Question

5.30 Husbands and wives, Part II. Exercise 5.6 presents a scatterplot displaying the rela- tionship between husbands’ and wives’ ages in a random sample of 170 married couples in Britain, where both partners’ ages are below 65 years. Given below is summary output of the least squares fit for predicting wife’s age from husband’s age.

60

40

20
20 40 60

Husband's age (in years)

(Intercept) age husband

Estimate 1.5740 0.9112

Std. Error 1.1501 0.0259

t value 1.37 35.25

Pr(>|t|) 0.1730 0.0000df = 168

1. (a) We might wonder, is the age difference between husbands and wives consistent across ages? If this were the case, then the slope parameter would be β1 = 1. Use the information above to evaluate if there is strong evidence that the difference in husband and wife ages differs for different ages.

2. (b) Write the equation of the regression line for predicting wife’s age from husband’s age.

3. (c) Interpret the slope and intercept in context.

4. (d) Given that R2 = 0.88, what is the correlation of ages in this data set?

5. (e) You meet a married man from Britain who is 55 years old. What would you predict his wife’s

   age to be? How reliable is this prediction?

6. (f) You meet another married man from Britain who is 85 years old. Would it be wise to use the same linear model to predict his wife’s age? Explain.

Answer 1

a)

Hypothesis:

The null and alternative hypotheses are defined as,

This is a two-tailed test.

Let the significance level = 0.05

Test statistic

The t statistic is used to test the hypothesis,

From the regression output summary,

p-value

The p-value is obtained from the t distribution table for the degree of freedom = n-2=168 and for the two-tailed test,

Conclusion:

Since the p-value = 0.0008 is less than 0.05 at a 5% significance level, the null hypothesis is rejected hence there is sufficient evidence to conclude that the slope parameter is significantly different from 1 which means the age difference between husbands and wives are not consistent across age.

b)

The regression is defined as,

where Y = age of wives, X = age pf husbands

From the regression output summary,

c)

Slope = 0.9112

Interpretation: For one more 1-year increase in the age of husbands, the age of wives increases by 0.9112 years.

Intercept: = 1.5740

Interpretation: If the age of the husband is zero years, the expected age of the wife will be 1.5740 years.

d)

e)

For X = 55 years,

From the regression equation,

The wife's age = 51.69 years

From the regression output summary,

The R square value tells how well the regression model fits the data values. the R square value is 0.88 which means the regression model explains the 88% of the variance of the data values and the age of husband significantly fit the regression model, (P-value =0.0000) hence we can say that model is reliable for the prediction.  

f)

The age = 85 years is outside the domain of the data values hence we can not use this regression model for the prediction.