Question

Suppose that ??, ??, … … ??, ??+? are all random variables from a ?or(?, ?^?) distribution. Additionally, suppose that we are able to observe (or measure) realizations of ??, ??, … … ??. Our goal is to use these ? observations to predict ??+?. An equivalent problem statement is that we want to derive a (?− ?a)% confidence interval for the random variable ??+? based on observed values of the first ? realizations of the normal random variable ??. Such an interval is known as the (?-?a)% prediction interval.

(a) Derive this prediction interval. (Hint: Your prediction interval should be based on the sample mean ?? and sample variance, ??^? that you have access to based on the first ? observations. Also, remember that a normal variable divided by a sample standard deviation typically follows a ?(?-?) distribution as we have seen before while deriving confidence intervals).

(b) The number of steps one of my colleagues walked in the last seven days was tracked using a cell phone app to be ?822, ?333, ?420, ?432, ?252, ?005, and ?752. Assuming that conditions remain the same, provide a prediction interval with ?5% confidence that will contain the number of steps walked by this person on the eighth day.

Answer 1