Question

The weight of adult female panthers in one habitat area are normally distributed with a mean of 82 pounds and a standard deviation of 9.8 pounds. What percent of the panthers weigh less than 76 pounds?

Answer 1

27.09% of the panthers weigh less than 76 pounds

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Mean of 82 pounds and a standard deviation of 9.8 pounds.

This means that
`\mu = 82, \sigma = 9.8`

What percent of the panthers weigh less than 76 pounds?

The proportion is the pvalue of Z when X = 76. So

`Z = (X - \mu)/(\sigma)`

`Z = (76 - 82)/(9.8)`

`Z = -0.61`

`Z = -0.61` has a pvalue of 0.2709

0.2709*100% = 27.09%

27.09% of the panthers weigh less than 76 pounds