Question

Math

Posted 3 months ago

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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
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

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: $X = 80 + 0.52 \times 9$

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.

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