Question

The mean investment that employees put into their companies 401k per year is $10,000 with standard deviation of $500 Assuming the investment follow a normal distribution, determine the following a. what proportion of employees put between $9, 500 and $11,000 into the 401 k per year b. What proportion of employee put more than $11, 500 into the 401 k per year? c. What proportional of employees put less than $11,000 into the 401k per year?d. What proportional of employees put more than $9,000 into the 401k per year?e. What proportional of employees put between than $11,000 and $11, 500 into the 401k per year?f. How much would an employees need to put into his or her 401 K to be in the upper 10% of investors?

Answer 1

a) 81.85% of employees put between $9, 500 and $11,000 into the 401 k per year

b) 0.13% of employee put more than $11, 500 into the 401 k per year

c) 97.72% of employees put less than $11,000 into the 401k per year.

d) 97.72% of employees put more than $9,000 into the 401k per year

e) 2.15% of employees put between than $11,000 and $11, 500 into the 401k per year

f) An employee would need to put $10,640 into his or her 401 K to be in the upper 10% of investors

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 = 10000, \sigma = 500`

a. what proportion of employees put between $9, 500 and $11,000 into the 401 k per year

This is the pvalue of Z when X = 11000 subtracted by the pvalue of Z when X = 9500. So

X = 11000

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

`Z = (11000 - 10000)/(500)`

`Z = 2`

`Z = 2` has a pvalue of 0.9772.

X = 9500

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

`Z = (9500 - 10000)/(500)`

`Z = -1`

`Z = -1` has a pvalue of 0.1587

0.9772 - 0.1587 = 0.8185

81.85% of employees put between $9, 500 and $11,000 into the 401 k per year

b. What proportion of employee put more than $11, 500 into the 401 k per year?

This is 1 subtracted by the pvalue of Z when X = 11500. So

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

`Z = (11500 - 10000)/(500)`

`Z = 3`

`Z = 3` has a pvalue of 0.9987

1 - 0.9987 = 0.0013

0.13% of employee put more than $11, 500 into the 401 k per year

c. What proportional of employees put less than $11,000 into the 401k per year?

This is the pvalue of Z when X = 11000. So

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

`Z = (11000 - 10000)/(500)`

`Z = 2`

`Z = 2` has a pvalue of 0.9772.

97.72% of employees put less than $11,000 into the 401k per year.

d. What proportional of employees put more than $9,000 into the 401k per year?

This is 1 subtracted by the pvalue of Z when X = 9000. So

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

`Z = (9000 - 10000)/(500)`

`Z = -2`

`Z = -2` has a pvalue of 0.0228.

1 - 0.0228 = 0.9772

97.72% of employees put more than $9,000 into the 401k per year

e. What proportional of employees put between than $11,000 and $11, 500 into the 401k per year?

This is the pvalue of Z when X = 11500 subtracted by the pvalue of Z when X = 11000. So

X = 11500

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

`Z = (11500 - 10000)/(500)`

`Z = 3`

`Z = 3` has a pvalue of 0.9987

X = 11000

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

`Z = (11000 - 10000)/(500)`

`Z = 2`

`Z = 2` has a pvalue of 0.9772.

0.9987 - 0.0972 = 0.0215

2.15% of employees put between than $11,000 and $11, 500 into the 401k per year

f. How much would an employees need to put into his or her 401 K to be in the upper 10% of investors?

This is the value of Z when X has a pvalue of 1-0.1 = 0.9. So it is X when Z = 1.28.

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

`1.28 = (X - 10000)/(500)`

`X - 10000 = 500*1.28`

`X = 10640`

An employee would need to put $10,640 into his or her 401 K to be in the upper 10% of investors