What is the relationship between diamond price and carat size? 307 diamonds were sampled and a straight-line relationship was hypothesized between y = diamond price (in dollars) and x = size of the diamond (in carats). The simple linear regression for the analysis is shown below:
Least Squares Linear Regression of PRICE
Predictor
Variables Coefficient Std Error T P
Constant -2298.36 158.531 -14.50 0.0000
Size 11598.9 230.111 50.41 0.0000
R-Squared 0.8925 Resid. Mean Square (MSE) 1248950
Adjusted R-Squared 0.8922 Standard Deviation 1117.56
The model was then used to create 95% confidence and prediction intervals for y and for E(Y) when the carat size of the diamond was 1 carat. The results are shown here:
95% confidence interval for E(Y): ($9091.60, $9509.40)
95% prediction interval for Y: ($7091.50, $11,510.00)
Which of the following interpretations is correct if you want to use the model to estimate E(Y) for all 1-carat diamonds?
◦ We are 95% confident that the average price of all 1-carat diamonds will fall between $7091.50 and $11,510.00.
◦ We are 95% confident that the price of a 1-carat diamond will fall between $9091.60 and $9509.40.
◦ We are 95% confident that the price of a 1-carat diamond will fall between $7091.50 and $11,510.00.
◦ We are 95% confident that the average price of all 1-carat diamonds will fall between $9091.60 and $9509.40.
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