Answered

You believe the population is normally distributed, but you do not know the standard deviation. You obtain the following sample of data:

\begin{tabular}{|r|r|r|r|r|}
\hline \begin{tabular}{c}
Column \\
A
\end{tabular} & \begin{tabular}{c}
Column \\
B
\end{tabular} & \begin{tabular}{c}
Column \\
C
\end{tabular} & \begin{tabular}{c}
Column \\
D
\end{tabular} & \begin{tabular}{c}
Column \\
E
\end{tabular} \\
\hline 48.3 & 60.9 & 78.9 & 82.6 & 55.7 \\
\hline 64.3 & 70.8 & 66.9 & 65.2 & 59.6 \\
\hline 60.3 & 62.2 & 53.3 & 68.1 & 77.4 \\
\hline 66.9 & 63.1 & 49.2 & 56.2 & 72.1 \\
\hline 74.5 & 90.2 & 73.5 & 63.1 & 68.4 \\
\hline 78.4 & 47.3 & 62.8 & 51.5 & 55.3 \\
\hline 72.1 & 79.4 & 61.9 & 64 & 58.2 \\
\hline 62.5 & 61.6 & 58.5 & 59.6 & 69 \\
\hline 66.4 & 49.2 & 56.2 & 64.9 & 77.9 \\
\hline 66.1 & 70.2 & 70.5 & 77 & 67.2 \\
\hline 73.5 & 65.2 & 71.8 & 66.9 & 69.3 \\
\hline
\end{tabular}

Note: To save vertical scrolling space, the data set is shown here with five columns. In Statcrunch, you will need to make sure all the data are in ONE column.

### Use Technology

#### What is the test statistic for this sample?
test statistic = [tex]$\square$[/tex] (Report answer accurate to 4 decimal places.)

#### What is the p-value for this sample?
p-value = [tex]$\square$[/tex] (Report answer accurate to 4 decimal places.)

The [tex]$p$[/tex]-value is:
- less than (or equal to) [tex]$\alpha$[/tex]
- greater than [tex]$\alpha$[/tex]

This test statistic leads to a decision to:
- reject the null
- accept the null
- fail to reject the null



Answer :

GinnyAnswer
Considering that the population is normally distributed and the standard deviation is unknown, we use the [tex]\( t \)[/tex]-test for the sample data provided.

Given the sample data:
[tex]\[48.3, 60.9, 78.9, 82.6, 55.7, 64.3, 70.8, 66.9, 65.2, 59.6, 60.3, 62.2, 53.3, 68.1, 77.4, 66.9, 63.1, 49.2, 56.2, 72.1, 74.5, 90.2, 73.5, 63.1, 68.4, 78.4, 47.3, 62.8, 51.5, 55.3, 72.1, 79.4, 61.9, 64, 58.2, 62.5, 61.6, 58.5, 59.6, 69, 66.4, 49.2, 56.2, 64.9, 77.9, 66.1, 70.2, 70.5, 77, 67.2, 73.5, 65.2, 71.8, 66.9, 69.3\][/tex]

### Step-by-Step Solution:
1. Calculate the sample mean ([tex]\(\bar{x}\)[/tex]):
Find the average value of the sample data.

2. Calculate the sample standard deviation (s):
This measures the dispersion of the data points in the sample.

3. Calculate the sample size (n):
Count the number of data points in the sample.

4. Calculate the standard error of the mean (SEM):
Use the formula [tex]\( \text{SEM} = \frac{s}{\sqrt{n}} \)[/tex].

5. Formulate the hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 0\)[/tex]
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu \neq 0\)[/tex]

6. Calculate the t-statistic:
Use the formula [tex]\( t = \frac{\bar{x} - \mu_0}{\text{SEM}} \)[/tex], where [tex]\(\mu_0\)[/tex] is the population mean under the null hypothesis. In this case, [tex]\(\mu_0 = 0\)[/tex].

7. Find the degrees of freedom (df):
[tex]\( \text{df} = n - 1 \)[/tex].

8. Determine the p-value:
Use the t-distribution with the calculated t-statistic and degrees of freedom to find the p-value.

### Solution Output:
- Test Statistic:
The test statistic calculated is [tex]\( t = 53.2690 \)[/tex].

- P-Value:
The p-value is [tex]\( 0.0000 \)[/tex].

### Comparison with Alpha ([tex]\(\alpha\)[/tex]) Level:
Given [tex]\(\alpha = 0.05\)[/tex]:
- Since the p-value [tex]\( (0.0000) \)[/tex] is less than or equal to [tex]\(\alpha \)[/tex] [tex]\( (0.05) \)[/tex], we conclude that the p-value is less than [tex]\(\alpha\)[/tex].

### Decision:
Based on the p-value,
- We reject the null hypothesis.

### Summary:
1. Test Statistic: [tex]\( t = 53.2690 \)[/tex]
2. P-Value: [tex]\( 0.0000 \)[/tex]
3. The p-value is less than [tex]\(\alpha\)[/tex].
4. This test statistic leads to a decision to reject the null hypothesis.

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