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Solved on Sep 25, 2023

For a survey of 18-year-old males, find the weights representing the 99th, 43rd, and first quartile percentiles given a mean of 167.5167.5 pounds and a standard deviation of 48.648.6 pounds.

STEP 1

Assumptions1. The survey is of18-year-old males. The mean weight is167.5 pounds3. The standard deviation is48.6 pounds4. The weights are normally distributed

STEP 2

We need to find the z-scores for the99th percentile,43rd percentile, and the first quartile (25th percentile). The z-score is the number of standard deviations a data point is from the mean. We can use the standard normal distribution table or a z-score calculator to find these values.
(a) For the99th percentile, the z-score is approximately2.33(b) For the43rd percentile, the z-score is approximately -0.2(c) For the first quartile (25th percentile), the z-score is approximately -0.67

STEP 3

Now, we can use the z-score formula to find the weights that correspond to these percentiles. The z-score formula isZ=(X−μ)/σZ = (X - \mu) / \sigmawhereZ = z-scoreX = value we're looking forμ = meanσ = standard deviationWe can rearrange the formula to solve for XX=Zσ+μX = Z\sigma + \mu

STEP 4

(a) Plug in the values for the z-score, standard deviation, and mean to find the weight that represents the99th percentile.
X=2.33×48.6+167.X =2.33 \times48.6 +167.

STEP 5

Calculate the weight that represents the99th percentile.
X=2.33×48.+167.5=280.4X =2.33 \times48. +167.5 =280.4The99th percentile weight is280.4 pounds.

STEP 6

(b) Plug in the values for the z-score, standard deviation, and mean to find the weight that represents the43rd percentile.
X=−0.2×48.6+167.5X = -0.2 \times48.6 +167.5

STEP 7

Calculate the weight that represents the43rd percentile.
X=−0.2×48.6+167.5=157.X = -0.2 \times48.6 +167.5 =157.The43rd percentile weight is157. pounds.

STEP 8

(c) Plug in the values for the z-score, standard deviation, and mean to find the weight that represents the first quartile.
X=−0.67×48.6+167.5X = -0.67 \times48.6 +167.5

STEP 9

Calculate the weight that represents the first quartile.
X=−.67×48.6+167.5=134.7X = -.67 \times48.6 +167.5 =134.7The first quartile weight is134.7 pounds.

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