Statistics Question

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.

1271

1208

1229

1299

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

Find the sample mean.

x-bar = (1271 + 1208 + 1229 + 1299 + 1268 + 1316 + 1275 + 1317 + 1275) / 9 = 1273 AD

Find the sample standard deviation. Take the sample mean and subtract it from the individual data points.

s = √∑(xi - x-bar) / (N - 1)

∑(x_i - x-bar)² = (1271 - 1273)² + (1208 - 1273)² + (1229 - 1273)² + ... + (1275 - 1273)²

= (-2)² + (-65)² + (-44)² + ... + (2)²

= 4 + 4225 + 1936 + 676 + 25 + 1849 + 4 1936 + 4

= 10,659

s² = 10659/8 = 1332 → s = 37 years

We need the t-distribution table since the standard deviation is unknown. Therefore, our degrees of freedom is 9 - 1 = 8 and the critical value is 1.860. Set up the confidence interval for the mean:

[x-bar ± t*(s/√n)] = [1273 ± 1.860*(37/√9)] → [1250,1296]

We are 90% confident that the mean of all tree ring dates from this archaeological site is between 1250 and 1296.