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.

1,2851,1871,2221,1941,2681,3161,2751,3171,275

Required:
a. Use a calculator with mean and standard deviation keys to find the sample mean year x and sample standard deviations.
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.)

Answer 1

a) The sample mean is 1260 and the standard deviation is 48.

b) The 90% confidence interval for the mean of all tree-ring dates from this archaeological site is (1230, 1290).

Explanation:

Question a:

Mean is the sum of all values divided by the number of values. So

`\overline{x} = (1285 + 1187 + 1222 + 1194 + 1268 + 1316 + 1275 + 1317 + 1275)/(9) = 1260`

Standard deviation is the square root of the sum of the differences squared between each value and the mean, divided by the one less than the sample size. So

`s = \sqrt{((1285-1260)^2 + (1187-1260)^2 + (1222-1260)^2 + (1194-1260)^2 + (1268-1260)^2 + ...)/(8)} = 48`

The sample mean is 1260 and the standard deviation is 48.

Question b:

We have the standard deviation for the sample, which means that the t-distribution is used to solve this question.

The first step to solve this problem is finding how many degrees of freedom,which is the sample size subtracted by 1. So

df = 9 - 1 = 8

90% confidence interval

Now, we have to find a value of T, which is found looking at the t table, with 8 degrees of freedom(y-axis) and a confidence level of
`1 - (1 - 0.9)/(2) = 0.95`. So we have T = 1.8595

The margin of error is:

`M = T(s)/(√(n)) = 1.8595(48)/(√(9)) = 30`

In which s is the standard deviation of the sample and n is the size of the sample.

The lower end of the interval is the sample mean subtracted by M. So it is 1260 - 30 = 1230

The upper end of the interval is the sample mean added to M. So it is 1260 + 30 = 1290

The 90% confidence interval for the mean of all tree-ring dates from this archaeological site is (1230, 1290).