# Question: The following data relate road width x and accident frequency

The following data relate road width x and accident frequency y. Road width (in feet) was treated as the independent variable, and values y of the random variable Y, in accidents per 108 vehicle miles, were observed.

Assume that Y is normally distributed with mean A + Bx and constant variance for all x and that the sample is random. Interpolate if necessary.

(a) Fit a least-squares line to the data, and forecast the accident frequency when the road width is 55 feet.

(b) Construct a 95 percent prediction interval for Y+, a future observation of Y, corresponding to x+ = 55 feet.

(c) Suppose that two future observations on Y, both corresponding to x+ = 55 feet, are to be made. Construct prediction intervals for both of these observations so that the probability is at least 95 percent that both future values of Y will fall into them simultaneously. If k predictions are to be made, such as given in part (d), each with probability 1 – α, then the probability is at least 1 – kα that all k future observations will fall into their respective intervals.

(d) Construct a simultaneous tolerance interval for the future value of Y corresponding to x+ = 55 feet with P = 0.90 and 1 – α = 0.95.

Assume that Y is normally distributed with mean A + Bx and constant variance for all x and that the sample is random. Interpolate if necessary.

(a) Fit a least-squares line to the data, and forecast the accident frequency when the road width is 55 feet.

(b) Construct a 95 percent prediction interval for Y+, a future observation of Y, corresponding to x+ = 55 feet.

(c) Suppose that two future observations on Y, both corresponding to x+ = 55 feet, are to be made. Construct prediction intervals for both of these observations so that the probability is at least 95 percent that both future values of Y will fall into them simultaneously. If k predictions are to be made, such as given in part (d), each with probability 1 – α, then the probability is at least 1 – kα that all k future observations will fall into their respective intervals.

(d) Construct a simultaneous tolerance interval for the future value of Y corresponding to x+ = 55 feet with P = 0.90 and 1 – α = 0.95.

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