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.
1320 1201 1257 1180 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.
See later post, but use ±.05 for Z score.
To find the sample mean and sample standard deviation using a calculator, follow these steps:
(a)
1. Enter the given data into the calculator.
2. Use the mean function on your calculator to calculate the sample mean (x̄).
3. Use the standard deviation function on your calculator to calculate the sample standard deviation (s).
Using the given data:
1320, 1201, 1257, 1180, 1268, 1316, 1275, 1317, 1275
Enter these values into your calculator and perform the following calculations:
Mean (x̄):
x̄ = (1320 + 1201 + 1257 + 1180 + 1268 + 1316 + 1275 + 1317 + 1275) / 9
x̄ ≈ 1261 A.D. (rounded to the nearest whole number)
Standard Deviation (s):
s = √((1/(n-1)) * Σ(xi - x̄)^2)
s ≈ √((1/8)*(59 + (-60)^2 + (-6)^2 + (-81)^2 + 7^2 + 55^2 + 14^2 + 56^2))
s ≈ √((1/8)*(59 + 3600 + 36 + 6561 + 49 + 3025 + 196 + 3136))
s ≈ √((1/8)*(16522))
s ≈ √(2065.25)
s ≈ 45.46 yr (rounded to the nearest whole number)
Therefore, the sample mean (x) is approximately 1261 A.D., and the sample standard deviation (s) is approximately 45 yr.
(b)
To find a 90% confidence interval for the mean of all tree ring dates from this archaeological site, use the following formula:
Confidence Interval = (x̄ - (z * (s/√n)), x̄ + (z * (s/√n)))
Where x̄ is the sample mean, s is the sample standard deviation, n is the sample size, and z is the z-score corresponding to the desired confidence level.
Since the sample size is 9 and we want a 90% confidence interval, the corresponding z-score is 1.645 (obtained from a z-table or calculator).
Using the values x̄ = 1261, s = 45, n = 9, and z = 1.645, we can calculate the confidence interval:
Confidence Interval = (1261 - (1.645 * (45/√9)), 1261 + (1.645 * (45/√9)))
Confidence Interval ≈ (1261 - (1.645 * 15), 1261 + (1.645 * 15))
Confidence Interval ≈ (1236, 1286) A.D.
Therefore, the 90% confidence interval for the mean of all tree ring dates from this archaeological site is approximately 1236 A.D. to 1286 A.D. (rounded to the nearest whole number).