Faculty of Business, Economics and Law

Bachelor of Business

Statistics for Business ECON862 Trimester 2 2020

ASSIGNMENT 3

• Answer all questions.

• There are 40 marks in total.

Section 1 – Statistical Inference and Confidence Intervals

Question 1 (8 Marks)

The manager of a leisure centre surveys a sample of 120 members on the average weekly amount of time spent using the centre’s gym, and whether they used the gym at least once in the last week.

The results of the sample are:

• Average time spent (minutes): ??¯ = 210

• Standard deviation (minutes): ?? = 82

• Number of members who used the gym at least once: 78

a) Construct a 90% confidence interval for the mean time spent in the gym and state the meaning of the confidence interval.

[2 marks]

b) Construct a 90% confidence interval for the proportion of members who used the gym at least once.

[2 marks] The manager wishes to conduct a similar survey on the use of the centre’s swimming pool.

c) What sample size does the manager need if he wants to estimate a 99% confidence interval for the mean amount time spent in the pool to within ±15 minutes, assuming a population standard deviation of 43 minutes?

[2 marks]

d) What sample size does he need to estimate a 99% confidence interval for the proportion of members who used the pool to within ±0.03?

[2 marks]

Section 2 – Hypothesis Testing: One-Sample Tests

Question 2 (6 Marks)

You work for a company that tests new pharmaceutical products. The current product is effective for 65% of the people who take it. You are currently testing a new product. The company would like the new product to be better than the existing product and improve symptoms for over 70% of the people who take it. From a sample of 350 subjects, 252 report an improvement in their symptoms from taking the new product.

a) State the null and alternative hypotheses that you would use to test the effectiveness of the new product.

[1 mark]

b) Determine to 5% significance whether the null hypothesis should be rejected.

[2 marks]

c) State whether the new product is over 70% effective and give reasons.

[1 mark]

d) If your conclusion were to turn out to be incorrect, state the type of error you will have made and explain the risks associated with this error in this context.

[2 marks] Question 3 (5 Marks)

Last season the mean amount spent in a store per customer was $41.76 with a population standard deviation of $17.05. The store manager wishes to determine to 2.5% significance if the sales for the current season have changed compared to last season. This season the standard deviation for all sales can be assumed to be the same as last season. The manager takes a sample of 75 customers, finding a mean of $43.19.

a) State the null and alternative hypothesis for this test.

[1 mark]

b) Determine, using the critical value approach, whether the mean amount spent this season has changed compared to last season.

[2 marks]

c) Find the p-value for the test and explain how the p-value relates to your conclusion in part b).

[2 marks]

Section 3 – Two Sample Tests

Question 4 (10 Marks)

A supermarket manager wishes to determine if crates of apples and crates of oranges that he gets from his suppliers are of the same mean weight. He takes a sample of 25 crates of apples to find a mean weight of 15.83kg with a sample standard deviation of 3.2kg, and a sample of 16 crates of oranges to find a mean weight of 17.1kg with a sample standard deviation of 2.6kg.

a) Assuming equal variances for both populations, test to a significance of 0.05 whether the mean weights of the populations of crates of apples and oranges are equal.

[3 marks]

b) Assuming unequal variances for both populations, test to a significance of 0.05 whether the mean weights of the populations of crates of apples and oranges are equal.

[4 marks]

c) Test whether the variances are equal at the 0.05 significance level.

[3 marks]

Section 4 – Analysis of Variance

Question 5 (6 Marks)

Test, at the 5% significance level, to discover whether differences exist between the population means given the statistics below. Remember to state the null and alternative hypotheses clearly.

Group 1 Group 2 Group 3

16 22 23

11 21 19

18 11 22

19 14 20

[6 marks]

Question 6 (5 Marks)

A researcher studying four different populations proposes the following:

??0 : ??1 = ??2 = ??3 = ??4

??1 : not all ???? are equal (?? = 1,2,3,4)

He takes samples from each of the populations, obtaining the following data:

Group 1 Group 2 Group 3 Group 4

Sample mean 104 99 85 96

Sample size 8 9 5 6

The within-group variation of the data (SSW) is equal to 108.

a) Using the Tukey-Kramer procedure, calculate the critical range for Groups 1 and 3, and for Groups 2 and 4 to a significance of 0.05.

[4 marks]

b) Explain whether or not the null hypothesis should be rejected and give evidence.

