Question

A videotape store has an average weekly gross of $1,158 with a standard deviation of $120. Let x be the store's gross during a randomly selected week. If this is a normally distributed random variable, then the number of standard deviations from $1,158 to $1,360 is:

Answer 1

The number of standard deviations from $1,158 to $1,360 is 1.68.

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 = 1158, \sigma = 120`

The number of standard deviations from $1,158 to $1,360 is:

This is Z when X = 1360. So

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

`Z = (1360 - 1158)/(120)`

`Z = 1.68`

The number of standard deviations from $1,158 to $1,360 is 1.68.