A sample of 17 joint specimens of a particular type gave a sample mean proportional limit stress of 8.45 MPa and a sample standard deviation of 0.74 MPa. (a) Calculate and interpret a 95% lower confidence bound for the true average proportional limit stress of all such joints. (Round your answer to two decimal places.)

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=8.45[/tex] represent the sample mean

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

s=0.74 represent the sample standard deviation

n=17 represent the sample size

2) Confidence interval

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=17-1=16[/tex]

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: "=-T.INV(0.025,16)".And we see that [tex]t_{\alpha/2}=2.12[/tex]

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

[tex]8.45-2.12\frac{0.74}{\sqrt{17}}=8.07[/tex]

[tex]8.45+2.12\frac{0.74}{\sqrt{17}}=8.83[/tex]

So on this case the 95% confidence interval would be given by (8.07;8.83)