QUESTION 1

625 The standard error of the mean (SE) is a measure of how much the sample mean is expected to vary from the true population mean. It is calculated using the formula: SE = Standard Deviation Given the information: For the initial sample: Sample mean ( X ˉ ) = 150 grams Sample standard deviation ( s ) = 30 grams Sample size ( n ) = 625 SE = 30 SE = 30 SE = 1.2 So, the standard error of the mean for the initial sample is 1.2 grams. Now, if the sample size is increased to 1,225, the standard error would be: SE new = SE new = 30 30 35 SE new = 0.8571 Calculate the confidence interval for the proportion of rail passengers in favor of the proposed new timetables, you can use the following formula: Confidence Interval = Sample Proportion ± Margin of Error The margin of error ( E ) is determined by the formula: E = Z × Where: Z is the Z-score associated with the desired confidence level. p is the sample proportion. n is the sample size. p ×(1− p ) n Sample Size 25 1225

Given information: Sample proportion ( p ^ ) = 0.55 (55% in favor) Sample size ( n ) = 400 Confidence level = 95% Find the Z-Score for a 95% Confidence Interval: For a 95% confidence interval, the Z-score is approximately 1.96. This value is commonly used for a 95% confidence level. Calculate the Margin of Error ( E ): E = 1.96 × Calculate the Confidence Interval: Confidence Interval = 0.55 ± E Let's calculate the margin of error and confidence interval: E = 1.96 × E ≈ 1.96 × E ≈ 1.96 × E ≈ 1.96 × 0.02485 E ≈ 0.0487 0.55×(1−0.55) 400 0.55×0.45 400 0.2475 400 0.00061875