Question

A blueberry farmer finds that the diameters of blueberries harvested on his farm follow a normal distribution with a mean diameter of 5.85 mm and a standard deviation of 0.24 mm.

The middle 20% of blueberries from this farm have diameters between 5.79 and ____ mm.

Answer 1

The middle 20% of blueberries from this farm have diameters between 5.79 and 5.91 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 = 5.85, \sigma = 0.24`

Middle 20%

50 - 20/2 = 40th percentile to the 50 + 20/2 = 60th percentile.

We want the 60th percentile, which is the value of X when Z has a pvalue of 0.6. So it is X when Z = 0.253.

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

`0.253 = (X - 5.85)/(0.24)`

`X - 5.85 = 0.24*0.253`

`X = 5.91`

So the answer is:

The middle 20% of blueberries from this farm have diameters between 5.79 and 5.91 mm