Suppose the method of tree ring dating gave the following dates A.D. for an archaeological excavation site.
1248 1187 1199 1236 1206 1241 1185 1288 1239
Find an 80% confidence interval for the mean of all tree ring dates from this archaeological site.
To find an 80% confidence interval for the mean of all tree ring dates from the archaeological site, you can follow these steps:
1. Calculate the sample mean (x̄) of the tree ring dates. To do this, add up all the dates and divide the sum by the total number of dates.
x̄ = (1248 + 1187 + 1199 + 1236 + 1206 + 1241 + 1185 + 1288 + 1239) / 9 = 1216.667
2. Calculate the sample standard deviation (s) of the tree ring dates. To do this, calculate the sum of squares of the differences between each date and the sample mean, divide it by the total number of dates minus 1, and then take the square root.
s = √[((1248 - 1216.667)^2 + (1187 - 1216.667)^2 + (1199 - 1216.667)^2 + (1236 - 1216.667)^2 + (1206 - 1216.667)^2 + (1241 - 1216.667)^2 + (1185 - 1216.667)^2 + (1288 - 1216.667)^2 + (1239 - 1216.667)^2) / (9 - 1)] ≈ 32.261
3. Determine the critical value from the t-distribution table for an 80% confidence level and degrees of freedom (n-1) = (9-1) = 8. The critical value for an 80% confidence level and 8 degrees of freedom is approximately 1.860. Let's denote this as "t*".
4. Calculate the margin of error (E) by multiplying the critical value (t*) by the sample standard deviation (s) divided by the square root of the sample size (n).
E = t* * (s / √n) = 1.860 * (32.261 / √9) ≈ 17.908
5. Calculate the lower and upper bounds of the confidence interval by subtracting and adding the margin of error to the sample mean.
Lower bound = x̄ - E = 1216.667 - 17.908 ≈ 1198.759
Upper bound = x̄ + E = 1216.667 + 17.908 ≈ 1234.575
Therefore, the 80% confidence interval for the mean of all tree ring dates from this archaeological site is approximately 1198.759 to 1234.575 A.D.