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

1187

1292

1250

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

Answers:

A)

x̄ = 1272 A.D.

s = 39 yr

B)

90% CI: (1248 A.D., 1296 A.D.)

Work:

A) Sample mean

Sum of values:

sum x = 1271+1187+1292+1250+1268+1316+1275+1317+1275 = 11451

Mean:

x̄ = 11451/9 =1272.3333

Rounded to nearest whole number: x̄ = 1272 A.D.

A) Sample standard deviation

formula: sqrt((sum(x - x̄)²/(n-1))

x	x − x̄ (exact)	x − x̄ (decimal)	(x − x̄)² (exact)	(x − x̄)² (decimal)
1271	−4/3	−1.3333	16/9	1.7778
1187	−256/3	−85.3333	65536/9	7281.7778
1292	59/3	19.6667	3481/9	386.7778
1250	−67/3	−22.3333	4489/9	498.7778
1268	−13/3	−4.3333	169/9	18.7778
1316	131/3	43.6667	17161/9	1906.7778
1275	8/3	2.6667	64/9	7.1111
1317	134/3	44.6667	17956/9	1995.1111
1275	8/3	2.6667	64/9	7.1111

Add the squared deviations:

1.7778 + 7281.7778 + 386.7778 + 498.7778 + 18.7778 + 1906.7778 + 7.1111 + 1995.1111 + 7.1111

= 12104.0000

Degrees of freedom: df=n-1

df=9-1=8

Sample variance:

s² = 12104/8 =1,513

Standard Deviation:

s = sqrt(1513) =38.8973

Rounded to nearest whole number:

s=39 years

(B) 90% confidence interval for the population mean

degrees of freedom: df= n-1

9-1=8

Standard error: SE= (s/sqrt(n))

38.8973/sqrt(9)=12.9658

Margin of error: ME = t(SE)

1.8595(12.9658)=24.1105

Confidence interval = x̄ ± ME

1272.3333 ± 24.1105 = (1248.2229, 1296.4438)

Round to nearest whole number:

lower limit = 1248 A.D.

upper limit = 1296 A.D.