Question

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

Explanation:

Step1: Recall z - statistic formula for sample mean

The formula for the z - statistic when dealing with a sample mean \(\bar{x}\) is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\), where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

Step2: Identify the values

We are given that \(\mu = 195\), \(\bar{x}=208\), \(\sigma = 12\), and \(n = 20\).

Step3: Calculate the standard error \(\frac{\sigma}{\sqrt{n}}\)

First, calculate \(\sqrt{n}=\sqrt{20}\approx4.4721\). Then, \(\frac{\sigma}{\sqrt{n}}=\frac{12}{4.4721}\approx2.6833\).

Step4: Calculate the z - statistic

Substitute the values into the z - statistic formula: \(z=\frac{208 - 195}{2.6833}=\frac{13}{2.6833}\approx4.84\).

Answer:

4.84