Final answer:
To construct the 95% confidence interval for the mean weight of the boxes of raisins, calculate the sample mean and standard deviation, then use the t-distribution with n-1 degrees of freedom to find the critical t-score.
The confidence interval is computed using the formula CI = mean ± (t-score * standard deviation / √n).
Step-by-step explanation:
To construct a 95% confidence interval for the mean weight of the boxes of raisins, we first need to calculate the sample mean (μ) and the sample standard deviation (S). Then, because the distribution is approximately normal and the sample size is small (n=12), we will use the t-distribution for our calculations.
First, calculate the sample mean by adding all the sample weights and dividing by the number of observations:
μ = (21.72 + 21.75 + 21.62 + ... + 22.15) / 12
Then, calculate the sample standard deviation. As we don't have the population standard deviation, we'll use the sample standard deviation formula. We subtract the mean from each weight, square the result, sum all these squares, divide by n-1, and take the square root of the result:
S = √[Σ(xi - μ)² / (n-1)]
Next, find the t-score that corresponds to a 95% confidence interval from the t-distribution table using degrees of freedom (df = n - 1 = 11).
The 95% confidence interval for the true mean can be calculated as:
CI = μ ± (t * S / √n)
After finding the t-score and calculating S, plug these values into the formula to get the upper and lower bounds of the confidence interval.