Chapter 12.3, Problem 30E

Reconsider the situation described in Exercise 7, in which x = accelerated strength of concrete and y = 28-day cured strength. Suppose the simple linear regression model is valid for x between 1000 and 4000 and that β₁ = 1.25 and σ = 350. Consider an experiment in which n = 7, and the x values at which observations are made are x₁ = 1000, x₂ = 1500, x₃ = 2000, x₄ = 2500, x₅ = 3000, x₆ = 3500, and x₇ = 4000.

1. a. Calculate ${\sigma }_{{\hat{\beta }}_1}$, the standard deviation of ${\hat{\beta }}_1$.
2. b. What is the probability that the estimated slope based on such observations will be between 1.00 and 1.50?
3. c. Suppose it is also possible to make a single observation at each of the n =11 values x₁ = 2000, x₂ = 2100,…, x₁₁ = 3000. If a major objective is to estimate β₁, as accurately as possible, would the experiment with n = 11 be preferable to the one with n = 7?