How heavy a load (pounds) is needed to pull apart pieces of Douglas fir 4 inches long and 1.5 inches square? Here are data from students doing a laboratory exercise:

22565 28373 25826 26468 27508
23401 24998 32436 25405 29235
29679 33661 27775 28517 30371
27793 32527 32748 26052 29398

We are willing to regard the wood pieces prepared for the lab session as an SRS of all similar pieces of Douglas fir. Suppose that the strength of pieces of wood like these follows a Normal distribution with standard deviation 3100 pounds.

Give a 95% confidence interval for the mean load required to pull the wood apart (give your answer with three decimal places).



95% confidence interval for the mean load is__________ to__________ .

27,845.945 to 30,845.945

To calculate the 95% confidence interval for the mean load required to pull the wood apart, we can use the formula:

Confidence Interval = X̄ ± Z * (σ / √n)

Where:
X̄ = sample mean
Z = z-score corresponding to the desired confidence level (95% confidence level corresponds to Z = 1.96)
σ = standard deviation of the population
n = sample size

First, we need to calculate the sample mean (X̄) and sample standard deviation (s) from the given data:

Sample mean (X̄) = (22565 + 28373 + 25826 + 26468 + 27508 + 23401 + 24998 + 32436 + 25405 + 29235 + 29679 + 33661 + 27775 + 28517 + 30371 + 27793 + 32527 + 32748 + 26052 + 29398) / 20 = 28155.2

Sample standard deviation (s) = √((Σ(xi - X̄)²) / (n - 1))
where xi = individual data point

= √(((22565 - 28155.2)² + (28373 - 28155.2)² + ... + (29398 - 28155.2)²) / (20 - 1))
= √(759209330.4 / 19)
= 3102.617

Now, we can calculate the confidence interval:

Confidence Interval = 28155.2 ± 1.96 * (3102.617 / √20)
= 28155.2 ± 1.96 * 693.986
= 28155.2 ± 1360.21

The 95% confidence interval for the mean load required to pull the wood apart is 26795.99 to 29514.41 pounds (rounded to three decimal places).

Therefore, the answer is:
95% confidence interval for the mean load is 26795.99 to 29514.41.

To find the 95% confidence interval for the mean load required to pull the wood apart, we can use the formula:

Confidence interval = Sample mean ± (Critical value) * (Standard deviation / √(Sample size))

- Sample mean: The average of the given data points. (Mean of the given data)
- Critical value: The z-score corresponding to the desired confidence level. For a 95% confidence interval, the critical value is approximately 1.96.
- Standard deviation: Given as 3100 pounds.
- Sample size: The number of data points in the given data set. In this case, it is 20.

Let's calculate the confidence interval:

Calculating the sample mean:
Add up all the given data points and divide by the sample size (20).

Sample mean = (22565 + 28373 + 25826 + 26468 + 27508 + 23401 + 24998 + 32436 + 25405 + 29235 + 29679 + 33661 + 27775 + 28517 + 30371 + 27793 + 32527 + 32748 + 26052 + 29398) / 20

Sample mean ≈ 27900.15 (rounded to two decimal places)

Using the z-score table or a statistical calculator, the critical value for a 95% confidence level is approximately 1.96.

Calculating the confidence interval:

Confidence interval = 27900.15 ± (1.96) * (3100 / √(20))

Confidence interval = 27900.15 ± (1.96) * (3100 / √20)

Confidence interval = 27900.15 ± (1.96) * 691.74

Confidence interval = 27900.15 ± 1355.58

Rounded to three decimal places, the 95% confidence interval for the mean load required to pull the wood apart is approximately:

(26544.592, 29255.708)

Therefore, the confidence interval for the mean load is 26544.592 pounds to 29255.708 pounds.