Shards of clay vessels were put together to reconstruct rim diameters of the original ceramic vessels found at the Wind Mountain archaeological site. A random sample of ceramic vessels gave the following rim diameters (in centimeters):

15.9 - 13.4 - 22.1 - 12.7 - 13.1 - 19.6 - 11.7 - 13.5 - 17.7 - 18.1

(A) With mean and sample deviation verify that x bar is about equal to 15.8 cm and s equal to 3.5 cm.

(B) Compute an 80% confidence interval for the population mean (mhu) of rim diameters for such ceramic vesselsfound at the Wind Mountain archaelogical site.

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[/tex] represent the sample mean for the sample

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

s represent the sample standard deviation

n represent the sample size

Part a

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 mean and the sample deviation we can use the following formulas:

[tex]\bar X= \sum_{i=1}^n \frac{x_i}{n}[/tex] (2)

[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)^2}{n-1}}[/tex] (3)

The mean calculated for this case is

The sample deviation calculated [tex]s=3.46 \approx 3.5[/tex]

Part b

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=10-1=9[/tex]

Since the Confidence is 0.80 or 80%, the value of [tex]\alpha=0.2[/tex] and [tex]\alpha/2 =0.1[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.1,9)".And we see that [tex]t_{\alpha/2}=1.383[/tex]

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

[tex]15.8-1.383\frac{3.5}{\sqrt{10}}=14.27[/tex]

[tex]15.8+1.383\frac{3.5}{\sqrt{10}}=17.33[/tex]

So on this case the 80% confidence interval would be given by (14.27;17.33)