1. This assignment consists of EIGHT (8) questions.
2. Answer ALL questions.
3. Write your answers on binder sheets.
4. ALL calculations/working MUST be clearly shown.
5. Silent, Cordless, Non–Programmable CALCULATORS are permitted.
6. Statistical Tables and Formula Sheets will be provided.
1. (a) List three (3) properties of the normal distribution. [3 marks]
(b) Find P 1.05 Z 1.5 . [3 marks]
(c) Scotia Bank has analyzed the monthly spending by its credit card customers and found that it is normally distributed with a mean of $100 (𝜇 = 100) and a standard deviation of $15 (𝜎 =15).
(i) What is the probability that a customer will spend over $115, that is, 𝑃 (𝑋𝑋> 115)? [3 marks]
(ii) What is the probability that a customer will spend less than $70, that is, 𝑃 (𝑋𝑋< 70)? [3 marks]
(iii) What is the probability that a customer will spend between $70 and $115, that is, 𝑃 (70<𝑋𝑋<115)? [3 marks]
(iv) What is minimum amount spent by the top 2.5% of customers? [3 marks]
2. (a) State THREE (3) reasons why it is sometimes necessary to sample. [3 marks]
(b) Scores on a Statistics test have a mean of 72 and a standard deviation of 12. Random samples of size 64 are drawn from this population and the mean of each sample is determined. Find the mean and standard error of the mean of the sampling distribution. [4 marks]
(c) State the central limit theorem for the mean. In this context, what sample size is considered large? [3 marks]
3. (a) An agriculturist measures the heights of 16 seedlings and from this sample obtains a mean
and standard deviation of 72.5 cm and 4.5 cm, respectively. a. Assuming the population is normally distributed, construct a 98 percent confidence interval for the mean height of seedlings in the population from which the sample was selected. [9 marks]
(b) Suppose that Francis Trading management team wants to estimate of the mean of annual income of the individuals in the marketing area of the new location in Nevis Street. It is known that the population standard deviation is $4500. How large a sample should be selected so that the estimate with a 99% confidence level is within $500 of the population mean? [5 marks]
4 When 490 college students were surveyed, 130 said they own their car. Construct a 95% confidence interval for the proportion of college students who say they own their cars. [9 marks]
5. An accountant claims to be able to complete a standard tax return in under an hour. For a random sample of 24 tax returns, the accountant averaged 63.2 minutes with a sample standard deviation of 7.7 minutes. Assuming that the population was normally distributed, use the critical value approach to investigate the validity of the claim made by the accountant. Conduct your test using the 10% level of significance. [14 marks]
6. An athletics coach claims that athletes in a community consume 30 regular sized bottles of drinking water per week on average (that is, Suppose you wanted to test if there is evidence to reject this claim).
(i) What would be your null hypothesis? (Be sure to state the null hypothesis algebraically). [1 mark]
(ii) What would be your alternative hypothesis? (Be sure to state the alternative hypothesis algebraically). [1 mark]
(iii) Would you use a two-tail test or a one tail test to test the validity of the claim made by the athletics coach? Give a reason for your choice. [2 marks]
(iv) In this context, what would be considered a Type I error? [2 marks]
(v) In this context, what would be considered a Type II error? [2 marks]
(vi) To test the claim made by the athletics coach that athletes consume 30 regular sized bottles of drinking water per week, you collected data on the water consumption habits of 36 athletes in a community. Suppose this random sample of 36 athletes yielded a sample mean of 32 regular sized bottles of drinking water per week and a sample standard deviation of 3 regular sized bottles of drinking water per week. Carry out a hypothesis test to investigate the validity of the claim made by the athletics coach at
𝛼 = 0.05. (Be sure to state your decision and conclusion). [8 marks]
7. The owner of Ocean View retail store randomly selected the following weekly data on profits and advertising cost
|
Advertising Cost ($) - X |
Profit ($) - Y |
|
0 |
200 |
|
50 |
270 |
|
250 |
420 |
|
150 |
300 |
|
125 |
325 |
Summary: Σx = 575, Σy = 1515, Σxy = 204125, Σx2= 103125, Σy2 = 484925
(a) Calculate the correlation coefficient, r. [3 marks]
(b) Interpret the correlation coefficient. [2 marks]
(c) Develop the least squares estimated regression equation based on the population regression model Y a bX
That is, write the simple regression equation, (providing the values of a and b) in the [4 marks]
(d) Interpret the y-intercept of the estimated regression equation from part (c) above. [2 marks]
(e) Interpret the slope of the estimated regression equation from part (c) above. [2 marks]
(f) Predict the profits for a week when $200 is spent on advertising. [2 marks]
(g) Predict the advertising cost for a week in which KB retail store made a profit of $350. [2 marks]
(h) What is the value of the coefficient of determination, that is, r 2 ? [2 marks]
(i) Interpret the value of the coefficient of determination found in part (h) above, that is, r 2 . [2 marks]
8. The CEO of a major office supply chain is interested in determining whether an employee’s educational level is related to their knowledge of their job. A job knowledge test was given to 500 employees. The results are given in the table below.
|
Test Scores |
Educational level |
||
|
Associate Degree |
Bachelor Degree |
Master Degree |
|
|
18-25 |
70 |
50 |
20 |
|
26-35 |
40 |
50 |
30 |
|
36-45 |
30 |
60 |
30 |
|
Over 45 |
20 |
80 |
20 |
Carry out a test to determine whether the job knowledge test score is independent of the employee’s education level at the 1% level of significance. (Be sure to state your null and alternative hypotheses, the type of test, the test statistic, the critical value, your decision and finally, state your conclusion.
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