Question

A sample of 100 workers received an average daily wage of $234. The distribution of the weekly wages is a bell-shaped normal curve. The standard deviation of the daily wages is $12.

(a) Approximately what percentage of workers will earn between $222 and $258?

(b) Approximately what percentage of the workers will earn less than $222 or more than $258?

(c) Approximately how many workers will earn between $222 and $258?

Answer 1

Solution:

Given:

Sample size = n = 100

Sample mean =

The distribution of the weekly wages is a bell-shaped normal curve.

standard deviation = s = 12

Part a)  Approximately what percentage of workers will earn between $222 and $258?

that is find:

P( 222 < X < 258) =..........?

Find number of standard deviations that x values lies from mean value.

and

thus x values 222 and 258 are 1 standard deviation below mean and 2 standard deviation above mean respectively.

According to Empirical rule:

1) 68% of the data falls within 1 standard deviation from mean

2) 95% of the data falls within 2 standard deviation from mean

3) 99.7% of the data falls within 3 standard deviation from mean

Thus 68% of the data falls within 1 standard deviation from mean, then its half of area is between 1 standard deviation below mean to mean ,

that is: 68%/2=32% area is between 1 standard deviation below mean to mean.

and 95% of the data falls within 2 standard deviation from mean, then its half of area is between 1 standard deviation above mean to mean ,

that is: 95%/2=47.5% area is between 2 standard deviation above mean to mean.

Thus total area from 1 standard deviation above mean to 2 standard deviation above mean = 32%+47.5%=79.5%

Thus Approximately 79.5 percentage of workers will earn between $222 and $258.

Part b) Approximately what percentage of the workers will earn less than $222 or more than $258?

We have 79.5 percentage of workers will earn between $222 and $258. then percentage of the workers will earn less than $222 or more than $258 = 100% - 79.5% = 20.5%

Part c) Approximately how many workers will earn between $222 and $258?

We have:

79.5 percentage of workers will earn between $222 and $258.

thus 79.5% of 100 workers = 79.5 = 80

Approximately 80 workers will earn between $222 and $258