To construct a 98% confidence interval for the mean of repeated measurements of the weight, we can use the formula:
Confidence interval = mean ± (t-value) x (standard error)
where the standard error is the standard deviation of the sample divided by the square root of the sample size, and the t-value is based on the degrees of freedom (n-1) and the desired level of confidence. Since we are given the sample mean, standard deviation, and sample size, we can plug in the values and solve for the confidence interval and margin of error.
First, we need to calculate the standard error:
Standard error = standard deviation / √n
Standard error = 0.020 gram / √40
Standard error = 0.00316 gram
Next, we need to find the t-value for a 98% confidence interval with 39 degrees of freedom. We can use a t-distribution table or calculator to find the t-value, which is approximately 2.423.
Substituting the values into the formula, we get:
Confidence interval = 10.230 ± (2.423)(0.00316)
Confidence interval = 10.230 ± 0.00766
Rounding to three decimal places, the 98% confidence interval for the mean weight is (10.222, 10.238) grams. The margin of error is half the width of the confidence interval, which is 0.00766/2 = 0.00383 grams