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DNSC 2001 – Business Analytics II – Fall 2020

Assignment 2 Solutions

Question 1

Does a horse's starting position affect the probability that it will win a race on a circular track? The following table lists the numbers of wins from each starting position (1 is closest to the inside rail) in 144

8-horse races:

Starting Position	1	2	3	4	5	6	7	8
Number of Wins	29	19	18	25	17	10	15	11

Assuming that starting positions were randomly assigned, use these data to test the null hypothesis that a horse's starting position does not affect its chance of winning.

We perform a chi-square goodness of fit test. The chi-square statistic is 16.33 with p-value 0.022. Because p < 0.05, we reject the null hypothesis (null: starting position doesn’t matter) at the conventional significance level of α = 0.05 and we conclude that the starting position matters.

Question 2

Consider the accompanying contingency table displaying the sample proportions that fell in the various combinations of categories (e.g., 13% of those in the sample were in the first category of both factors).

1 2 3

.13	.19	.28
.07	.11	.22

a. Suppose the sample consisted of n = 100 people. Use the chi-squared test for independence with significance level .10.

Table of observed counts ni :

1 2 3

Row Total

Column Total

20 30 50

100

Table of estimated expected counts Ei :

1 2 3

Row Total

Column Total

20 30 50

100

2 (13 −12)2 (22 − 20)2

TS: X = + + = .6806.

12 20

RR: Reject Ho if X2 > X.10, 2 = 4.605.

Because X2 = .6806 is not larger than 4605 then, we fail to reject the null hypothesis of independence.

b. Repeat the above assuming that the sample size was n = 1000.

Each observation count here is 10 times what is was in Question a, and the same is true of the estimated expected counts so now X2 = 6.806 > 4.605, and Ho is rejected. With the much larger sample size, the departure from what is expected under Ho , the independence hypothesis, is statistically significant – it cannot be explained just by random variation.

c. What is the smallest sample size n for which these observed proportions would result in rejection of the independence hypothesis?

The observed counts are .13n, .19n, .28n, .07n, .11n, .22n, whereas the estimated expected

(.60n)(.20n)

hypotheses will be rejected at level . 10 iff .006806n > 4.605, i.e., iff n > 676.6, so the minimum n = 677.

Question 3

The National Management Association reports that during the past year, there has been a substantial increase in the use of "flextime" in the work place. Last year, a sample of 100 businesses was taken which indicated that 22% had implemented the use of "flextime." This year, a second survey of 100 showed that 29% were using flextime. At the 0.05 level, does this represent an increase in the proportion?

a. State the null and alternative hypotheses.

b. What is the value of the test statistic?

c. Specify the rejection region?

d. State the conclusion.

a. State the null and alternative hypotheses.

H0: π1-π2 =0, Ha: π1-π2<0

b. What is the value of the test statistic?

TS: z =-1.139

c. Specify the rejection region?

RR: Reject H0 ifz<-1.645.

d. State the conclusion.

Since=-1.139 is not smaller than-1.645, we fail to reject H..There has not been a significant increase in the proportion of flextime.

Question 4

A comparison of the price-earnings (P/E) ratio for the top and bottom 100 companies in valuation is being prepared. A financial advisor randomly sampled each group to determine whether there is any difference in P/E ratios of the two groups of companies. Let 1 = a top 100 company and 2 = a bottom 100 company. Assume equal population variances and that the populations are normally distributed. The advisor is to use a 0.01 significance level. The data were randomly selected and are summarized below:

Top 100 company

Bottom 100 company

n = 6

y1 = 18.83

s = 128.17

n = 6

y2 = 10.67

s = 8.67

a. State the null and alternative hypotheses.

b. Calculate the standard error of the estimate.

c. What is the value of the test statistic? What is the rejection region?

d. What is the conclusion?

a. State the null and alternative hypotheses.

H,:A-μ=0, H,:μ-μ =0

b. Calculate the standard error of the estimate.

4.776

c. What is the value of the test statistic? What is the rejection region?

TS:1=1.71; RR: Reject H, ift<-3.169 or ift>3.169

d. What is the conclusion?

Since r=1.71 is not larger than 3.169, we fail to reject H, and conclude that there is no significant difference in the average P/E ratio of the two groups

Question 5

A pharmaceutical manufacturer has been researching new formulas to provide quicker relief of minor pains. His laboratories have produced three different formulas, which he wanted to test. Fifteen people who complained of minor pains were recruited for an experiment. Five were given formula 1, five were given formula 2, and the last five were given formula 3. Each was asked to take the medicine and report the length of time until some relief was felt. The results are shown below. Do these data provide sufficient evidence to indicate that differences in the time of relief exist among the three formulas? Use = .05.

Time in Minutes Until Relief is Felt

Formula 1

Formula 2

Formula 3

Source of Variation

SS df MS F

Between

Within

36.4

42.0

18.2

3.5

5.2

Total

78.4

H0 : 1 = 2 = 3

Ha : The population means are not equal

Test statistic: F = 5.2; Critical value = F05, 2, 12 = 3.89. Since F = 5.2 > 3.885, we reject H0 , and conclude that the population means are not equal.

Question 6

A large annuity company holds many industry group stocks. Among the industries are banks, business services and construction. Seven companies from each industry group are randomly sampled to test the hypothesis that the mean price per share is the same among industries. The data are:

Banks

Business Services

Construction

n = 7