Question

Suppose the method of tree-ring dating gave the following dates A.D. for an archaeological excavation site.

1264 1201 1286 1173 1197 1249 1273 1210 1183

Find a 95% confidence interval for the mean of all tree-ring dates from this archaeological site. Suppose that the sample mean round to the whole number. Round your answer to the whole number.

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1194 to 1258

1193 to 1260

1198 to 1260

1196 to 1260

1192 to 1255

Answer 1

SOLUTION:

From given data,

Suppose the method of tree-ring dating gave the following dates A.D. for an archaeological excavation site.

1264

1201

1286

1173

1197

1249

1273

1210

1183

Find a 95% confidence interval for the mean of all tree-ring dates from this archaeological site.

We calculate sample mean and standard deviation from given data,

Mean = = / = (1264+1201+1286+1173+1197+1249+1273+1210+1183) / 9

= 11036/ 9 = 1226.222

Standard deviation = s = sqrt(( ( - )²)/(n-1))

s = sqrt(((1264-1226.222)^2+(1201-1226.222)^2+(1286-1226.222)^2+(1173-1226.222)^2+(1197-1226.222)^2+(1249-1226.222)^2+(1273-1226.222)^2+(1210-1226.222)^2+(1183-1226.222)^2)/(9-1))

s = sqrt( 14161.555556/ 8)

s = 42.0737

95% confidence interval

95% = 95/100 = 0.95

= 1-0.95 = 0.05

/2 = 0.05/2 = 0.025

Degree of freedom = df = n-1 = 9-1 = 8

Critical value = t_/2,df = t_0.025,8 = 2.306

Margin of error = E = t_/2,df * (s/sqrt(n)) = 2.306 * (42.0737/sqrt(9)) = 32.3406507

Limits of 95% confidence interval are given by :

Lower limit = -E =  1226.222-32.3406507 = 1193.8   1194

Upper limit = +E =  1226.222​​​​​​​+32.3406507 = 1258.5   1258

95% confidence interval is :  E = 1226.22232.3406507 = (1194 , 1258)