Question

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.

Answer 1

Answer:

a) In order to calculate the mean and the sample deviation we can use the following formulas:  

$\bar X= \sum_{i=1}^n \frac{x_i}{n}$ (2)  

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

The mean calculated for this case is

The sample deviation calculated $s=3.46 \approx 3.5$

b) $15.8-1.383\frac{3.5}{\sqrt{10}}=14.27$    

$15.8+1.383\frac{3.5}{\sqrt{10}}=17.33$    

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

Step-by-step 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 represent the sample size  

Part a

The confidence interval for the mean is given by the following formula:

$\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$   (1)

In order to calculate the mean and the sample deviation we can use the following formulas:  

$\bar X= \sum_{i=1}^n \frac{x_i}{n}$ (2)  

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

The mean calculated for this case is

The sample deviation calculated $s=3.46 \approx 3.5$

Part b

In order to calculate the critical value $t_{\alpha/2}$ we need to find first the degrees of freedom, given by:

$df=n-1=10-1=9$

Since the Confidence is 0.80 or 80%, the value of $\alpha=0.2$ and $\alpha/2 =0.1$, 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 $t_{\alpha/2}=1.383$

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

$15.8-1.383\frac{3.5}{\sqrt{10}}=14.27$    

$15.8+1.383\frac{3.5}{\sqrt{10}}=17.33$    

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