Question

Test of Independence - Introductory Statistics - OpenStax Frequencies of Hair Colors for Various Body Types Blonde Brunette Red Head Short and Slender 86 57 Short and Pudgy 1 61 106 74 Tall and Slender 158 9978 Tall and Heavy 108 79 49 What can be concluded at the a-0.05 significance level? a. What is the correct statistical test to use? Paired t-test Goodness-of-Fit Independence Homogeneity b. What are the null and alternative hypotheses? HO: Hair color and body type are independent. The distribution of hair color is the same for each body type. The distribution of hair color is not the same for each body type. Hair color and body type are dependent. Hair color and body type are independent. The distribution of hair color is the same for each body type. Hair color and body type are dependent. The distribution of hair color is not the same for each body type. c. The test statistic for this data- (Please show your answer to 2 decimal places.) d. The p-value for this sample (Please show your answer to 4 decimal places.) e. The p-value is Select an answer a 1. Based on this, we should reject the null accept the null fail to reject the null 8. Thus, the final conclusion is... There is sufficient evidence to conclude that hair color and body type are dependent There is insufficient evidence to conclude that the distribution of hair color is not the same for each body type There is sufficient evidence to conclude that the distribution of hair color is not the same for each body type

Answer 1

(a)

Correct option:

Independence

(b)

Correct option:

Hair color and body type are independent

(c)

Correct option:

Hair color and body type are dependent

(d)

The Test Statistic ($\chi ^{2}$) for this data is got as follows:

	Blonde	Brunette	Red Head	Total
Short and slender	86 (61.81) [9.46]	86 (97.10)[1.27]	57 (70.09)[2.44]	229
Short and pudgy	61 (65.05) [0.25]	106 (102.19) [0.14]	74 (73.76) [0.00]	241
Tall and slender	58 (63.43) [0.47]	99 (99.64) [0.00]	78 (71.92) [0.51]	235
Tall and heavy	49 (63.70) [3.39]	108 (100.07) [0.63]	79 (72.23) [0.63]	236
Total	254	399	288	941

Adding all the $\chi ^{2}$ values given in square bracket, we get:

The Test Statistic ($\chi ^{2}$) for this data = 19.21

Thus,

The Test Statistic for this data 19.21

(e)

Degrees of Freedom = (r - 1) X (c - 1)

                                 = (4 - 1) X (3 - 1) = 6

By Technology, p - value = 0.0038

So,

The p - value is less than $\alpha$ .

(f)

Correct option:

Reject the null

(g)

Correct option:

There is sufficient evidence to conclude that the hair color and body type are dependent.