Question

In a study conducted to investigate browsing activity by shoppers, each shopper was initially classified as a nonbrowser, light browser, or heavy browser. For each shopper, the study obtained a measure to determine how comfortable the shopper was in a store. Higher scores indicated greater comfort. Suppose the following data were collected.

Nonbrowser	Light Browser	Heavy Browser
4	5	5
5	6	7
6	5	5
3	4	7
3	7	4
4	4	6
5	6	5
4	5	7

(a)Use α = 0.05 to test for differences among comfort levels for the three types of browsers.

State the null and alternative hypotheses.

H₀: Not all the population means are equal.
H_a: μ_NB = μ_LB = μ_HB

H₀: μ_NB ≠ μ_LB ≠ μ_HB
H_a: μ_NB = μ_LB = μ_HB    

H₀: μ_NB = μ_LB = μ_HB
H_a: Not all the population means are equal.

H₀: μ_NB = μ_LB = μ_HB
H_a: μ_NB ≠ μ_LB ≠ μ_HB

H₀: At least two of the population means are equal.
H_a: At least two of the population means are different.

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 =

State your conclusion.

Reject H₀. There is sufficient evidence to conclude that the mean comfort scores are not all the same for the three groups.

Do not reject H₀. There is sufficient evidence to conclude that the mean comfort scores are not all the same for the three groups.    

Reject H₀. There is not sufficient evidence to conclude that the mean comfort scores are not all the same for the three groups.

Do not reject H₀. There is not sufficient evidence to conclude that the mean comfort scores are not all the same for the three groups.

(b)Use Fisher's LSD procedure to compare the comfort levels of nonbrowsers and light browsers. Use α = 0.05.

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

LSD =

Find the absolute difference between sample means for the comfort levels of nonbrowsers and light browsers.

x_NB − x_LB

=

What is your conclusion?

There is a significant difference between the population means for the comfort levels of nonbrowsers and light browsers.

There is not a significant difference between the population means for the comfort levels of nonbrowsers and light browsers.

Answer 1

	Original observations				Squared observations
Nonbrowser	Light Browser	Heavy Browser		Nonbrowser	Light Browser	Heavy Browser
4	5	5		16	25	25
5	6	7		25	36	49
6	5	5		36	25	25
3	4	7		9	16	49
3	7	4		9	49	16
4	4	6		16	16	36
5	6	5		25	36	25
4	5	7		16	25	49
34	42	46	122	152	228	274	654

We have n= 24

            t=3

            r=8

Correction factor=

Correction factor=

Correction factor=620.1667

Total sum of squares= sum of squares of every observation - correction factor

Total sum of squares= 654 – 620.1667

Total sum of squares= 33.83

Treatment sum of squares=

Treatment sum of squares=

Treatment sum of squares=

Treatment sum of squares=

Error sum of squares= Total SS – Treatment SS

Error sum of squares= 33.83 – 9.33

Error sum of squares= 24.5

ANOVA Table

Source of Variation	df	SS	MS	F	p value
Groups	3-1=2	9.33	4.667	4.00	0.034
Error	21	24.5	1.167
Total	24-1=23	33.83

Answer(a):

We have to test

H₀: μ_NB = μ_LB = μ_HB
H_a: Not all the population means are equal.

The correct option is C.

The test statistic to test the above hypothesis is F statistic in the above ANOVA table.

The value of test statistic, F = 4.00

The p-value = 0.034

Conclusion:

The above p-value is less that α=0.05 and it suggest that we have sufficient evidence against null hypothesis to reject it, so we reject the null hypothesis and conclude that the mean comfort scores are not all the same for the three groups.

Hence the correct option is A.

Answer(b):

The Fisher’s LSD can be obtained as below:

We have α=0.05, so the table value of t at 0.025 with 21df is 2.079614

Also we have, MSE= 1.167 and r = 8

So,

Sample mean for nonbrowsers,

Sample mean for Light browsers,

Absolute difference in means,

We have the absolute difference of 1 which is less than the calculated Fisher’s LSD and it indicates that there is no significant difference between the population means for the comfort levels of nonbrowsers and light browsers.

Hence the correct option is B.