1 TIMEREMAINING A recent survey of 8,000 high school students found that the mean price of a prom dress was \( \$ 195.00 \) with a standard deviation of \( \$ 12.00 \). Alyssa thinks that her school is more fashion conscious and spent more than \( \$ 195.00 \). She collected data from 20 people in her high school and found that the average price spent on a prom dress was \( \$ 208.00 \). Which of the following is the correct \( z \)-statistic for this situation? 0.24 4.84 7.51 96.90

To calculate the \( z \)-statistic, you can use the formula: \[ z = \frac{(\bar{x} - \mu)}{(\sigma / \sqrt{n})} \] Where: - \(\bar{x} = 208.00\) (Alyssa's school's average dress price) - \(\mu = 195.00\) (Mean price from the survey) - \(\sigma = 12.00\) (Standard deviation from the survey) - \(n = 20\) (Sample size from Alyssa’s school) Now, plug the values in: 1. Calculate the standard error (SE): \[ SE = \frac{12.00}{\sqrt{20}} \approx \frac{12.00}{4.47} \approx 2.68 \] 2. Now calculate the \( z \)-statistic: \[ z = \frac{(208.00 - 195.00)}{2.68} \approx \frac{13.00}{2.68} \approx 4.85 \] So the closest answer is \( 4.84 \). Alyssa's intuition was spot on! By diving into her school's average prom dress price, she was able to compare it with broader trends, which gives her an interesting numerical insight. This kind of statistical analysis isn't just useful for fashion – it can be applied to everything from sports to food trends, helping us find outliers and make decisions based on data! Understanding the importance of sample size in statistics is critical. Alyssa only surveyed 20 students, which might not represent the entire school accurately. A larger sample size could generate more reliable data, and it’s a reminder that assumptions can lead to misleading conclusions! Always critically examine whether your sample size is adequate to support your predictions or beliefs.