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STATISTICS FOR BUSINESS

NBS-5108B

1. Question (Total 20 marks)

In an electronics factory, three engineers are responsible for checking microchips. Engineer 1 checks 25% of all microchips, Engineer 2 checks 15% of all microchips, and Engineer 3 checks 60% of all microchips. Engineer 1 incorrectly identifies as not faulty 5% of microchips. Engineer 2 incorrectly identifies as not faulty 2% of microchips. Engineer 3 incorrectly identifies as not faulty 10% of microchips.

a) A microchip has just passed control. If the microchip has been incorrectly identified as not faulty, how should the probabilities that it has been checked by one of the three engineers (Engineer 1, Engineer 2 or Engineer 3) be revised to take into account this information?

[12 marks] 

b) Tabulate: (1) the prior probabilities that a randomly selected microchip has been checked by Engineer 1, 2 or 3; (2) the conditional probabilities that a microchip is incorrectly identified as not faulty if checked by Engineer 1, 2, or 3; (3) the joint probabilities that a microchip is incorrectly identified as not faulty and it has been checked by Engineer 1, 2, or 3; and (4) the posterior/revised probabilities that a randomly selected microchip has been checked by Engineer 1, 2 or 3 knowing that it has been incorrectly identified as not faulty.

[8 marks]

2. Question (Total 14 marks)

A bike store conducted a survey of its customers who bought their three most popular bike brands: Ribble, Endura, and Giant. They asked whether they are satisfied with their purchase or not. Here are the data:

a) What is the probability that a randomly selected surveyed customer is not satisfied with their purchase?

[2 marks] 

c) What is the probability that a randomly selected surveyed customer has bought a Giant bike and is not satisfied with their purchase?

[2 marks] 

d) Given that a randomly surveyed customer has bought an Endura bike, what is the probability that they are satisfied with their purchase?

[5 marks] 

e) Given that a randomly selected surveyed customer is not satisfied with their purchase, what is the probability that they bought a Ribble bike? 

[5 marks]

3. Question (Total 30 marks)

This question uses the dataset gym.sav that captures data from 90 members of a gym in relation to their reason for visiting the gym, the frequency of their visits to the gym and the total time they spent there, their gender, and whether they visit the gym with company.

The dataset contains 5 variables:

Reason: The reason for which a member visits the gym

Accompanied: Whether a member visits the gym with company or not

Frequency: How many times a week a member visits the gym

Gender: The gender of the gym member

TotalTime: The total time a member spends in the gym during a visit in minutes

a) Use SPSS to produce point estimates of the mean amount of time gym members spent in the gym during a visit across the different frequency of visits groups-“less than 1 day/week”, “1 to 3 days/week” and “more than 3 days/week”. Comment on your findings. Include your SPSS output in your answer.

[5 marks]

The gym manager believes that members who visit the gym with different weekly frequency, spent on average different amounts of time in the gym.

f) Which statistical test would you use to assess the manager’s belief? Explain why this test is appropriate. Provide the null and alternative hypothesis for the test. Define any symbols you use. Detail any assumptions you make.   

[6 marks] 

Use SPSS to perform this test.

g) Is the homogeneity of variances assumption met or not? Explain how you concluded whether it has been met or not. Include any SPSS output table you might refer to in your answer.

[6 marks] 

h) Do the data provide significant evidence to support the manager’s belief at the 5% significance level? Write down your results and conclusions providing statistical evidence. Include any SPSS output tables you might refer to in your answer.

[5 marks] 

Depending on whether the results indicate a significant difference on the mean amount of time spent in the gym by members who have different frequency of visiting the gym, the gym manager would wish to investigate between which of the groups there is a statistically significant difference on the mean amount of time spent during a visit.

i) Which procedure would you use to assess the manager’s belief? Explain why this procedure is appropriate.

[3 marks]

If appropriate depending on your conclusions in question 2d, use SPSS to perform this test.

j) Between which pairs of groups there is a statistically significant difference at 5% significance level in the average amount of time spent in the gym during a visit? Write down your results and conclusions providing statistical evidence. Include any SPSS output tables you might refer to in your answer.

[5 marks]

4. Question (Total 20 marks)

This question continues to use the dataset gym.sav.

The gym manager wishes to further explore the collected data and would like to find out whether female members who visit the gym for different reasons spent on average different amount of time during a visit.

a) Which statistical test would you use to assess the manager’s belief? Explain why this test is appropriate. Provide the null and alternative hypothesis for the test. Define any symbols you use. Detail any assumptions you make.   

[5 marks] 

Use SPSS to perform this test.

k) Do the data provide significant evidence to support the manager’s belief at the 5% significance level? Write down your results and conclusions providing statistical evidence. Include any SPSS output tables you might refer to in your answer.

[5 marks]

The gym manager would also like to find out whether the female members spent on average different amounts of time when visiting the gym depending on whether or not they are visiting with company.

l) Which statistical test would you use to assess the manager’s belief? Explain why this test is appropriate. Provide the null and alternative hypothesis for the test. Define any symbols you use. Detail any assumptions you make.   

[5 marks] 

Use SPSS to perform this test.

m) Do the data provide significant evidence to support the manager’s belief at the 5% significance level? Write down your results and conclusions providing statistical evidence. Include any SPSS output tables you might refer to in your answer.

[5 marks] 

5. Question (Total 16 marks) 

The data file data salaries.sav contains data from a random sample of 66 companies from either the USA or Japan who employ data analysts.  The dataset has the following variables:

company: An ID number for the company.

degree: Whether a degree in a mathematical subject is required.

salary: The starting salary in thousands of US dollars.

Country: Whether the company is based in the USA or Japan.

a) Use SPSS to produce point estimates of the mean starting salary for companies based in the USA and in Japan. Comment on your findings. Include your SPSS output in your answer.

[2 marks]

A researcher wishes to investigate whether starting salaries in the USA and in Japan are the same or different.

n) Which statistical test would you use to investigate this? Explain why this test is appropriate. Provide the null and alternative hypothesis for the test. Define any symbols you use. Detail any assumptions you make.   

[4 marks] 

Use SPSS to perform this test.

o) Is the homogeneity of variances assumption met or not? Explain how you concluded whether it has been met or not. Include any SPSS output table you might refer to in your answer.

[5 marks] 

p) What evidence do data provide at the 5% significance level? Write down your results and conclusions providing statistical evidence. Include any SPSS output tables you might refer to in your answer.

[5 marks]