To find the [tex]\(z\)[/tex]-score, we will follow a series of steps using the provided formula:
[tex]\[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \][/tex]
where:
- [tex]\(\bar{x}\)[/tex] is the sample mean.
- [tex]\(\mu\)[/tex] is the population mean.
- [tex]\(\sigma\)[/tex] is the population standard deviation.
- [tex]\(n\)[/tex] is the sample size.
Given:
- The sample mean [tex]\(\bar{x} = \$31,650\)[/tex]
- The population mean [tex]\(\mu = \$29,056\)[/tex]
- The population standard deviation [tex]\(\sigma = \$3,339\)[/tex]
- The sample size [tex]\(n = 10\)[/tex]
1. Calculate the standard error of the mean (SEM), which is the standard deviation of the sample mean:
[tex]\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} \][/tex]
Substitute the given values into the equation:
[tex]\[ \text{SEM} = \frac{3339}{\sqrt{10}} \][/tex]
2. Next, calculate the difference between the sample mean and the population mean:
[tex]\[ \bar{x} - \mu = 31650 - 29056 \][/tex]
3. Now, calculate the [tex]\(z\)[/tex]-score using the formula:
[tex]\[ z = \frac{\bar{x} - \mu}{\text{SEM}} \][/tex]
Putting it all together:
[tex]\[ z = \frac{31650 - 29056}{\frac{3339}{\sqrt{10}}} \][/tex]
4. When you perform these calculations, the result is:
[tex]\[ z \approx 2.46 \][/tex]
Hence, the correct answer is:
a.) 2.46
