Compute the 95% confidence interval of the mean height. Round your answers to 3 decimal places. You should get an interval of the form (lower bound, upper bound). You'll enter your answers in the next two problems.

The heights of young adult females in the United States are said to have a population standard deviation of [tex]\sigma = 2.6 inches[/tex]. A sample was taken of n = 85 young adult females at school and the mean computed to be bar [tex]\bar x = 65.4 inches[/tex].

Previous concepts

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=65.4 [/tex] represent the sample mean

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

[tex]\sigma=2.6[/tex] represent the sample standard deviation

n=85 represent the sample size

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 0.95 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]65.4-1.96\frac{2.6}{\sqrt{85}}=64.847[/tex]

[tex]65.4+ 1.96\frac{2.6}{\sqrt{85}}=65.953[/tex]

So on this case the 95% confidence interval would be given by (64.847;65.953)