Compute a 95% t–confidence interval

• To practice for the exam use the t and z-tables supplied at the end of this file. Be sure to learn to use these tables. Note the t and z-tables give left tail probabilities and the χ2-table gives right tail critical values. 1. Basketball. Suppose that against a certain opponent the number of points the MIT basketaball team scores is normally distributed with unknown mean θ and unknown variance, σ2. Suppose that over the course of the last 10 games between the two teams MIT scored the following points: 59, 62, 59, 74, 70, 61, 62, 66, 62, 75 (a) Compute a 95% t–confidence interval for θ. Does 95% confidence mean that the probability θ is in the interval you just found is 95%? (b) Now suppose that you learn that σ2 = 25. Compute a 95% z–confidence interval for θ. How does this compare to the interval in (a)? (c) Let X be the number of points scored in a game. Suppose that your friend is a confirmed Bayesian with a priori belief θ ∼ N(60, 16) and that X ∼ N(θ, 25). He computes a 95% probability interval for θ, given the data in part (a). How does this interval compare to the intervals in (a) and (b)? (d) Which of the three intervals constructed above do you prefer? Why?

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