Martin Hegmann
Feb 03, 2025
How would you solve this stat problem without the calculator?
Hi, my teacher only teaches us how to solve problems using a calculator, but I want to learn how to solve this one by hand. I used the normal cumulative distribution function (CDF) for this question: A simple random sample (SRS) of 400 American adults is asked, "What do you think is the most serious problem facing our schools?" Suppose that, in fact, 30% of all adults would respond "drugs" if asked this question. The sample proportion, denoted as p-hat, will vary in repeated sampling. We can assign probabilities to different values of p-hat using the normal density curve with a mean of 0.3 and a standard deviation of 0.023.
Using this density curve, I need to find the probabilities for the following events: a) At least half of the sample believes that drugs are the most serious problem facing schools. b) Less than 25% of the sample believes that drugs are the most serious problem. c) The sample proportion is between 0.25 and 0.35.
I obtained the following results using a calculator: a) normalcdf(0.5, 1, 0.3, 0.023) = 1.7^-18 b) normalcdf(0, 0.25, 0.3, 0.023) = 0.015 c) normalcdf(0.25, 0.35, 0.3, 0.023) = 0.97.
Could someone please show me how to solve this by hand? I read that I need to use the z-score and other concepts, but I would appreciate a detailed explanation. Thank you!
1 Answers
a) At least half of the sample believes that drugs are the schools' most serious problem. normalcdf(0.5,1,0.3,0.023)=1.7^-18
P(X ≥ 0.5) = P((X - 0.3)/0.023 ≥ (0.5 - 0.3)/0.023) = P(Z ≥ 8.69) = P(Z ≤ - 8.69) = 0
b) normalcdf(0,0.25,0.3,0.023)=0.015
P(X ≤ 0.25) = P((X - 0.3)/0.023 ≤ (0.25 - 0.3)/0.023) = P(Z ≤ - 2.17) = 0.015003
c)The sample proportion is between 0.25 and 0.35.
P(0.25 ≤ X ≤ 0.35) = P((0.25 - 0.3)/0.023 ≤ (X - 0.3)/0.023 ≤ (0.35 - 0.3)/0.023) =
P( - 2.17 ≤ Z ≤ 2.17) = P(Z ≤ 2.17) - P(Z ≤ - 2.17) = 0.984997 - 0.015003 = 0.969994
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