Question

A journal article reports that a sample of size 5 was used as a basis for calculating a 95% CI for the true average natural frequency (Hz) of delaminated beams of a certain type. The resulting interval was (229.266, 233.002). You decide that a confidence level of 99% is more appropriate than the 95% level used. What are the limits of the 99% interval? [Hint: Use the center of the interval and its width to determine x and s.] (Round your answers to three decimal places.)

Answer 1

`Lower = 231.134- 3.098=228.036`

`Upper = 231.134+ 3.098=234.232`

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` represent the sample mean for the sample

`\mu` population mean (variable of interest)

s represent the sample standard deviation

n=5 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)

And for this case we know that the 95% confidence interval is given by:

`\bar X=(233.002 +229.266)/(2)= 231.134`

And the margin of error is given by:

`ME = (233.002 -229.266)/(2)= 1.868`

And the margin of error is given by:

`ME= t_(\alpha/2) (s)/(√(n))`

The degrees of freedom are given by:

`df = n-1 = 5-1=4`

And the critical value for 95% of confidence is
`t_(\alpha/2)= 2.776`

So then we can find the deviation like this:

`s = (ME √(n))/(t_(\alpha/2))`

`s = (1.868* √(5))/(2.776)= 1.506`

And for the 99% confidence the critical value is:
`t_(\alpha/2)= 4.604`

And the margin of error would be:

`ME = 4.604 *(1.506)/(√(5))= 3.098`

And the interval is given by:

`Lower = 231.134- 3.098=228.036`

`Upper = 231.134+ 3.098=234.232`