Question

destiny performs a hypothesis test on a population with μ = 115 and σ = 30. her sample size is 25 with a mean of 120.4. which of the following correctly depicts the z - statistic for this data? 0.04, 0.90, 3.53, 3.93

Explanation:

Step1: Recall z - statistic formula

The formula for the z - statistic in a hypothesis test for the sample mean 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 \(\mu = 115\), \(\sigma=30\), \(n = 25\), and \(\bar{x}=120.4\).

Step3: Calculate the standard error

First, calculate the standard error \(\frac{\sigma}{\sqrt{n}}\). Substitute \(\sigma = 30\) and \(n = 25\) into the formula: \(\frac{30}{\sqrt{25}}=\frac{30}{5}=6\).

Step4: Calculate the z - statistic

Substitute \(\bar{x}=120.4\), \(\mu = 115\), and the standard error (6) into the z - statistic formula: \(z=\frac{120.4 - 115}{6}=\frac{5.4}{6}=0.90\).

Answer:

0.90 (corresponding to the option "0.90")