The 99% prediction interval for the test score of a person who spent 7 hours preparing for the test is (52.01, 87.02). This means that we are 99% confident that the actual test score will fall within this range.
Calculate the standard error of the prediction:
The standard error of the prediction (SEP) is different from the standard error of estimate (SEE) because it takes into account the uncertainty in the prediction for a single new data point, not just the average error across all data points.
The formula for SEP is:
SEP = SEE * sqrt(1 + (1/n))
where:
SEE is the standard error of estimate (given as 5.40 in this case)
n is the number of data points (5 in this case)
Plugging in the values, we get:
SEP = 5.40 * sqrt(1 + (1/5)) ≈ 6.10
Find the t-statistic:
We need to find the t-statistic for a 99% prediction interval with 5 degrees of freedom (n-1).
You can use a t-table or a calculator to find this value.
For a 99% prediction interval with 5 degrees of freedom, the t-statistic is approximately 2.921.
Calculate the prediction interval:
The prediction interval is calculated using the following formula:
Prediction interval = Predicted score ± t-statistic * SEP
where:
Predicted score is the score predicted by the regression equation for 7 hours of preparation (44.845 + 3.524 * 7 ≈ 69.51)
Plugging in the values, we get:
Prediction interval = 69.51 ± 2.921 * 6.10 ≈ (52.01, 87.02)
Therefore, the 99% prediction interval for the test score of a person who spent 7 hours preparing for the test is (52.01, 87.02).
This means that we are 99% confident that the actual test score will fall within this range.