Question

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.

Answer 1

`0.87-1.66(0.295)/(√(140))=0.829`

`0.87+1.66(0.295)/(√(140))=0.911`

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.

Explanation:

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".

`\bar X=0.87` represent the sample mean

`\mu` population mean (variable of interest)

`s=\sqrt[0.87]=0.295` 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:

`\bar X \pm t_(\alpha/2)(s)/(√(n))` (1)

In order to calculate the critical value
`t_(\alpha/2)` we need to find first the degrees of freedom, given by:

`df=n-1=140-1=139`

Since the Confidence is 0.90 or 90%, the value of
`\alpha=0.1` and
`\alpha/2 =0.05`, 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
`t_(\alpha/2)=1.66`

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

`0.87-1.66(0.295)/(√(140))=0.829`

`0.87+1.66(0.295)/(√(140))=0.911`

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.