To solve this hypothesis testing problem using the critical value approach, let us work through the steps systematically:

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1. Hypotheses

• Null Hypothesis ($H_0$): $\mu = 20$
• Alternative Hypothesis ($H_a$): $\mu \neq 20$ (two-tailed test)

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2. Significance Level

• $\alpha = 0.05$ (5% significance level)

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3. Test Statistic

The test statistic provided is: $t \approx 4.841$

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4. Degrees of Freedom

The degrees of freedom for a $t$-test is calculated as: $df = n - 1 = 15 - 1 = 14$

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5. Critical Value

For a two-tailed test at $\alpha = 0.05$ and $df = 14$, we find the critical values from a $t$-distribution table (or calculator):

• $t_{critical} = \pm 2.145$

This means that:

• Reject $H_0$ if $t < -2.145$ or $t > 2.145$

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6. Decision Rule

Compare the test statistic $t = 4.841$ with the critical values:

• Since $4.841 > 2.145$, the test statistic falls in the rejection region.

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7. Conclusion

Reject $H_0$. At the 5% significance level, the data provide sufficient evidence to conclude that the population mean $\mu$ is not equal to 20.

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Let me know if you'd like further clarifications or if you have any other questions!

Follow-Up Questions

1. What is the formula for the test statistic $t$?
2. How do we interpret the significance level $\alpha$?
3. What changes in the analysis if this were a one-tailed test?
4. How do degrees of freedom affect the critical value in a $t$-test?
5. How can the $p$-value approach be used for the same test?

Tip:

Always check if the sample size is large enough to assume normality when performing $t$-tests. For smaller samples, ensure the population is approximately normal.