Wiki Page, Math 13 - Perez Sandbox : ANSWER(S): Module 17 - Distribution of Sample Proportions (6 of 6 discussion 3)

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2026-01-17T21:14:49-08:00

Here is an example of a correct response to the prompt from the previous Learn by Doing activity.

Conclusion:

Answer: The probability is about 0.28.

Interpretation: If it is true that 30% of smokers started smoking before the age of 16, then there is a 28% chance that the percentage from a random sample of 100 smokers is off by more than 5%. In other words, there is a 28% chance that the sample proportion is either less than 0.25 or greater than 0.35.

Here is the detailed work that lead to the correct response above.

Check normality conditions:

and

(The number of successes is greater than 10.)

(The number of failures is greater than 10.)

So the conditions are met. Therefore, we can use a normal model. A normal model allows us to use a Z-score and the simulator to find the probability.

Find the Z-score:

We want the probability that the sample proportion under- or over- estimates the population proportion by more than 5%. Therefore, we want the probability that the sample proportion is less than 25% or greater than 35%. (How did we get these numbers? 30% – 5% is 25% and 30% + 5% is 35%.)

We need to find the Z-scores for the sample proportions 0.25 and 0.35:

Without doing any more calculations, we can use the symmetry in the normal curve to determine that the Z-score for 0.25 will be negative 1.09.

Find the probability using the standard normal model:

We want the probability that Z is less than -1.09 or greater than 1.09, so we want the probability described by the two tails. The probability for one tail is 0.1379 (area to the right of Z=1.09), which is about 0.14. Because of the symmetry in the curve, the two tails are the same size, so we can just double 0.14 to find the answer: the probability for both tails is about 2(0.14) = 0.28.