https://file.aiquickdraw.com/mathbot/file/17269476249969dags2n3.jpg

The image presents a dataset of paired values \(x_i\) and \(y_i\), along with a series of statistical problems:

1. \(x_i = \{3, 12, 6, 20, 14\}\)
2. \(y_i = \{55, 40, 55, 10, 15\}\)

The tasks include:

a. Estimating the standard deviation of \( \hat{y}^* \) when \( x = 8 \) (to 4 decimal places).

b. Developing a 95% confidence interval for the expected value of \( y \) when \( x = 8 \) (to 2 decimals).

c. Estimating the standard deviation of an individual value of \( y \) when \( x = 8 \) (to 4 decimal places).

d. Developing a 95% prediction interval for \( y \) when \( x = 8 \) (to 2 decimals).

The answers provided in the image were:
- Part a: 9.79 (incorrect)
- Part b: (12.97, 75.03) (incorrect)
- Part c: 13.115 (incorrect)
- Part d: (7.61, 80.39) (incorrect)

Would you like me to guide you through the calculations for each part and help you find the correct values?

The data set includes paired values (x_i, y_i): x_i = {3, 12, 6, 20, 14} and y_i = {55, 40, 55, 10, 15}. Estimate the standard deviation of y-hat when x = 8, develop a 95% confidence interval for the expected value of y when x = 8, estimate the standard deviation of an individual value of y when x = 8, and develop a 95% prediction interval for y when x = 8.

Explore linear regression analysis with this example, which estimates the standard deviation, confidence interval, and prediction interval for y when x = 8 using a dataset. Understand key concepts such as standard deviation, confidence intervals, and prediction intervals for accurate statistical inference.

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