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.

1285

1194

1313

1264

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 :

We are given a data of sample size n = 9

1285,1194,1313,1264,1268,1316,1275,1317,1275

Using this, first we find sample mean() and sample standard deviation(s).

=   

= (1285 + 1194.......+ 1275)/9

= 1279 A.D.

Now ,

s=   

Using given data, find Xi - for each term.Take square for each.Then we can easily find s.

s = 39 yr

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 = 8  

    =    =  0.05,8 = 1.86

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

The margin of error is given by

E =  /2,d.f. * ( / n )

= 1.86* ( 39/ 9 )

= 24.17

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

( - E ) <   <  ( + E)

( 1279 - 24.17 )   <   <  ( 1279 + 24.17 )

1255 <   < 1304

Required 90% confidence interval is ( 1255 , 1304 )

Lower limit = 1255  A.D.

Upper limit = 1304 A.D.