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
Step 1: Identify the given values and the appropriate formula. We are given: • Sample mean (x) = 6.5 visits • Sample standard deviation (s) = 1.8 visits • Sample size (n) = 14 women • Confidence level = 95% Since the sample size is small (n < 30) and the population standard deviation is unknown, we use the t-distribution to calculate the confidence interval for the population mean (). The formula is: x ± t_/2, df · (s)/(sqrt(n)) Step 2: Determine the degrees of freedom and the critical t-value. The degrees of freedom (df) are n - 1. df = 14 - 1 = 13 For a 95% confidence interval, = 1 - 0.95 = 0.05. So, /2 = 0.025. Using a t-distribution table for df = 13 and /2 = 0.025, the critical t-value (t_0.025, 13) is approximately 2.160. Step 3: Calculate the standard error of the mean. The standard error (SE) is (s)/(sqrt(n)). SE = (1.8)/(sqrt(14)) SE ≈ (1.8)/(3.741657) SE ≈ 0.4811 Step 4: Calculate the margin of error. The margin of error (ME) is t_/2, df · SE. ME = 2.160 · 0.4811 ME ≈ 1.039176 Step 5: Calculate the confidence interval. The confidence interval is x ± ME. Lower bound: 6.5 - 1.039176 ≈ 5.460824 Upper bound: 6.5 + 1.039176 ≈ 7.539176 Rounding to two decimal places, the 95% confidence interval is: (5.46, 7.54) The 95% confidence interval for the population mean number of visits is (5.46, 7.54) visits. Send me the next one 📸