Question

Question 1     

In a survey of the population of employed workers in the UK, respondents are asked how many hours they usually work per week in their main job. The average of the responses is 25 hours, with a standard deviation of 8 hours. Suppose that working hours in the population follow a normal distribution with these values of the mean and standard deviation.

1. What percentage of people in the population work between 22 and 35 hours per week?
2. The middle 70% of people in the population have weekly working hours between which two figures?
3. What is the number of hours such that only 22% of people work that long or longer per week?
4. What is the number of hours that only 35% of people work that or less per week?

Answer 1

From the given information

X=number of hours workers usually work per week

X follows Normal distribution with

Mean = 25 hours

Standard deviation = 8 hours

We know that

Z=(x-mean) /standard deviation

Follows standard normal distribution.

A)

p( 22 < x < 35) * 100

=​​​​​​* 100

=p(-0.375 < z < 1.25) * 100

=(p(z < 1.25) - p(z < - 0.375))*100

=(0. 8944 - 0.3538)*100

=54.6 %

B)

Let The middle 70% observations are between a and b such that

P(a < x < b) = 0.7

Using symmetry of normal distribution we can write

2*p(x < a) =2*p(x > b) = 0.3

2*p(x < a) = 0.3

P(x < a) =0.15

P( z < (a-25)/8) = 0.15

Using standard normal table

(a-25)/8 = - 1.04

That gives

a = 16.68

lly for b

P( x > b) = 0.15

P( z > (b-25)/8)= 0.15

(b-25)/8 = 1.04

b= 33.32

C)

Let 22% of worker have longer working time than t hours such that

P( x > t) = 0.22

P( z > (t-25)/8))=0.22

Using standard normal table

(t - 25)/8 = 0.7722

That gives

t =31.1776 hours

D)

Let m be the working hours such that only 35% of workers work that or less than that of hours pre week

Such that

P( x < m) = 0.35

P( z < (m-25)/8) = 0.35

Using standard normal distribution table

(m - 25)/8 = - 0.3853

That gives

m = 21.9176 hours