Question

Sherds of clay vessels were put together to reconstruct rim diameters of the original ceramic vessels 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) Use a calculator with mean and sample standard deviation keys to find the sample mean x and sample standard deviation s. (Round your answers to one decimal place.)

x =	cm
s =	cm

(b) Compute a 95% confidence interval for the population mean μ of rim diameters for such ceramic vessels found at the Wind Mountain archaeological site. (Round your answers to one decimal place.)

lower limit	cm
upper limit	cm

Answer 1

Solution:

x	x2
15.9	252.81
13.4	179.56
22.1	488.41
12.7	161.29
13.1	171.61
19.6	384.16
11.7	136.89
13.5	182.25
17.7	313.29
18.1	327.61
$\sum$x=157.8	$\sum$x2=2597.88

a ) The sample mean is $\bar x$

Mean $\bar x$     = ($\sum$x / n) )

= (15.9+13.4+22.1+12.7+13.1+19.6+11.7+13.5+17.7+18.1/ 10 )

=157.8 / 14

= 15.78

Mean $\bar x$    = 15.78

The sample standard is S

  S = $\sqrt{}$ ( $\sum$ x² ) - (( $\sum$ x)² / n ) n -1

= $\sqrt{}$ (2597.88( (157.8)² / 10 ) 9

   = $\sqrt{}$ ( 2597.88- 2490.084 / 9)

=$\sqrt{}$ (107.796/ 9)

= $\sqrt{}$ 11.9773

= 3.4608

The sample standard is = 3.46

b ) Degrees of freedom = df = n - 1 = 10- 1 = 9

At 95% confidence level the t is ,

$\alpha$  = 1 - 95% = 1 - 0.95 = 0.05

$\alpha$ / 2 = 0.05 / 2 = 0.025

t_{$\alpha$ /2,df} = t_0.025,9 = 2.262

Margin of error = E = t_{$\alpha$/2,df} * (s /$\sqrt$n)

= 2.262* (3.46 / $\sqrt$ 10)

= 2.47

Margin of error = 2.47

The 95% confidence interval estimate of the population mean is,

$\bar x$ - E < $\mu$ < $\bar x$ + E

15.78 - 2.47 < $\mu$ < 15.78 + 2.47

13.31 < $\mu$ < 18.25

(13.31, 18.25 )