To address the hypothesis test, let's break down each step involved.
### 1. Formulating the Hypotheses
The null and alternative hypotheses are given as:
- [tex]\( H_0: \mu = 71.2 \)[/tex]
- [tex]\( H_a: \mu < 71.2 \)[/tex]
where [tex]\(\mu\)[/tex] represents the population mean.
### 2. Significance Level
The significance level ([tex]\(\alpha\)[/tex]) is specified as 0.005.
### 3. Data Collection
We are provided with the following array of data points:
[tex]\[
\begin{array}{|r|r|r|r|r|}
\hline
48.3 & 60.9 & 78.9 & 82.6 & 55.7 \\
64.3 & 70.8 & 66.9 & 65.2 & 59.6 \\
60.3 & 62.2 & 53.3 & 68.1 & 77.4 \\
66.9 & 63.1 & 49.2 & 56.2 & 72.1 \\
74.5 & 90.2 & 73.5 & 63.1 & 68.4 \\
78.4 & 47.3 & 62.8 & 51.5 & 55.3 \\
72.1 & 79.4 & 61.9 & 64 & 58.2 \\
62.5 & 61.6 & 58.5 & 59.6 & 69 \\
66.4 & 49.2 & 56.2 & 64.9 & 77.9 \\
66.1 & 70.2 & 70.5 & 77 & 67.2 \\
73.5 & 65.2 & 71.8 & 66.9 & 69.3 \\
\hline
\end{array}
\][/tex]
### 4. Calculations
#### A. Sample Size
The sample size ([tex]\(n\)[/tex]) is 55.
#### B. Sample Mean ([tex]\(\bar{x}\)[/tex])
The mean of the sample data is calculated to be approximately 64.7818.
#### C. Sample Standard Deviation (s)
The sample standard deviation is approximately 8.1565.
#### D. Test Statistic (t)
The test statistic is calculated using the formula:
[tex]\[
t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
\][/tex]
Substituting the values:
[tex]\[
t = \frac{64.7818 - 71.2}{\frac{8.1565}{\sqrt{55}}} \approx -4.5778
\][/tex]
#### E. Degrees of Freedom
The degrees of freedom (df) is [tex]\(n - 1 = 54\)[/tex].
### 5. P-value Calculation
The P-value for the one-tailed test is found using the t-distribution with 54 degrees of freedom at the obtained t-statistic:
[tex]\[
\text{P-value} \approx 0.0
\][/tex]
### 6. Conclusion
Since the P-value (0.0) is less than the significance level (0.005), we reject the null hypothesis.
### Summary
- Test Statistic: [tex]\( t = -4.5778 \)[/tex]
- P-value: [tex]\( \text{P-value} = 0.0 \)[/tex]
- Conclusion: The P-value is less than or equal to [tex]\(\alpha\)[/tex].
The test statistic [tex]\(-4.5778\)[/tex] is accurately reported to 4 decimal places. The p-value is [tex]\(0.0\)[/tex], indicating strong evidence against the null hypothesis. Hence, we conclude that the population mean is likely less than 71.2 at the 0.005 significance level.
