MS5312 Business Statistics with R

MS5312 Business Statistics with R
2008-2009 final exam paper

Question 1 (25 marks)

(a) A production line makes bottles of shampoo. When operating correctly, it produces bottles whose contents weigh, on average, 20 ounces. A random sample of 9 bottles from the production line gave the following weights (in ounces).

21.4 19.7 18.7 21.6 21.8 20.1 18.7 22.3 21.9

Assume that the population distribution is normal.

(i) Test the null hypothesis that the production line is operating correctly against the alternative that it is not operating correctly at the 5% level, and draw your conclusion. (7 marks)
(ii) Find a tight range of the p-value in subpart (i), and use it to draw your conclusion? (3 marks)
(iii) Construct a 99% confidence interval for the population mean weight of the bottles of shampoo. (4 marks)
(iv) Assume that the population standard deviation is known to be 0.6 ounces. What sample size is needed to estimate the population mean weight to within 2 ounces with 90% confidence? (3 marks)


(b) Many academic researchers have developed models based on financial ratios in order to predict whether a firm will go bankrupt next year. In a reliability study of one such model, the records indicate that the model had correctly predicted 85% of firms that went bankrupt and correctly predicted 82% of firms that did not go bankrupt. Suppose that 4 firms out of 100 go bankrupt in a year,

(i) what is the probability that the model predicts bankruptcy? (3 marks)
(ii) what is the probability that a firm will go bankrupt given that the model predicts bankruptcy? (2 marks)
(iii) what is the probability that the prediction of the model is correct? (3 marks)



Question 2 (25 marks)

(a) In a television commercial, the manufacturer of a toothpaste claims that 4 dentists out of 5 recommend the ingredients in his product. To test that claim, a consumer-protection group randomly samples 400 dentists and asks each one whether he or she would recommend toothpaste that contained the ingredients. The responses can only be in two forms: 'Agree or 'Disagree'. The sample shows that 240 dentists choose 'Agree' and 160 dentists choose 'Disagree'.

(i) At the 5% significance level, can the group infer the over-statement in the TV commercial? (6 marks)
(ii) Determine a 95% approximate confidence interval for the proportion of dentists' recommendation for the ingredients. (4 marks)
(iii) Given that the proportion of dentist recommendation for the ingredients is between 60% and 80%, and if the estimation error of this proportion must be within 0.05 at the 90% confidence, what is the sample size? (4 marks)


(b) The BDW car dealership sells one sports model, the FX500. Of the customers who buy this model, 50% choose fire-engine red as the color, 30% choose snow white and 20% choose jet black.

(i) What is the probability that at least 2 of the next 10 customers who buy the FX500s will choose black? (4 marks)
(ii) What is the probability that at least 40 of the next 100 customers who buy the FX500s will choose white? (5 marks)
(iii) A customer who buys a red, white or black FX500 on average puts a premium of $3000, $2000 or $1500 on his/her car, respectively. What is the expected value of a premium? (2 marks)



Question 3 (20 marks)

A sample of 12 homes sold last week in Shatin was selected. The home sizes x (in thousands of square feet) and the corresponding selling prices y (in millions of dollars) were recorded as follows:

x 0.9 1.3 1.5 1.2 1.1 1.4 1.0 0.8 1.3 1.2 0.9 1.1
y 4.4 4.8 5.3 5.4 4.8 5.2 4.1 3.2 7.2 4.5 3.1 4.0

(a) Test through both linearity and strength whether there is a straight-line relationship between the two variables 'size' and 'price'.

(i) What are the respective null hypotheses and alternative hypotheses? (2 marks)
(ii) Calculate the respective values of the test statistic for the given data set. (4 marks)
(iii) What will be your conclusions if the significance level of the test is 0.05? (2 marks)

(b) Find a straight-line equation for the data so that we are able to estimate or predict the price for a given size. (4 marks)
(c) What will be the estimated or predicted price if the size is 1150 square feet? (2 marks)
(d) Interpret the slope of the fitted straight-line equation. (2 marks)
(e) What are the values of the determination coefficient and correlation coefficient? (2 marks)
(f) For the given data set, what percentage of the observed total variation in prices can be explained by the estimated straight-line relationship with sizes? (2 marks)



Question 4 (30 marks)

(a) To celebrate their first anniversary, Peter decided to buy a pair of diamond earrings for his wife Mary. He was shown 9 pairs with marquise germs weighing approximately 2 carats per pair. Because of differences in the colors and qualities of the stones, the prices varied from set to set. The average price was $2990, with a sample standard deviation of $370.
He also looked at 6 pairs with pear-shaped stones of the same two-carat approximate weight. These earrings had an average price of $3065, with a standard deviation of $805. Assume that both population distributions are normal and their variances are equal,

(i) find a 95% confidence interval for the difference of the mean price between marquise diamonds (regarded as population 1) and pear-shaped (as population 2) diamonds. (4 marks)
(ii) can Peter conclude at a significance level of 5% that marquise diamonds cost less, on average, than pear-shaped diamonds? (6 marks)


(b) To test whether the mean times to mix a batch of a certain material done by three manufacturers are the same, we use the following mixing times (in minutes).

Manufacturer 1 Manufacturer 2 Manufacturer 3
32 44 33
30 43 36
30 44 35
26 46 36





(i) Obtain the six unknown values and the overall sample mean. (3 marks)
(ii) At the 5% level of significance, test whether the population mean time to mix a batch of the material differs for the three manufacturers. (5 marks)
(iii) Complete an ANOVA table for this problem. (2 marks)


(c) A diligent statistics student wants to see if it is reasonable to assume that some sales data have been sampled from a normal population before performing a hypothesis test on the mean sales. She collected 200 sales data, computed and s = 9, and tabulated the data as follows:

Sales ≤ 65 66 - 70 71 - 75 76 - 80 81 - 85 ≥ 86
Observed number 10 20 40 50 40 40

The following figure reveals the probabilities (obtained using a normal distribution with and ) that sales will be less than or equal to 65, between 66 and 70, between 71 and 75, between 76 and 80, between 81 and 85, and greater than or equal to 86 are 0.0823, 0.1210, 0.1864, 0.2206, 0.1864, and 0.2033, respectively.


(i) Notice that the degree of freedom for the goodness-of-fit test is 3 not 5. Give a brief explanation. (1 mark)
(ii) At the 5% level of significance, does the observed distribution follow a normal distribution? (9 marks)