The distribution of resting pulse rates of all students at Santa Maria high school was approximately normal with mean $\mu=80$ beats per minute and standard deviation $\sigma=9$ beats per minute.

The school nurse plans to provide additional screening to students whose resting pulse rates are in the top $30 \%$ of the students who were tested.

What is the minimum resting pulse rate at that school for students who will receive additional screening?
Choose 1 answer:
(A) 76 beats per minute
(B) 85 beats per minute
(C) 87 beats per minute
(D) 89 beats per minute
(ㄷ) 94 beats per minute

Question

The distribution of resting pulse rates of all students at Santa Maria high school was approximately normal with mean $\mu=80$ beats per minute and standard deviation $\sigma=9$ beats per minute.

The school nurse plans to provide additional screening to students whose resting pulse rates are in the top $30 \%$ of the students who were tested.

What is the minimum resting pulse rate at that school for students who will receive additional screening?
Choose 1 answer:
(A) 76 beats per minute
(B) 85 beats per minute
(C) 87 beats per minute
(D) 89 beats per minute
(ㄷ) 94 beats per minute

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the percentile corresponding to the top 30% of the distribution. ‖ ‖ step 2 ⋮ Use the standard normal distribution table or a calculator to find the z-score that corresponds to the 70th percentile (since the top 30% is equivalent to the bottom 70%). ‖ ‖ step 3 ⋮ The z-score for the 70th percentile is approximately 0.52. ‖ ‖ step 4 ⋮ Use the z-score formula to solve for the pulse rate $X$ that corresponds to this z-score: $X = \mu + z\sigma$. ‖ ‖ step 5 ⋮ Substitute the given values into the formula: $X = 80 + 0.52 imes 9$. ‖ ‖ step 6 ⋮ Calculate the minimum pulse rate: $X = 80 + 4.68 = 84.68$. ‖ ‖ step 7 ⋮ Round up to the nearest whole number since we are looking for the minimum rate that is in the top 30%, which gives us 85 beats per minute. ‖ ***B*** ∻Key Concept∻ ⚹Percentiles and z-scores in a normal distribution⚹ ∻Explanation∻ ⚹To find the minimum value that falls within a certain percentile of a normal distribution, we calculate the corresponding z-score and then use the mean and standard deviation to find the specific value.⚹◊What is the formula for calculating the z-score in a normal distribution?⍭ Generate me a similar question◊

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