Question

The weight of laboratory grasshoppers follows a Normal distribution, with a mean of 90 grams and a standard deviation of 2 grams. What percentage of the grasshoppers weigh between 86 grams and 94 grams

Answer 1

95.44% of the grasshoppers weigh between 86 grams and 94 grams.

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 90 grams and a standard deviation of 2 grams.

This means that
`\mu = 90, \sigma = 2`

What percentage of the grasshoppers weigh between 86 grams and 94 grams?

The proportion is the p-value of Z when X = 94 subtracted by the p-value of Z when X = 86. So

X = 94

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

`Z = (94 - 90)/(2)`

`Z = 2`

`Z = 2` has a p-value of 0.9772.

X = 86

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

`Z = (86 - 90)/(2)`

`Z = -2`

`Z = -2` has a p-value of 0.0228.

0.9772 - 0.0228 = 0.9544

0.9544*100% = 95.44%

95.44% of the grasshoppers weigh between 86 grams and 94 grams.