Question

An article presents measures of penetration resistance for a certain fine-grained soil. Fifteen measurements, expressed as a multiple of a standard quantity, had a mean of 2.64 and a standard deviation of 1.32. Find a 95% confidence interval for the mean penetration resistance for this soil. Round the answers to three decimal places.

Answer 1

`2.64-2.14(1.32)/(√(15))=1.911`

`2.64+2.14(1.32)/(√(15))=3.369`

So on this case the 95% confidence interval would be given by (1.911;3.37)

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=2.64` represent the sample mean for the sample

`\mu` population mean (variable of interest)

s=1.32 represent the sample standard deviation

n=15 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=15-1=14`

Since the Confidence is 0.95 or 95%, the value of
`\alpha=0.05` and
`\alpha/2 =0.025`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,14)".And we see that
`t_(\alpha/2)=2.14`

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

`2.64-2.14(1.32)/(√(15))=1.911`

`2.64+2.14(1.32)/(√(15))=3.369`

So on this case the 95% confidence interval would be given by (1.911;3.37)