General revision
Question 1: State whether the following statements are true or false and give a brief explanation
1. If the value of the linear correlation coefficient is zero, then these two variables are independent.
2. The least square regression line minimizes the sum of errors.
3. The covariance between any two variables decides the strength of their relation.
4. The regression coefficient decides the trend and the strength of the relation between the dependent and the independent variables.
Question 2: Define the following terms briefly.
1. The determinant coefficient.
2. The deterministic model and the probabilistic model.
Question 3: Answer the following questions briefly.
1. Explain the meaning of independent and dependent variables for a regression model. Explain the difference between a simple and a multiple regression model.
2. Compare between the roles of both the regression coefficient and the correlation coefficient. Can the values of regression coefficient and the correlation coefficient have different signs? Explain.
3. The similarities and the difference between the normal, the standard normal.
4. Can the values of the regression coefficient and the correlation coefficient have different signs? Explain.
5. If the correlation coefficient between two variables equals zero, then the two variables are independent. Comment.
6. Why is the random error included in a regression model?
Applied Problems
Question1: The following table gives the experience (in years) and the number of computers sold during the previous three months by seven salespersons.
|
Experience ( years)
|
4
|
12
|
9
|
6
|
10
|
16
|
7
|
|
Computer sold {unit)
|
19
|
42
|
28
|
31
|
39
|
35
|
21
|
(a) Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear relationship between the two variables?
(b) Find the suitable least squares regression line.
(c) Give a brief interpretation of the values of the Y-intercept and slope calculated in the previous part.
(d) Calculate the correlation coefficient and the determinant coefficient. Explain what they mean.
(e) Predict the number of computers sold during the past three months by salespersons with eleven years experience.
Question 2: S jeans manufacturer knows that a large budget for television advertising (x) of his product will create a demand (y) for it among department store buyers. In a sample of eight years we obtain the following data.
∑x = 920 ∑ x2 = 124500 ∑y = 885 ∑y2 = 115075
The regression coefficient of y on x (b) = 0.94
Using the previous information calculate
(1) The correlation coefficient (r) and comment on the result.
(2) The determinant coefficient (R2) and comment on the result.
(3) Predict the demand (y) when the budget for television advertising (x) equals 10.
Question 3: Scores on the sales aptitude test are approximately normally distributed, with a mean of 500 and a variance of 625. The head of a personal has decided to extend job to those in the top 15% and to discard the files of those in the bottom 60%.
1) What is the cutoff for a job offer?
2) What is the cutoff for discarding?
Question 4: A charity believes that when it puts out an appeal for charitable donations the donations it receives will normally distributed with a mean $ 50 and standard deviation $ 6, and it is assumed that donations will be independent of each other.
1) Find the probability that the first donation it receives will be greater than $ 60.
2) Find the value X such that 5% of donations are more than $ X.
Question 5: The management of a supermarket wants to adopt a new promotional policy of giving free gift to every customer who spends more than a certain amount per visit at this supermarket. The expectation of the management is that after this promotional policy is advertised, the expenditure for all customers at this supermarket will be normally distributed with mean $95 and a standard deviation of $21.
a) If the management wants to give free gifts to at most 8% of the customers, what should the amount be above which a customer would receive a free gift?
b) In a sample of 100 customers, what are the number of customers whose expenditure is between $74 and $137?
c) In a sample of 25 customers, what is the probability of choosing a customer whose mean of expenditure is between $90 and $101?
Question 6: The following data represents the grades of a sample of 12 students in statistics and economics as follows
|
Math grade
|
A
|
C
|
B
|
B
|
D
|
B
|
F
|
F
|
D
|
C
|
B
|
C
|
|
Economic Grade
|
B
|
F
|
C
|
B
|
C
|
A
|
C
|
F
|
B
|
B
|
A
|
C
|
Calculate the correlation between the student's grade in math and economics. What its type and strength.
Question 7: In a study of the relation between the expanding budget of advertisement on a specific product (X) in thousand of pounds and the quantity sold (Y), in tons, of a product. A sample of the advertising budget and the quantity sold in eight successive months is drawn from on of the companies which are dealing in such a product and the following information is obtained.
Using the previous information calculate:
1) The correlation coefficient (r) and the determinant coefficient ( ). Comment on the result.
2) Find the regression line of Y on X and interpret the meaning of its intercept and regression coefficient.
3) Predict the number of tones sold from this product if the expended advertising budget in one month is 25 thousand pounds.
Question 8: The food and drug administration (FDA) approves 20% of drugs submitted for its approval. In a random sample of 15 drugs submitted for the FDA's approval.
a) Find the probability that the number of drugs that are approved is at most three.
b) What is the mean and the standard deviation of the number of drugs submitted for the FDA's approval?
c) If the sample size of drugs is changed to be 30 drugs, what is the probability that the number of drugs that are approved is between 15 and 25 drugs?
Question 9: The balances of all saving accounts at a local bank have a normal distribution with its mean equal to $ 12450 and standard deviation equal to $ 4160. Find the probability that the mean of a sample of 50 saving accounts selected from this bank will be
(a) More than $ 11500.
(b) Within $ 1500 of the population mean.
(c) Find the number of accounts which will have an account of more than the population mean by at least $ 1000.
Question 10: Fast Auto service provides oil and lube for cars. It is known that the mean time taken for oil and lube service at this garage is 15 minutes per car with a standard deviation of 2.4 minutes. The management wants to promote the business guaranteeing a maximum waiting time for customers. If a customer's car is not served with that period, the customers will receive a 50% discount on the charge. The company wants to limit this discount to at most 5% of the customers. What should the maximum guaranteed waiting time be? Assume that the times taken for oil and lube service for all cars have a normal distribution.
Question 11: In a study of the relation between the experience (in years) (x) and the number of computers sold (y) during the last three months, we draw a sample of seven sales persons and obtain the following results:
The correlation coefficient ( r ) = 0.89
Find:
a) Find the suitable least squares regression line.
b) Give a brief interpretation of the values of the Y-intercept and slope calculated in the previous part.
c) Calculate the determinant coefficient and interpret its value.
d) Predict the number of computers sold during the past three months by salespersons with twenty years experience.