Question

A recent study indicated that women took an average of 8.6 weeks of unpaid leave from their jobs after the birth of a child. Assume that this distribution follows the normal probability distribution with a standard deviation of 2.0 weeks. We select a sample of 35 women who recently returned to work after the birth of a child. What is the likelihood that the mean of this sample is at least 8.8 weeks?

 

 

Answer 1

It is given that, average number of unpaid leaves from their jobs after the birth of the child,µ = 8.6.

 

The standard deviation (σ) is 2.0 weeks.

 

Number of women recently returned to work after the birth of a child, n = 35.

 

The cumulative probability for finding means in Excel based on a known population mean can be found the same way as cumulative for random variable, except that the standard error is used in the “=norm.dist” function instead of standard deviation.

 

Determine the standard error of the distribution of the sample mean using the following formula,

se = σ/√n

 

Here, n is the sample size and σ is the population standard deviation.

 

Substitute the values 2.0 for σ and 35 for n in the above equation.

se = 2.0/√35

     = 0.338

 

The standard error is 0.338.

 

Find the likelihood that the mean of this sample is at least 8.8 weeks.

 

To find the right tail cumulative probability of a sample mean of at least 8.8 weeks use the complementary rule.

P(X̅ ≥ 8.8) = 1 – P(X̅ < 8.8)

 

First find probability of sample mean up to 8.8 weeks leave. The normal distribution function in Excel can be used to find probability of normal distributions. Follow the steps as below:

• Use the formula “=norm.dist(sample mean, population mean, se, cumulative)

• For x enter 5, Sample mean is 8.8, the population mean is 8.6, se is the standard error 0.338,

• For cumulative enter 1 or “True”

 

This returns the left tail cumulative probability which is 0.72. Now, apply the complementary rule

P(X̅ ≥ 8.8) = 1 – P(X̅ < 8.8)

                  = 1 – 0.72

                 = 0.28

 

The right tail probability is 0.28. It means that 28% of the time the mean leave will exceed 8.8 weeks.

The right tail probability is 0.28. It means that 28% of the time the mean leave will exceed 8.8 weeks.