Question

The mean of a normally distributed group of weekly incomes of a large group of executives is $1,000 and the standard deviation is $100. What is the z-score (value of z) for an income of $1,100

Answer 1

The z-score (value of z) for an income of $1,100 is 1.

Explanation:

We are given that the mean of a normally distributed group of weekly incomes of a large group of executives is $1,000 and the standard deviation is $100.

Let X = group of weekly incomes of a large group of executives

So, X ~ N(
`\mu=1,000 ,\sigma^(2) = 100^(2)`)

The z-score probability distribution for a normal distribution is given by;

Z =
`(X-\mu)/(\sigma)` ~ N(0,1)

where,
`\mu` = mean income = $1,000

`\sigma` = standard deviation = $100

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) 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.

Now, we are given an income of $1,100 for which we have to find the z-score (value of z);

So, z-score is given by =
`(X-\mu)/(\sigma)` =
`(1,100-1,000)/(100)` = 1

Hence, the z-score (value of z) for an income of $1,100 is 1.