Question

The lifetime of a certain type of calculators is normally distributed. If the mean is 400 hours and the standard deviation is 50 hours, for a group of 5000 batteries, how many are expected to last

1.between 350 hrs to 450 hours
2.more than 300 hours
3.less than 300 hours

Answer 1

1. 3413

2. 4886

3. 114

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 is 400 hours and the standard deviation is 50 hours

This means that
`\mu = 400, \sigma = 50`

1. Between 350 and 450 hours:

The proportion is the pvlaue of Z when X = 450 subtracted by the pvalue of Z when X = 350.

X = 450:

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

`Z = (450 - 400)/(50)`

`Z = 1`

`Z = 1` has a pvalue of 0.8413.

X = 350:

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

`Z = (350 - 400)/(50)`

`Z = -1`

`Z = -1` has a pvalue of 0.1587.

0.8413 - 0.1587 = 0.6826

Out of 5000:

0.6826 out of 5000. So

0.6826*5000 = 3413

3413 are expected to last between 350 hours and 450 hours.

2.more than 300 hours

The proportion is one subtracted by the pvalue of Z when X = 300.

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

`Z = (300 - 400)/(50)`

`Z = -2`

`Z = -2` has a pvalue of 0.0228

1 - 0.0228 = 0.9772

Out of 5000:

0.9772of 5000 is

0.9772*5000 = 4886

4886 are expected to last more than 300 hours.

3.less than 300 hours

4886 are expected to last more than 300 hours, so 5000 - 4886 = 114 are expected to last less than 300 hours.