Question

dion is performing a hypothesis test in which the population mean is 92 and the standard deviation is 2. his sample size is 7 with a mean of 93.5. which of the following correctly depicts the z - statistic for this data?
0.28
0.36
1.98
2.63

Explanation:

Step1: Recall z - statistic formula

The formula for the z - statistic in a hypothesis test for the mean (when the population standard deviation \(\sigma\) is known) 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 \(\mu = 92\), \(\sigma=2\), \(n = 7\), and \(\bar{x}=93.5\).

Step3: Substitute the values into the formula

First, calculate the denominator \(\frac{\sigma}{\sqrt{n}}=\frac{2}{\sqrt{7}}\approx\frac{2}{2.6458}\approx0.756\).
Then, calculate the numerator \(\bar{x}-\mu=93.5 - 92 = 1.5\).
Now, calculate the z - statistic: \(z=\frac{1.5}{\frac{2}{\sqrt{7}}}=\frac{1.5\sqrt{7}}{2}\).
\(\sqrt{7}\approx2.6458\), so \(1.5\times2.6458 = 3.9687\), and \(\frac{3.9687}{2}\approx1.98\).

Answer:

1.98