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.

1264

1285

1313

1187

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

Answer: 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.

1264 1285 1313 1187 1268 1316 1275 1317 1275

Solution:

a) the sample mean year x and sample standard deviation s.

Sample mean:

Mean, x̄ = Σx/n

x̄ = Σ(1264 +1285 +.....+ 1275)/9

x̄ = 11500/9

x̄ = 1277.77

Mean, x̄ = 1278

Sample standard deviation, s

S = √(Σ(x-x​​​​2​​​)/n-1)

S = √(1264-1278)+......(1275-1278)/9-1

S = √12834/8

S = 40.05309

Therefore,

Mean, x̄ = 1278 A.D

S = 40 yr

(b) 90% confidence interval for the mean of all tree ring dates from this archaeological site.

df = n- 1 = 9 - 1 = 8

At 90% confidence interval

t(α/2,df) = 1.859 = 1.860

Therefore,

E = tα * S/√n

E = 1.860 * 40/√9

E = 74.40/ 3

E=24.80

Therefore,

(x̄ – E < μ < x̄ + Ε)

(1278 - 24.80 < μ < 1278 + 24.80 )

(1253.20 < μ < 1302.80)

(1253 < μ < 1303 )

Therefore,

Lower Limit = 1253 AD

Upper Limit = 1303 AD