Question

1. Basketball. Suppose that against a certain opponent the number of points the MIT basketaball team scores is normally distributed with unknown mean $ heta$ and unknown vari- ance, $sigma^2$. 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 $ heta$. Does 95% confidence mean that the probability $ heta$ is in the interval you just found is 95%? (b) Now suppose that you learn that $sigma^2 = 25$. Compute a 95% $z$-confidence interval for $ heta$. 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 $ heta sim N(60, 16)$ and that $X sim N( heta, 25)$. He computes a 95% probability interval for $ heta$, 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?

          1. Basketball. Suppose that against a certain opponent the number of points the MIT
basketaball team scores is normally distributed with unknown mean $	heta$ and unknown vari-
ance, $sigma^2$. 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 $	heta$. Does 95% confidence mean that the
probability $	heta$ is in the interval you just found is 95%? 
(b) Now suppose that you learn that $sigma^2 = 25$. Compute a 95% $z$-confidence interval for
$	heta$. 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 $	heta sim N(60, 16)$ and that $X sim N(	heta, 25)$. He computes
a 95% probability interval for $	heta$, 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?

1. Basketball. Suppose that against a certain opponent the number of points the MIT
basketaball team scores is normally distributed with unknown mean heta and unknown vari-
ance, sigma^2. 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 $ heta$. Does 95% confidence mean that the
probability heta is in the interval you just found is 95%?
(b) Now suppose that you learn that sigma^2 = 25. Compute a 95% $z$-confidence interval for
heta. 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 heta sim N(60, 16) and that X sim N( heta, 25). He computes
a 95% probability interval for $ heta$, 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?

Added by Emily B.

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