This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions to the questions: a) Calculate a 96% confidence interval for the mean weight of all cabbages. Step 1: Calculate the sample mean and sample standard deviation. Given: Sample size, n = 100 Sum of weights, x = 16380 Sum of squared weights, x^2 = 2,686,310 Sample mean: x = ( x)/(n) = (16380)/(100) = 163.8 Sample variance: s^2 = (1)/(n-1) ( x^2 - (( x)^2)/(n) ) s^2 = (1)/(100-1) ( 2,686,310 - ((16380)^2)/(100) ) s^2 = (1)/(99) ( 2,686,310 - (268304400)/(100) ) s^2 = (1)/(99) ( 2,686,310 - 2,683,044 ) s^2 = (3266)/(99) ≈ 33.00 Sample standard deviation: s = sqrt(33.00) ≈ 5.7446 Step 2: Determine the critical z-value. For a 96% confidence level, = 1 - 0.96 = 0.04. The critical value is z_/2 = z_0.02. Looking up the z-table for an area of 1 - 0.02 = 0.98, we find z_0.02 ≈ 2.05. Step 3: Calculate the margin of error. The margin of error (E) is given by: E = z_/2 (s)/(sqrt(n)) E = 2.05 × (5.7446)/(sqrt(100)) E = 2.05 × (5.7446)/(10) E = 2.05 × 0.57446 ≈ 1.1776 Step 4: Construct the confidence interval. The confidence interval is x ± E: 163.8 ± 1.1776 Lower bound: 163.8 - 1.1776 = 162.6224 Upper bound: