Question

The American Association of Individual Investors (AAII) On-Line Discount Broker Survey polls members on their experiences with discount brokers. As part of the survey, members were asked to rate the quality of the speed of execution with their broker as well as provide an overall satisfaction rating for electronic trades. Possible responses (scores) were no opinion (0), unsatisfied (1), somewhat satisfied (2), satisfied (3), and very satisfied (4). For each broker summary scores were computed by calculating a weighted average of the scores provided by each respondent. Suppose a portion of the survey results follow.

Brokerage	Speed	Satisfaction
A	3.4	3.5
B	3.3	3.4
C	3.4	3.9
D	3.6	3.8
E	3.2	2.9
F	3.8	2.8
G	3.8	3.6
H	2.6	2.5
I	2.7	2.3
J	4.0	4.0
K	2.5	2.5

These ratings data on x = the quality of the speed of execution and y = overall satisfaction with electronic trades provided the estimated regression equation ŷ = 0.078 + 0.946x. At the 0.05 level of significance, test whether speed of execution and overall satisfaction are related. (Use the F test.)

State the null and alternative hypotheses.

H₀: β₁ ≠ 0
H_a: β₁ = 0H₀: β₀ ≠ 0
H_a: β₀ = 0    H₀: β₁ = 0
H_a: β₁ ≠ 0H₀: β₁ ≥ 0
H_a: β₁ < 0H₀: β₀ = 0
H_a: β₀ ≠ 0

Set up the ANOVA table. (Round your for F to two decimal places and all other values to three decimal places.)

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	F	p-value
Regression
Error
Total

Find the value of the test statistic. (Round your answer to two decimal places.)

Find the p-value. (Round your answer to three decimal places.)

p-value =

What is your conclusion?

Reject H₀. We cannot conclude that the relationship between speed of execution and overall satisfaction is significant.Reject H₀. We conclude that the relationship between speed of execution and overall satisfaction is significant.     Do not reject H₀. We cannot conclude that the relationship between speed of execution and overall satisfaction is significant.Do not reject H₀. We conclude that the relationship between speed of execution and overall satisfaction is significant.

Answer 1

The null and alternative hypotheses.

H₀: β₁ = 0
H_a: β₁ ≠ 0

ANOVA table.

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	F	p-value
Regression	2.328	1	2.328	14.02	0.0046
Error	1.492	9	0.166
Total	3.82	10

From the data,

SS Total = SSyy = 3.82

SS Regression = SS²xy / SSxx = 2.46² / 2.6 = 2.328

SS Error = SS Total - SS Regression = 3.82 - 2.328 = 1.492

DF Regression = Number of predictors = 1

DF Error = Number of Observations - Number of predictors - 1 = 11 - 1 - 1 = 9

DF Total = Number of Observations - 1 = 11 - 1 = 10

MS  Regression = SS Regression/ DF Regression = 2.328 / 1 = 2.328

MS Error = SS Error / DF Error = 1.492 / 9 = 0.166

F = MS Regression / MS Error = 2.328 /0.166 = 14.02

P-value = P(F > 14.02, df = 1, 9) = 0.0046

test statistic F = 14.02

p-value = 0.0046

Since p-value is less than 0.05 significance level,

.Reject H₀. We conclude that the relationship between speed of execution and overall satisfaction is significant.