Find the necessary confidence interval for a population mean μ for the following values. (Round your answers to three decimal places.) a 90% confidence interval, n = 140, x = 0.87, s2 = 0.087 to Interpret the interval that you have constructed. There is a 90% chance that an individual sample mean will fall within the interval. In repeated sampling, 10% of all intervals constructed in this manner will enclose the population mean. There is a 10% chance that an individual sample mean will fall within the interval. 90% of all values will fall within the interval. In repeated sampling, 90% of all intervals constructed in this manner will enclose the population mean.

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=0.87[/tex] represent the sample mean

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

[tex]s=\sqrt[0.87]=0.295[/tex] represent the sample standard deviation

n=140 represent the sample size

Solution to the problem

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

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

In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:

[tex]df=n-1=140-1=139[/tex]

Since the Confidence is 0.90 or 90%, the value of [tex]\alpha=0.1[/tex] and [tex]\alpha/2 =0.05[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.05,139)".And we see that [tex]t_{\alpha/2}=1.66[/tex]

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

[tex]0.87-1.66\frac{0.295}{\sqrt{140}}=0.829[/tex]

[tex]0.87+1.66\frac{0.295}{\sqrt{140}}=0.911[/tex]

So on this case the 90% confidence interval would be given by (0.829;0.911)

And the best interpretation for this case is:

In repeated sampling, 90% of all intervals constructed in this manner will enclose the population mean.