ECO2121 Basic Statistics: Problem Set 5

3. In hypothesis testing, the hypothesis tentatively assumed to be true is

a. the alternative hypothesis

b. the null hypothesis

c. either the null or the alternative

d. None of the other answers are correct.

4. In hypothesis testing, the alternative hypothesis is

a. the hypothesis tentatively assumed true in the hypothesis-testing procedure b. the hypothesis concluded to be true if the null hypothesis is rejected

c. the maximum probability of a Type I error

d. All of the answers are correct.

5. The level of significance is the

a. maximum allowable probability of Type II error

b. maximum allowable probability of Type I error

c. same as the confidence coefficient

d. same as the p-value

6. The ANOVA procedure is a statistical approach for determining whether or not a. the means of two samples are equal

b. the means of two or more samples are equal

c. the means of more than two samples are equal

d. the means of two or more populations are equal

7. An ANOVA procedure is applied to data obtained from 6 samples where each sample contains 20 observations. The degrees of freedom for the critical value ofF are

a. 6 numerator and 20 denominator degrees of freedom

b. 5 numerator and 20 denominator degrees of freedom

c. 5 numerator and 114 denominator degrees of freedom

d. 6 numerator and 20 denominator degrees of freedom

13. As the goodness offit for the estimated regression equation increases,

a. the absolute value of the regression equation’s slope increases

b. the value of the regression equation’s y intercept decreases

c. the value of the coefficient of determination increases

d. the value of the correlation coefficient increases

14. Regression analysis was applied between sales (in $1000) and advertising (in $100) and the following regression function was obtained.

= 500 + 4x

Based on the above estimated regression line if advertising is $10,000, then the point estimate for sales (in dollars) is

a. $900

b. $900,000

c. $40,500

d. $505,000

15. Larger values of r2 imply that the observations are more closely grouped about the

a. average value of the independent variables

b. average value of the dependent variable

c. least squares line

d. origin

II. Fill in the Blank (10 points in total)

1. (7 pts, 1 pt each) In a completely randomized experimental design involving five treatments, thirteen observations were recorded for each of the five treatments. The following information is provided.

SSTR = 200 (Sum Square Between Treatments)

SST = 800 (Total Sum Square)

.    The sum of squares within treatments (SSE) is ,

.    The number of degrees of freedom corresponding to between treatments is ,

.    The number of degrees of freedom corresponding to within treatments is ,

.    The mean square between treatments (MSTR) is ,

.    The mean square within treatments (MSE) is ,

.    If at a 5% level of significance we want to determine whether or not the means of the five populations are equal, the critical value ofF is ,

.    The conclusion of the test is that the five means ,

2. (3 pts) The following is part of an ANOVA table, which was the results of three treatments and a total of 15 observations.

Source of Variation

Sum of

Squares

Degrees of Freedom

Mean

Square

F

Total                                            ?                ?

III. Free Response (75 pts in total)

Show detailed steps for the questions in this section.

1. (17 pts) A carpet company advertises that it will deliver your carpet within 15 days of purchase. A sample of 49 past customers is taken. The average delivery time in the sample was 16.2 days. The

population standard deviation is 5.6 days.

a.    (2 pts) State the null and alternative hypotheses.

b.    (5 pts) Using a critical value, test the null hypothesis at the 5% level of significance.

c.    (5 pts) Using ap-value, test the hypothesis at the 5% level of significance.

(5 pts) Compute the probability of a Type II error if the true average delivery time is 17 days d.

after purchase.

2. (13 pts)  Nancy believes that the average running time of movies is equal to 140 minutes. A sample of 4 movies was taken and the following running times were obtained. Assume the distribution of the

population is normally distributed.

150           150           180           170

a.     (2 pts) State the null and alternative hypotheses.

b.     (5 pts) Using a critical value, test the hypothesis at the 10% level of significance.

c.     (5 pts) Using ap-value, test the hypothesis at the 10% level of significance.

d.     (1 pt) Could a Type II error have been committed in this hypothesis test?

3. (8 pts)  A new soft drink is being market tested. A sample of 400 individuals participated in the taste test and 80 indicated they like the taste.

a. (5 pts) At a 5% significance level, test to determine if at least 22% of the population will like the new soft drink.

b. (3 pts) Determine the p-value.

4. (7 pts) In order to determine whether or not a driver's education course improves the scores on a

driving exam, a sample of 6 students were given the exam before and after taking the course. The results are shown below.

Let d = Score After - Score Before.

Score

Student                                        Before the Course

1                                                  83

2                                                    89

3                                                  93

4                                                  77

5                                                     86

6                                                  79

Score

After the Course

87

88

91

77

93

83

a. (2 pts) Compute the test statistic.

b. (5 pts) At 95% confidence using the p-value approach, test to see if taking the course actually increased scores on the driving exam.

5. (15 pts) A company has recorded data on the daily demand for its product (y in thousands of units) and  the unit price (x in hundreds of dollars). A sample of 15 days demand and associated prices resulted in the following data.

a.

b.

c.

d.

Using the above information, develop the least-squares estimated regression line and write the equation. (5 pts)

Compute the coefficient of determination. (2 pts)

Perform an F test and determine whether or not there is a significant relationship between demand and unit price. Let α = 0.05. (5 pts)

Would the demand ever reach zero? If yes, at what price would the demand be zero? (3 pts)

6. (15 pts)  Shown below is a portion of a computer output for regression analysis relating y (dependent variable) and x (independent variable).

ANOVA

a. b.

c. d.

e.

What has been the sample size for the above? (3 pts)

Perform a t-test and determine whether or not x andy are related. Use α = 0.05. (3 pts)    Perform an F-test and determine whether or not x andy are related. Use α = 0.05. (3 pts) Compute the coefficient of determination. (3 pts)

Interpret the meaning of the value of the coefficient of determination that you found in d. Be very specific. (3 pts)