Question

A leading magazine (like Barron's) reported at one time that the average number of weeks an individual is unemployed is 28 weeks. Assume that the length of unemployment is normally distributed with population mean of 28 weeks and the population standard deviation of 9 weeks. Suppose you would like to select a random sample of 35 unemployed individuals for a follow-up study. Round the answers of following questions to 4 decimal places.

What is the distribution of X? X ~ N

What is the distribution of ¯x? ¯x ~ N

What is the probability that one randomly selected individual found a job more than 30 weeks?

For 35 unemployed individuals, find the probability that the average time that they found the next job is more than 30 weeks.

For part d), is the assumption of normal necessary? yes or no

Answer 1

a) Let X be the number of weeks an individual is unemployed

X follow Normal with mean =28 weeks , and standard deviation = 9 weeks

that can be written as ,

b) Let be the average of number of weeks of unemployment for a sample of n=35

According to Central limit theorem ,

follow Normal with mean =population mean =28 weeks ,

and standard error= population standard deviation/ = weeks

that can be written as ,

c) To find P( X > 30)

then ,

z is a standard normal variate with mean =0 , standard deviation =1

= P(z > 0.22)

   = 0.4129 (using z table z to or - to z)

Probability that an individual found job after more than 30 weeks =0.4129

(d)

To find P( > 30)

then ,

z is a standard normal variate with mean =0 , standard deviation =1

= P(z > 1.31)

  =  0.0951 ( using z table: z to or - to z)

Probability that average time  found is after more than 30 weeks =0.0951

e) According to central limit theorem the sampling distribution of sample mean, follow normal with mean = population mean and standard error =  population standard deviation/ (sample size) , even if the parent population(X) is not normal, if the sample size is large ( more than 30) .

Here the sample size , n= 35 >30

Thus to calculate P( > 30) in part (d) , assumption of normality of X is not necessary

Answer is NO