To solve this hypothesis testing problem, let’s go through the required steps. The question requires us to determine whether there is evidence that statistics students study less than the recommended 2 hours per unit.

Given Information

1. Population mean $\mu_0 = 2$ (recommended hours).
2. Sample data: $1.2, 2, 1.1, 1, 1.8, 2.3, 0.2, 1.4, 2, 2.1, 1.2, 1.9$.
3. Population standard deviation $\sigma = 0.9$ hours.
4. Sample size $n = 12$.
5. Significance level $\alpha = 0.05$.
6. We are conducting a one-tailed test since we are testing if the mean study time is less than 2 hours.

Steps for Hypothesis Testing

Step 1: State the Hypotheses

• Null Hypothesis $H_0$: $\mu = 2$
• Alternative Hypothesis $H_1$: $\mu < 2$

Step 2: Calculate the Sample Mean

The sample mean $\bar{x}$ is calculated as follows: $\bar{x} = \frac{1.2 + 2 + 1.1 + 1 + 1.8 + 2.3 + 0.2 + 1.4 + 2 + 2.1 + 1.2 + 1.9}{12} = 1.5$

Step 3: Compute the Test Statistic

Since we know the population standard deviation $\sigma$, we use the z-test statistic formula: $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$ Substitute the values: $z = \frac{1.5 - 2}{0.9 / \sqrt{12}} = \frac{-0.5}{0.2598} \approx -1.92$

Step 4: Determine the Critical Value and p-Value

For a one-tailed test at a $\alpha = 0.05$ significance level:

• The critical z-value for $\alpha = 0.05$ is approximately $-1.645$.
• Since $-1.92 < -1.645$, we reject the null hypothesis.

The p-value for $z = -1.92$ is approximately 0.0274.

Step 5: Conclusion

Since the p-value (0.0274) is less than the significance level (0.05), we have sufficient evidence to conclude that the mean study time per unit for statistics students is less than 2 hours.

Filling in the Answers

Based on the above calculations:

1. H0: $\mu = 2$
2. H1: $\mu < 2$
3. Test Statistic: $z = -1.92$
4. p-Value: Approximately 0.0274
5. Conclusion: There is sufficient evidence to conclude that the population mean study time per unit for statistics students is less than 2 hours.

Additional Fields

• p-Value Interpretation: If the mean study time per unit is equal to 2 hours, there is approximately a 2.74% chance of observing a sample mean this low or lower purely by random chance.
• Level of significance interpretation: There’s a 5% chance of incorrectly rejecting the null hypothesis if the true mean is actually 2 hours.

Would you like any further clarification on this solution, or help with another question?

Relative Questions

1. How do you calculate the test statistic if the population standard deviation is unknown?
2. What is the importance of setting a significance level before conducting the test?
3. How does a one-tailed test differ from a two-tailed test in hypothesis testing?
4. Why is the p-value interpretation critical in understanding hypothesis test results?
5. What steps should be taken if the sample size is very small and the population standard deviation is unknown?

Tip

Always verify if the problem requires a one-tailed or two-tailed test before calculating the critical values and making conclusions.