Answer :
Commute time of 30 minutes is approximately 67.51 to 69.19.
(a) To predict the mean well-being index composite score of individuals with a commute time of 30 minutes, we can use the equation of the least-squares regression line. Plugging in x = 30 into the equation y = -0.0445x + 69.6846, we get:
y = -0.0445(30) + 69.6846
y = -1.335 + 69.6846
y ≈ 68.35
Therefore, the predicted mean well-being index composite score for individuals with a commute time of 30 minutes is approximately 68.35.
(b) To construct a 90% confidence interval for the mean well-being index composite score of individuals with a commute time of 30 minutes, we need to consider the standard error of the estimate. The standard error of the estimate, denoted as SE, is given as 0.4362.
A confidence interval can be calculated using the formula:
Lower Bound = predicted y - (critical value * SE)
Upper Bound = predicted y + (critical value * SE)
Since we want to construct a 90% confidence interval, we need to find the critical value associated with a 90% confidence level. This critical value can be obtained from statistical tables or software and is typically denoted as t*.
Assuming that our sample size is large enough and follows normal distribution properties, we can use a t-distribution for this calculation.
For a 90% confidence level, with n-2 degrees of freedom (where n is the number of data points), the critical value t* is approximately 1.833.
Plugging in the values into the formula, we get:
Lower Bound = 68.35 - (1.833 * 0.4362)
Upper Bound = 68.35 + (1.833 * 0.4362)
Lower Bound ≈ 67.51
Upper Bound ≈ 69.19
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