to get a better understanding of our data, which will help us interpret them correctly and answer questions such as “How many cars have low air pressure?”

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