Question

An ethologist is interested in how long it takes a certain species of water shrew to catch its prey. He only has access to a sample of 100 shrews. On multiple occasions each day he lets a dragonfly loose inside the cage of the shrews and times how long it takes until the shrews catch the dragonfly. After months of research the ethologist concludes 1) that the mean prey catching time was 30 seconds, 2) the standard deviation was 5.5 seconds and 3) that the scores he has collected are normally distributed. What is the percentage of shrews that: a) catch a dragonfly in less than 18 seconds; b) catch a dragonfly in between 22 and 45 seconds; c) take longer than 45 seconds to catch a dragonfly? d) take less than 30 seconds to catch its prey;

Answer 1

a) 1.46%.

b) 92.33%.

c) 0.32%.

d) 50%

Explanation:

When the distribution is normal, we use 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 question, we have that:

`\mu = 30, \sigma = 5.5`

a) catch a dragonfly in less than 18 seconds;

This is the pvalue of Z when X = 18. So

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

`Z = (18 - 30)/(5.5)`

`Z = -2.18`

`Z = -2.18` has a pvalue of 0.0146

So the percentage of shrews is 1.46%.

b) catch a dragonfly in between 22 and 45 seconds;

This is the pvalue of Z when X = 45 subtracted by the pvalue of Z when X = 22.

X = 45

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

`Z = (45 - 30)/(5.5)`

`Z = 2.73`

`Z = 2.73` has a pvalue of 0.9968

X = 22

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

`Z = (22 - 30)/(5.5)`

`Z = -1.45`

`Z = -1.45` has a pvalue of 0.0735

0.9968 - 0.0735 = 0.9233

So the answer is 92.33%.

c) take longer than 45 seconds to catch a dragonfly?

From b), when X = 45, Z = 2.73 has a pvalue of 0.9968

1 - 0.9968 = 0.0032

So the answer for this item is 0.32%.

d) take less than 30 seconds to catch its prey;

This is the pvalue of Z when X = 30.

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

`Z = (30 - 30)/(5.5)`

`Z = 0`

`Z = 0` has a pvalue of 0.5

So the answer for d) is 50%.