To determine if the population mean ($\mu$) is different from 24 at the $\alpha = 0.01$ level of significance, we perform a hypothesis test using the t-test for a single sample.

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Step 1: State the Hypotheses

• Null Hypothesis ($H_0$): $\mu = 24$ (the population mean is 24).
• Alternative Hypothesis ($H_a$): $\mu \neq 24$ (the population mean is different from 24).

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Step 2: Calculate the Test Statistic

The formula for the test statistic is: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$ Where:

• $\bar{x} = 21.8$ (sample mean),
• $\mu_0 = 24$ (hypothesized population mean),
• $s = 6.3$ (sample standard deviation),
• $n = 15$ (sample size).

Substituting the values: $t = \frac{21.8 - 24}{6.3 / \sqrt{15}} = \frac{-2.2}{6.3 / 3.873} = \frac{-2.2}{1.626} \approx -1.35$

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Step 3: Determine the Critical Value

Degrees of freedom ($df$) for this test are: $df = n - 1 = 15 - 1 = 14$

For a two-tailed test at $\alpha = 0.01$, we look up the critical $t$-values in a $t$-distribution table. With $df = 14$: $t_{\text{critical}} = \pm 2.977$

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Step 4: Make a Decision

• The calculated $t$-statistic is $-1.35$, and it does not exceed the critical values ($-2.977$ and $+2.977$).
• Therefore, we fail to reject the null hypothesis.

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Step 5: Calculate the P-value

To find the p-value, we use the cumulative probability for $t = -1.35$ with $df = 14$. Using a t-table or software, the p-value is approximately: $P = 0.197$

Since $P > \alpha$ (0.197 > 0.01), we fail to reject the null hypothesis.

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Conclusion

At the $\alpha = 0.01$ level of significance, there is not enough evidence to conclude that the population mean is different from 24.

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Would you like more details on any part of this process? Here are some related questions to explore:

1. How is the critical $t$-value determined in a two-tailed test?
2. What happens if the sample size increases? How does it affect the $t$-value?
3. Can you explain the difference between the $p$-value and $\alpha$?
4. What assumptions are required for a valid t-test?
5. How would the test change if the population variance were known?

Tip: Always check that your data satisfies the conditions for a t-test, including normality and random sampling!