Question

Given are five observations for two variables, x and y.

xᵢ | 3 7 10 16 20
yᵢ | 7 16 9 26 21

develop the 90% confidence and prediction intervals when x=14 .
s = _____

Answer 1

To find the 90% confidence interval and prediction interval when x=14, you need to calculate the standard deviation (s). The confidence interval is (8.8, 22.8) and the prediction interval is (1.1, 30.5).

Step-by-step explanation:

To find the 90% confidence interval and prediction interval when x=14, we need to calculate the standard deviation, denoted as s. The given data for x and y are:

x: 3, 7, 10, 16, 20

y: 7, 16, 9, 26, 21

First, we calculate the mean of x and y:

mean(x) = (3+7+10+16+20)/5 = 11.2

mean(y) = (7+16+9+26+21)/5 = 15.8

Next, we calculate the sample standard deviation s:

s = sqrt(((3-11.2)^2 + (7-11.2)^2 + (10-11.2)^2 + (16-11.2)^2 + (20-11.2)^2)/4) = sqrt(62.4) ≈ 7.9

To find the 90% confidence interval, we use the formula:

Confidence interval = mean(y) ± (critical value) * (s / sqrt(n))

For a 90% confidence interval, the critical value is approximately 1.645. Substituting the values:

Confidence interval = 15.8 ± (1.645) * (7.9 / sqrt(5))

Confidence interval = 15.8 ± (1.645) * (7.9 / sqrt(5))

Confidence interval = (8.8, 22.8)

To find the prediction interval, we use the formula:

Prediction interval = mean(y) ± (critical value) * (s * sqrt(1 + 1/n))

For a 90% prediction interval, the critical value is approximately 1.645. Substituting the values:

Prediction interval = 15.8 ± (1.645) * (7.9 * sqrt(1 + 1/5))

Prediction interval = 15.8 ± (1.645) * (7.9 * sqrt(1.2))

Prediction interval = (1.1, 30.5)