Module 1, v1
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Name
Sep 30, 2022
Math 37, Fall 2022 - Module 2 Assessment, Sec 02
Problem
1
2
3
4
Total
Total Point Value Student’s Score
5
5
5
5
20
(1) According to our opening week survey, Math 37 stress scores were approximately normal
with a mean of about 18 and a standard deviation of about 6. Based on these results,
approximately what percentage of Math 37 students have moderate stress scores between
14 and 26?
Solution: The z-scores for 26 and 14 are (26 − 18)/6 = 1.33 and (14 − 18)/6 ≈ −0.67,
respectively, so the desired probability is
Φ(1.33) − Φ(−0.67) = 0.9082 − (1 − 0.7486) = 0.9082 − 0.2514 ≈ 0.66.
So 66% have moderate stress levels.
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(2) Suppose that the number of goals scored per game in NCAA women’s water polo games
is approximately normal with a mean of 19 and a standard deviation of 7 goals per game.
What would be the cutoffs for the middle 70% of the distribution of goals scored per
game?
Solution: The cutoffs for the middle 70% of all z-scores would be Φ−1 (0.85) = 1.04 and
−1.04 so the cutoffs for goals scored would be the solutions to −1.04 = (x1 − 19)/7 and
1.04 = (x2 − 19)/7 or about 12-26.
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(3) People of color (POC) make up 41% of the US population according to the 2020 census.
Suppose that we select a random sample of 200 people from the US.
(a) What is the standard error of the sample proportion of POC?
(b) What is the probability that the sample proportion of POC will be less than 0.3?
Solution: The SE is
p
0.41(1 − 0.41)/200 ≈ 0.035.
Then z = (0.3 − 0.41)/0.035 ≈ −3.14 and the probability is 1 − Φ(3.14) = 1 − 0.9991 =
0.0009.
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(4) You are playing a game where the expected value and standard deviation of your winnings per round are $5 and $3, respectively. Suppose you play 36 rounds of this game
and assume that the results are independent. Find the 99th percentile of the sample
mean amount that you will win per round AND the 99th percentile of the total amount
you will win.
Solution: The 99th percentile of z− scores is 2.33 so the 99th percentile of sample
means is the solution to
√
2.33 = (x − 5)/(3/ 36)
or x ≈ 6.165. Converting this back into a total, tells us the 99th percentile for amount
won is 6.165 ∗ 36 ≈ 221.94.