Question

In: Statistics and Probability

The method of tree ring dating gave the following years A.D. for an archaeological excavation site....

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.

Solutions

Expert Solution

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.


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