4. Below is a scatterplot and regression print-out recording the ages of husband

Question

4. Below is a scatterplot and regression print-out recording the ages of husband/wife pairs for country X (an unknown place on Earth) 60 40 Intencept 0.9112 HE 20- 20 40 60 Husband's age (in years) a. Before looking at the data, a friend of yours makes the claim that husbands in country X tend to be twice as old as their wives. Naturally, you disagree. Frame these claims in terms of hypotheses about the linear regression model. b. Make a decision about your hypotheses from part a using the printout. c. In the table, what value would your friend need to change 35.25 to if he wanted the number 0.0000 to change to 0.06 (and hence, win the argument)? d. Suppose a random husband from country X is known to be 38 years old. What does the model predict for the wife's age?

Explanation / Answer

Solution

Let x = husband’s age and y = wife’s age. Then, then linear regression model is:

y = ?0 + ?1x + ?, ………………………………………..(1)

where ? is the error term, which is assumed to be Normally distributed with mean 0 and variance ?2.

Part (a)

If the claim is valid, then ?1cap must be ½. So, testing the claim would be:

To test H0: ?1 = ½ Vs H0: ?1 ? ½ ANSWER

Part (b)

Test Statistic is: t = (?1cap – ½)/SE(?1cap)

= (0.9112 – 0.5)/0.0259 [for 0.9112 – 2.0)/0.0259 refer to the analysis print-out given as part of the question]

= 15.8764

Under H0, t ~ tn – 2 = t168 [for 168, refer to df in the analysis print-out given]

So, p-value = P(| t168 | > 15.8764) = 1.45E-24[using Excel Function of t-distribution]

Since this p-value is very very small, H0 is rejected. Thus, we conclude that

there is not sufficient evidence to suggest the claim is valid. ANSWER

Part (c)

We want the value of t such that P(| t168 | > t) = 0.06.

Using Excel Function of t-distribution inverse, t = 1.8936 ANSWER

Part (d)

Referring to the analysis print-out given as part of the question, the estimated regression equation is: Estimated wife age (ycap) = 1.5740 + 0.9112.Husband age (x).

Substituting x = 38, Estimated wife age = 36.2 ANSWER

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