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 1236 1292 1222 1268 1316 1275 1317 1275
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.)
Upper Limit:
Lower Limit:
To find the confidence interval for the mean of all tree ring dates from this archaeological site, we can use the formula:
Confidence Interval = sample mean ± margin of error
First, we need to find the sample mean. The sample mean is simply the average of all the tree ring dates provided:
Sample mean = (1320 + 1236 + 1292 + 1222 + 1268 + 1316 + 1275 + 1317 + 1275) / 9 = 1270.11 (rounded to two decimal places)
Next, we need to find the margin of error. The formula for calculating the margin of error is:
Margin of error = critical value * (standard deviation / √n)
Since the population distribution is assumed to be approximately normal, we can use a t-distribution to find the critical value. For a 90% confidence level with a sample size of 9 (n-1), the critical value is approximately 1.83.
To find the standard deviation, we need to calculate the sample variance first:
Sample variance = Σ(x - x̄)^2 / (n - 1)
where x is each individual tree ring date, x̄ is the sample mean, and n is the sample size.
Substituting the given values into the formula:
Sample variance = [(1320 - 1270.11)^2 + (1236 - 1270.11)^2 + (1292 - 1270.11)^2 + (1222 - 1270.11)^2 + (1268 - 1270.11)^2 + (1316 - 1270.11)^2 + (1275 - 1270.11)^2 + (1317 - 1270.11)^2 + (1275 - 1270.11)^2] / (9 - 1)
Calculating the above expression:
Sample variance = 135052.22
Finally, we can calculate the standard deviation:
Standard deviation = √(sample variance) = √(135052.22) = 367.60 (rounded to two decimal places)
Now we can substitute the values into the formula to calculate the margin of error:
Margin of error = 1.83 * (367.60 / √9) = 183.84 (rounded to two decimal places)
Finally, we can find the confidence interval:
Lower limit = sample mean - margin of error = 1270.11 - 183.84 = 1086 (rounded to the nearest whole number)
Upper limit = sample mean + margin of error = 1270.11 + 183.84 = 1454 (rounded to the nearest whole number)
Therefore, the 90% confidence interval for the mean of all tree ring dates from this archaeological site is approximately 1086 to 1454.
To find the 90% confidence interval for the mean of all tree ring dates, we can use the t-distribution since the sample size is small (less than 30) and the population standard deviation is unknown.
1. Calculate the sample mean (x̄) of the tree ring dates:
- Add up all the values: 1320 + 1236 + 1292 + 1222 + 1268 + 1316 + 1275 + 1317 + 1275 = 11511
- Divide by the number of values (sample size): 11511 / 9 = 1279
2. Calculate the sample standard deviation (s) of the tree ring dates:
- Calculate the sum of squares of the differences between each value and the mean:
(1320 - 1279)^2 + (1236 - 1279)^2 + (1292 - 1279)^2 + (1222 - 1279)^2 + (1268 - 1279)^2 + (1316 - 1279)^2 + (1275 - 1279)^2 + (1317 - 1279)^2 + (1275 - 1279)^2
= 18080 + 1849 + 169 + 583 + 121 + 1296 + 16 + 144 + 16
= 23874
- Divide the sum of squares by (n - 1), where n is the sample size: 23874 / (9 - 1) = 2984.25
- Take the square root of the result: √2984.25 ≈ 54.66
3. Calculate the t-value for a 90% confidence level with (n - 1) degrees of freedom:
- Since the sample size is 9, the degrees of freedom is 8.
- You can find the t-value from the t-distribution table or use a calculator. The t-value for a 90% confidence level with 8 degrees of freedom is approximately 1.86.
4. Calculate the margin of error (E):
- E = t-value * (s / √n)
= 1.86 * (54.66 / √9)
≈ 1.86 * (54.66 / 3)
≈ 1.86 * 18.22
≈ 33.90
5. Calculate the confidence interval:
- The lower limit is the sample mean minus the margin of error: 1279 - 33.90 ≈ 1245
- The upper limit is the sample mean plus the margin of error: 1279 + 33.90 ≈ 1312
Therefore, the 90% confidence interval for the mean of all tree ring dates from this archaeological site is approximately 1245 to 1312.