Question

The length of time, M hours that a phone battery will work before it needs recharging is normally distributed with a mean of 90 hours, and a standard deviation of 10 hours. Find the interquartile range for this distribution.

Answer 1

The interquartile range is between 83.25 hours and 96.75 hours

Explanation:

Problems of normally distributed samples can be 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 = 90, \sigma = 10`

Find the interquartile range for this distribution.

That is, the middle 50%, from the 25th to the 75th percentile.

25th percentile:

X when Z has a pvalue of 0.25. So X when Z = -0.675

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

`-0.675 = (X - 90)/(10)`

`X - 90 = -0.675*10`

`X = 83.25`

75th percentile:

X when Z has a pvalue of 0.75. So X when Z = 0.675

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

`0.675 = (X - 90)/(10)`

`X - 90 = 0.675*10`

`X = 96.75`

The interquartile range is between 83.25 hours and 96.75 hours