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

Question

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

Answer 1

We do not do your work for you. Once you have attempted to answer your questions, we will be happy to give you feedback on your work. Although it might require more time and effort, you will learn more if you do your own work. Isn't that why you go to school?