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Answer :
The p-value is approximately 0.018, indicating significant evidence to reject the null hypothesis of an average age under 18.
To test whether the average age of AP Statistics students in Drew's high school is under 18, we can perform a one-sample t-test.
1. Calculate the sample mean and sample standard deviation of the ages.
2. Use the following formula to calculate the t-statistic:
[tex]\[ t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \][/tex]
Where:
[tex]- \(\bar{x}\)[/tex] is the sample mean
[tex]- \(\mu\)[/tex] is the hypothesized population mean (18 in this case)
[tex]- \(s\)[/tex] is the sample standard deviation
[tex]- \(n\)[/tex] is the sample size
4. Use the T.DIST.2T function in Excel to calculate the p-value. The syntax is: `T.DIST.2T(t, df)`, where `t` is the t-statistic and `df` is the degrees of freedom (which is [tex]\(n - 1\)[/tex] for a one-sample t-test).
Let's perform these steps:
1. Calculate the sample mean [tex](\(\bar{x}\)):[/tex]
[tex]\[\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i\][/tex]
2. Calculate the sample standard deviation[tex](\(s\)).[/tex]
3. Calculate the t-statistic:
[tex]\[t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}\][/tex]
Given:
[tex]- \(n = 20\)[/tex]
[tex]- \(\mu = 18\)[/tex]
- Population standard deviation [tex](\(\sigma\))[/tex] is given as [tex]\(1.23\)[/tex]
Let's calculate:
[tex]\[ \bar{x} = \frac{1}{20} (17 + 19 + 18 + \ldots + 18 + 15) \][/tex]
[tex]\[ \bar{x} = 17.6 \][/tex]
[tex]\[ s = 1.716 \][/tex] (approximated from the provided data)
[tex]\[ t = \frac{17.6 - 18}{\frac{1.716}{\sqrt{20}}} \][/tex]
t ≈ -2.251
Now, we use the T.DIST.2T function in Excel to find the p-value. The degrees of freedom [tex](\(df\))[/tex] for a one-sample t-test is [tex]\(n - 1 = 20 - 1 = 19\).[/tex]
Entering `=T.DIST.2T(-2.251, 19)` into an Excel cell gives the p-value.
The p-value is approximately [tex]\(0.018\).[/tex] Therefore, rounded to three decimal places, the p-value is [tex]\(0.018\)[/tex]. This indicates that there is significant evidence to reject the null hypothesis that the true average age of AP Statistics students in Drew's high school is not under 18.
Complete Question:
Drew wondered if the average age of students in AP Statistics classes in his high school is under 18. He randomly selected 20 AP Statistics students in his school, and the data set is given below. Using the information from a previous survey, he has concluded that the population standard deviation for the age of AP Statistics students in his school is 1.23. Assume that the ages of students in AP Statistics classes in this high school are normally distributed. Use Excel to test whether the true age of AP Statistics students in his school is under 18. Identify the p-value, rounded to three decimal places.
Student Ages:
17
19
18
17
15
18
16
17
16
17
18
19
18
18
16
18
18
16
18
15
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