Question

A study on “Do left-handed people live shorter lives than right-handed people?” examined a sample of 20 death records and contacted to the next of kin to determine handedness. Note that there are many possible definitions of ”left-handed.” The researchers examined the effects of different definitions on the results of their analysis and found that their conclusions were not sensitive to the exact definition used. For the results presented here, people were defined to be right-handed if they wrote, drew, and threw a ball with the right hand. All others were defined to be left-handed. People were classified by gender (female or male), and a 2 × 2 ANOVA was run with the age at death as the response variable. The F statistics were 2.36 (handedness), 37.44 (gender), and 2.10 (interaction). The following marginal mean ages at death (in years) were reported: 77.39 (Right-handed females), 71.32 (Right-handed males), 75.00 (left-handed females), and 66.03 (left-handed males).

a. For each of the F statistics given above find the degrees of freedom and an approximate p-value.

b. Summarize and conclude the results of these tests.

Answer 1

a)

Answer:

Source of variance	df	F	p-value
handedness	1	2.36	0.1286
gender	1	3.44	0.0675
interaction	1	2.1	0.1514
within	76	.	.

Explanation:

The F statistic

$F_{\text{handedness}}=2.36$

$F_{\text{gender}}=37.44$

Finteraction 2.10

The degree of freedom

$df_{\text{handedness}}=a-1=2-1=1$

$df_{\text{gender}}=b-1=2-1=1$

$df_{\text{interaction}}=(a-1)(b-1)=1$

$df_{\text{within}}=ab(r-1)=2\times 2\times (20-1)=76$

where

a = number levels for variable handedness = 2

b = number levels for variable gender = 2

r = total number of rows (total number of data points) = 20

The p-value

For handedness

The p-value for F = 2.36 is obtained from the F distribution table for numerator degree of freedom = 1 and the denominator degree of freedom = 76

P-value (handedness)= 0.1286

For gender

The p-value for F = 3.44 is obtained from the F distribution table for numerator degree of freedom = 1 and the denominator degree of freedom = 76

P-value (gender)= 0.0675

For interaction

The p-value for F = 2.1 is obtained from the F distribution table for numerator degree of freedom = 1 and the denominator degree of freedom = 76

P-value (interaction)= 0.1514

b)

Answer:

handedness: There is not sufficient evidence to conclude that handedness has a significant main effect on age of death.

gender: There is not sufficient evidence to conclude that gender has a significant main effect on age of death.

interaction: There is not sufficient evidence to conclude that the interaction of handedness and gender has a significant main effect on age of death.

Explanation:

Hypothesis

The Null and Alternative Hypothesis are,

H₀₁: There is no difference in the means of factor handedness

H_A1: There is a significant difference in the means of factor handedness

H₀₂: There is no difference in means of factor gender

H_A2: There is a significant difference in means of factor gender

H₀₃: There is no interaction between factors handedness and gender

H_A3: There is a significant interaction between factors handedness and gender

Decision Rule:

Let the significance level for the hypothesis test = 0.05

The decision rule for the F tests states that reject the null hypothesis if the p-value is less than the significance level = 0.05

Decision

Source of variance	p-value		Significance level	Whether significant?
handedness	0.1286	>	0.05	No
gender	0.0675	>	0.05	No
interaction	0.1514	>	0.05	No

Conclusion:

handedness: There is not sufficient evidence to conclude that handedness has a significant main effect on age of death.

gender: There is not sufficient evidence to conclude that gender has a significant main effect on age of death.

interaction: There is not sufficient evidence to conclude that the interaction of handedness and gender has a significant main effect on age of death.