Question

The method of tree ring dating gave the following years A.D. for an archaeological excavation site. Assume that the population of x values has an approximately normal distribution.

1222

1208

1215

1257

1268

1316

1275

1317

1275

(a) Use a calculator with mean and standard deviation keys to find the sample mean year x and sample standard deviation s. (Round your answers to the nearest whole number.)

x =	A.D.
s =	yr

(b) Find a 90% confidence interval for the mean of all tree ring dates from this archaeological site. (Round your answers to the nearest whole number.)

lower limit	A.D.
upper limit	A.D.

Answer 1

Solution:

Sample size n = 9

1222,1208,1215,1257,1268,1316,1275,1317,1275

a)

Sample mean = (1222 + 1208 + ......+ 1275)/9 =  1261.4444444444 = 1261

= 1261

To find sample SD s , we prepare a table.

	x	x2
	1222	1493284
	1208	1459264
	1215	1476225
	1257	1580049
	1268	1607824
	1316	1731856
	1275	1625625
	1317	1734489
	1275	1625625
SUM	11353	14334241

Sample variance s² =

= [1/(9 - 1)][14334241- (11353²/9) ]

= 1632.7777777778

Now ,

sample standard deviation s = variance = 1632.7777777778 = 40

s = 40

b)

Note that, Population standard deviation() is unknown..So we use t distribution.

Our aim is to construct 90% confidence interval.   

c = 0.90

= 1- c = 1- 0.90 = 0.10

  /2 = 0.10 2 = 0.05

Also, d.f = n - 1 = 9 - 1 = 8

    =    = t_0.05,8 = 1.860

( use t table or t calculator to find this value..)

The margin of error is given by

E = t/2,d.f. * (s / n)

=  1.860* (40 / 9)

= 8.27

Now , confidence interval for mean() is given by:

( - E ) <   <  ( + E)

(1261 - 8.27)   <   <  (1261 + 8.27)

1253 <   < 1269

Required 90% confidence interval is (1253 , 1269)