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Keith sampled 10 private universities in Colorado and recorded the tuition cost. Keith wants to test the hypothesis that the tuition of these 10 universities is more expensive than the national average tuition, which is [tex]\[tex]$29,056[/tex] with a population standard deviation of [tex]\$[/tex]3,339[/tex].

The formula for the [tex]z[/tex]-score is:
[tex]
z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
[/tex]

If the sample mean is [tex]\$31,650[/tex], what is the [tex]z[/tex]-score? Answers are rounded to the hundredths place.

A. 2.46
B. 0.13



Answer :

GinnyAnswer
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

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