Question

Defiance Tool and Die has 293 sales offices throughout the world. The VP of Sales is studying the usage of its copy machines. A random sample of six of the sales offices revealed the following number of copies made in each selected office last week.

 

Develop a 95% confidence interval for the mean number of copies per week.

Answer 1

The given information is related to number of copies made in each selected office last week.

 

It is given that, number of observations, n = 6 and total sales in offices throughout the world N = 293.

 

Here, the sample size is small (n < 30) and population standard deviation (σ) is unknown. So, apply one sample t confidence intervals.

 

The confidence interval requires the average and standard deviation to be found first. Determine the mean (x̅) and standard deviation (s)using Excel functions is shown below:

 

Mean:

= average(826,931, 1126, 918, 1011, 1101)

= 986

 

Standard deviation:

= stdev(826, 931, 1126, 918, 1011, 1101)

= 115

 

Find the 95% confidence interval for the mean number of copies per week.

 

Determine the 95% confidence interval (Cl) for the population mean is shown below:

95% Cl = [x̅ - t × s/√n{√(N – n/N – 1)}, x̅ + t s/√n{√(N – n)/(N – 1)}] …. (1)

 

Here, x̅ is the sample mean, s is the sample standard deviation, n is the number of observations, t is the critical value.

 

Determine the degrees of freedom using the following formula,

df = n – 1

     = 6 – 1

     = 5

 

From the t-distribution table, the t critical value at 5% level of the significance at 5 degrees of freedom is 2.571.

 

Substitute the values 986 for x̅, 115 for s, 2.571 for t, 6 for n and 293 for N in equation (1).

 

95%Cl = [986 – 2.571 × 115/√6{√(293 – 6)/(293 – 1)}, 986 + 2.571 × 115/√6{(293 – 6)/(293 – 1)}]

            = (986 – 119.6668, 986 + 119.6668

           = (866.3332, 1105.6668)

 

Therefore, the 95% confidence interval for the mean number of copies per week is lies between 866.3332 and 1105.6668.

Therefore, the 95% confidence interval for the mean number of copies per week is lies between 866.3332 and 1105.6668.