Question

The thumb length of fully grown females of a certain type of frog is normally distributed with a mean of 8.59 mm and a standard deviation of 0.63 mm. Calculate the probability that a randomly selected frog of this type has thumb length longer than 9.08 mm.

Answer 1

21.77% probability that a randomly selected frog of this type has thumb length longer than 9.08 mm.

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore 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 pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 8.59, \sigma = 0.63`

Calculate the probability that a randomly selected frog of this type has thumb length longer than 9.08 mm.

This is 1 subtracted by the pvalue of Z when X = 9.08. So

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

`Z = (9.08 - 8.59)/(0.63)`

`Z = 0.78`

`Z = 0.78` has a pvalue of 0.7823

1 - 0.7823 = 0.2177

21.77% probability that a randomly selected frog of this type has thumb length longer than 9.08 mm.