Sales personnel for Skillings Distributors submit weekly reports listing the customer contacts made during the week.

A sample of 75 weekly reports showed a sample mean of 19.5 customer contacts per week.

The sample standard deviation was 5.2.

Provide 90% and 95% confidence intervals for the population mean number of weekly customer contacts for the sales personnel:

a. 90% Confidence, to 2 decimals

b. 95% Confidence, to 2 decimals

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

[tex]\bar X=19.5[/tex] represent the sample mean for the sample

[tex]\mu[/tex] population mean (variable of interest)

[tex]\sigma=5.2[/tex] represent the population standard deviation

n=75 represent the sample size

2) Confidence interval 90%

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (1)

Since the Confidence is 0.90 or 90%, the value of [tex]\alpha=0.1[/tex] and [tex]\alpha/2 =0.005[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.005,0,1)".And we see that [tex]z_{\alpha/2}=1.64[/tex]

Now we have everything in order to replace into formula (1):

[tex]19.5-1.64\frac{5.2}{\sqrt{75}}=18.515[/tex]

[tex]19.5+1.64\frac{5.2}{\sqrt{75}}=20.485[/tex]

So on this case the 90% confidence interval would be given by (18.52;20.49)

3) Confidence interval 95%

Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that [tex]z_{\alpha/2}=1.96[/tex]

Now we have everything in order to replace into formula (1):

[tex]19.5-1.96\frac{5.2}{\sqrt{75}}=18.323[/tex]

[tex]19.5+1.96\frac{5.2}{\sqrt{75}}=20.677[/tex]

So on this case the 95% confidence interval would be given by (18.32;20.68)