[1 mark]

~end~

Bachelor of Business

Statistics for Business ECON862 Trimester 2 2020

ASSIGNMENT 3

• Answer all questions.

• There are 40 marks in total.

Section 1 – Statistical Inference and Confidence Intervals

Question 1 (8 Marks)

The manager of a leisure centre surveys a sample of 120 members on the average weekly amount of time spent using the centre’s gym, and whether they used the gym at least once in the last week.

The results of the sample are:

• Average time spent (minutes): ??¯ = 210

• Standard deviation (minutes): ?? = 82

• Number of members who used the gym at least once: 78

a) Construct a 90% confidence interval for the mean time spent in the gym and state the meaning of the confidence interval.

[2 marks]

b) Construct a 90% confidence interval for the proportion of members who used the gym at least once.

[2 marks] The manager wishes to conduct a similar survey on the use of the centre’s swimming pool.

c) What sample size does the manager need if he wants to estimate a 99% confidence interval for the mean amount time spent in the pool to within ±15 minutes, assuming a population standard deviation of 43 minutes?

[2 marks]

d) What sample size does he need to estimate a 99% confidence interval for the proportion of members who used the pool to within ±0.03?

[2 marks]

Section 2 – Hypothesis Testing: One-Sample Tests

Question 2 (6 Marks)

You work for a company that tests new pharmaceutical products. The current product is effective for 65% of the people who take it. You are currently testing a new product. The company would like the new product to be better than the existing product and improve symptoms for over 70% of the people who take it. From a sample of 350 subjects, 252 report an improvement in their symptoms from taking the new product.

a) State the null and alternative hypotheses that you would use to test the effectiveness of the new product.

[1 mark]

b) Determine to 5% significance whether the null hypothesis should be rejected.

[2 marks]

c) State whether the new product is over 70% effective and give reasons.

[1 mark]

d) If your conclusion were to turn out to be incorrect, state the type of error you will have made and explain the risks associated with this error in this context.

[2 marks] Question 3 (5 Marks)

Last season the mean amount spent in a store per customer was $41.76 with a population standard deviation of $17.05. The store manager wishes to determine to 2.5% significance if the sales for the current season have changed compared to last season. This season the standard deviation for all sales can be assumed to be the same as last season. The manager takes a sample of 75 customers, finding a mean of $43.19.

a) State the null and alternative hypothesis for this test.

[1 mark]

b) Determine, using the critical value approach, whether the mean amount spent this season has changed compared to last season.

[2 marks]

c) Find the p-value for the test and explain how the p-value relates to your conclusion in part b).

[2 marks]

Section 3 – Two Sample Tests

Question 4 (10 Marks)

A supermarket manager wishes to determine if crates of apples and crates of oranges that he gets from his suppliers are of the same mean weight. He takes a sample of 25 crates of apples to find a mean weight of 15.83kg with a sample standard deviation of 3.2kg, and a sample of 16 crates of oranges to find a mean weight of 17.1kg with a sample standard deviation of 2.6kg.

a) Assuming equal variances for both populations, test to a significance of 0.05 whether the mean weights of the populations of crates of apples and oranges are equal.

[3 marks]

b) Assuming unequal variances for both populations, test to a significance of 0.05 whether the mean weights of the populations of crates of apples and oranges are equal.

[4 marks]

c) Test whether the variances are equal at the 0.05 significance level.

[3 marks]

Section 4 – Analysis of Variance

Question 5 (6 Marks)

Test, at the 5% significance level, to discover whether differences exist between the population means given the statistics below. Remember to state the null and alternative hypotheses clearly.

Group 1 Group 2 Group 3

16 22 23

11 21 19

18 11 22

19 14 20

[6 marks]

Question 6 (5 Marks)

A researcher studying four different populations proposes the following:

??0 : ??1 = ??2 = ??3 = ??4

??1 : not all ???? are equal (?? = 1,2,3,4)

He takes samples from each of the populations, obtaining the following data:

Group 1 Group 2 Group 3 Group 4

Sample mean 104 99 85 96

Sample size 8 9 5 6

The within-group variation of the data (SSW) is equal to 108.

a) Using the Tukey-Kramer procedure, calculate the critical range for Groups 1 and 3, and for Groups 2 and 4 to a significance of 0.05.

[4 marks]

b) Explain whether or not the null hypothesis should be rejected and give evidence.

[1 mark]

~end~

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