1. Test the Nation on CBC:“Test the Nation” was the biggest

survey ever conducted to see just how smart Canadians are. In this live two-hour special,
1.5 million Canadians participated in a real-time interactive IQ test. Viewers took the test
in the comfort of their own homes, on the internet or with pen and paper, while seven
teams – tattoo artists, millionaires, fitness instructors, surgeons, mayors, talk jocks and
celebrities – exercised their grey matter in CBC’s Toronto studio. Suppose that 25% of
Canadians scored above average (that is, scoring 110 or more). You took a random sample
of n=81 individuals and recorded their IQ scores. Let bp denote the sample proportion
of individuals who scored above average.
a. What is the probability that bp lies between 19% and 22%?
b. Ninety percent of the time, the sample proportion bp is smaller than what number?
c. Would a value of bp = 0:35 be considered unusual? Justify your answer.

2. Let X be a normally distributed random variable with mean m and standard deviation o:
a. Show that
P(m - 1:96o < X < m + 1:96o) = 0:95:
Hint: standardize!
b. Find a value B � (Greek letter “beta”) such that
P(m - 2:5o < X < m + Bo) = 0:95:
Hint: again, standardize!
b. The above problem states that X lies in the interval (m - 1:96; m + 1:96o) with the
same probability as it lies in the interval (m - 2:5; m + �Bo): Find the lengths of both
of these intervals, using the numeric value of � which you obtained above. Which
interval is narrower?
c. Can you find an interval (a; b) such that
P(a < X < b) = 0:95
but which is narrower than the interval (m - 1:96; m + 1:96o)? Justify your answer
heuristically

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