Question

The amount of time all students in a very large undergraduate statistics course take to complete an examination is distributed continuously and normally. The probability a student selected at random takes at least 55.50 minutes to complete the examination equals 0.6915. The probability a student selected at random takes no more than 71.52 minutes to complete the examination equals 0.8997.

a) Determine the value for the mean (u) of the associated distribution
b) Determine the value for the standard deviation (o) of the associated distribution.

Answer 1

a) The mean is
`\mu = 60`

b) The standard deviation is
`\sigma = 9`

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 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.

The probability a student selected at random takes at least 55.50 minutes to complete the examination equals 0.6915.

This means that when X = 55.5, Z has a pvalue of 1 - 0.6915 = 0.3085. This means that when
`X = 55.5, Z = -0.5`

So

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

`-0.5 = (55.5 - \mu)/(\sigma)`

`-0.5\sigma = 55.5 - \mu`

`\mu = 55.5 + 0.5\sigma`

The probability a student selected at random takes no more than 71.52 minutes to complete the examination equals 0.8997.

This means that when X = 71.52, Z has a pvalue of 0.8997. This means that when
`X = 71.52, Z = 1.28`

So

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

`1.28 = (71.52 - \mu)/(\sigma)`

`1.28\sigma = 71.52 - \mu`

`\mu = 71.52 - 1.28\sigma`

Since we also have that
`\mu = 55.5 + 0.5\sigma`

`55.5 + 0.5\sigma = 71.52 - 1.28\sigma`

`1.78\sigma = 71.52 - 55.5`

`\sigma = ((71.52 - 55.5))/(1.78)`

`\sigma = 9`

`\mu = 55.5 + 0.5\sigma = 55.5 + 0.5*9 = 55.5 + 4.5 = 60`

Question

The mean is
`\mu = 60`

The standard deviation is
`\sigma = 9`