Part 3 of 5 Points: 0 of 1 Save Is there a relation between the age difference between husband/wives and the percent of a country that is literate? Researchers found the least-squares regression between age difference (husband age minus wife age), $y$, and literacy rate (percent of the population that is literate), $x$, is $\hat{y}=-0.0592 x+8.5$. The model applied for $24 \leq x \leq 100$. Complete parts (a) through (e) below. (b) Does it make sense to interpret the y-intercept? Explain. Choose the correct answer below. A. Yes-it makes sense to interpret the $y$-intercept because an $x$-value of 0 is within the realm of possibilities. B. No-it does not make sense to interpret the $y$-intercept because a y-value of 0 is outside the scope of the model. C. No-it does not make sense to interpret the $y$-intercept because an $x$-value of 0 is outside the scope of the model. D. No-it does not make sense to interpret the $y$-intercept because a $y$-value of 0 is impossible. E. No-it does not make sense to interpret the $y$-intercept because an $x$-value of 0 is impossible. (c) Predict the age difference between husband/wife in a country where the literacy rate is 42 percent. years (Round to one decimal place as needed.)
Solution
Step 1 ：The y-intercept is the value of y when x is 0. In this case, x represents the literacy rate. So, the y-intercept would represent the age difference between husband and wife when the literacy rate is 0. Since the model is only valid for literacy rates between 24 and 100, it does not make sense to interpret the y-intercept because a literacy rate of 0 is outside the scope of the model. So, the correct answer is C. No-it does not make sense to interpret the $y$-intercept because an $x$-value of 0 is outside the scope of the model.
Step 2 ：To predict the age difference between husband and wife in a country where the literacy rate is 42 percent, we need to substitute x = 42 into the regression equation to find the corresponding y value, which represents the predicted age difference. The calculation is as follows: \(\hat{y}=-0.0592 \times 42+8.5\), which gives \(\hat{y}=6.0136\). So, the predicted age difference between husband and wife in a country where the literacy rate is 42 percent is approximately \(\boxed{6.0}\) years.
