Assume That Both Populations Are Normally Distributed.(a) Test Whether $\mu_1 \neq \mu_2$ At The $\alpha=0.05$ Level Of Significance For The Given Sample Data.(b) Construct A \$95\%$[/tex\] Confidence Interval About

Introduction

In statistical analysis, hypothesis testing and confidence interval construction are two fundamental techniques used to make inferences about a population based on a sample of data. In this article, we will assume that both populations are normally distributed and perform hypothesis testing and construct a confidence interval for the given sample data.

Hypothesis Testing

Null and Alternative Hypotheses

The null hypothesis, denoted as $H_0$, is a statement of no effect or no difference, while the alternative hypothesis, denoted as $H_1$, is a statement of an effect or a difference. In this case, we want to test whether the population means are equal, so the null and alternative hypotheses are:

• $H_0$: $\mu_1 = \mu_2$
• $H_1$: $\mu_1 \neq \mu_2$

Test Statistic and Critical Region

The test statistic is a numerical value that is used to determine whether the null hypothesis should be rejected. For a two-sample t-test, the test statistic is given by:

$$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$$

where $\bar{x}_1$ and $\bar{x}_2$ are the sample means, $s_p$ is the pooled standard deviation, and $n_1$ and $n_2$ are the sample sizes.

The critical region is the region of the test statistic where the null hypothesis is rejected. For a two-tailed test, the critical region is given by:

$$t < -t_{\alpha/2} \text{ or } t > t_{\alpha/2}$$

where $t_{\alpha/2}$ is the critical value from the t-distribution with $n_1 + n_2 - 2$ degrees of freedom.

P-Value and Decision Rule

The p-value is the probability of observing a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. The decision rule is to reject the null hypothesis if the p-value is less than the significance level, $\alpha$.

For this example, we will use a significance level of $\alpha = 0.05$. We will calculate the test statistic and p-value using the given sample data and then make a decision about the null hypothesis.

Sample Data

Sample	Mean	Standard Deviation	Sample Size
1	10.2	2.1	20
2	12.5	3.2	25

Test Statistic and P-Value Calculation

Using the formula for the test statistic, we get:

$$t = \frac{10.2 - 12.5}{\sqrt{\frac{1}{20} + \frac{1}{25}}} = -2.31$$

The p-value is calculated using a t-distribution table or software package. For this example, the p-value is approximately 0.023.

Decision Rule

Since the p-value (0.023) is less than the significance level (0.05), we reject the null hypothesis.

Conclusion

Based on the hypothesis testing, we have evidence to suggest that the population means are not equal.

Confidence Interval Construction

Formula for Confidence Interval

The formula for a confidence interval is given by:

$$\bar{x}_1 - \bar{x}_2 \pm t_{\alpha/2} \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$$

where $\bar{x}_1$ and $\bar{x}_2$ are the sample means, $t_{\alpha/2}$ is the critical value from the t-distribution with $n_1 + n_2 - 2$ degrees of freedom, and $n_1$ and $n_2$ are the sample sizes.

Confidence Interval Calculation

Using the formula for the confidence interval, we get:

$$10.2 - 12.5 \pm 2.09 \sqrt{\frac{1}{20} + \frac{1}{25}} = -2.3 \pm 0.63$$

The 95% confidence interval is therefore:

$$-2.93, -1.67$$

Conclusion

Based on the confidence interval, we can be 95% confident that the true difference between the population means lies between -2.93 and -1.67.

Conclusion

Introduction

In the previous article, we discussed hypothesis testing and confidence interval construction for normally distributed populations. In this article, we will answer some frequently asked questions about hypothesis testing and confidence interval construction.

Q: What is the difference between a hypothesis test and a confidence interval?

A: A hypothesis test is used to determine whether a population parameter is equal to a specified value or not, while a confidence interval is used to estimate the value of a population parameter.

Q: What is the significance level, and how is it used in hypothesis testing?

A: The significance level, denoted as $\alpha$, is the maximum probability of rejecting the null hypothesis when it is true. It is used to determine the critical region of the test statistic and to make a decision about the null hypothesis.

Q: What is the p-value, and how is it used in hypothesis testing?

A: The p-value is the probability of observing a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. It is used to determine whether the null hypothesis should be rejected.

Q: What is the difference between a one-tailed and a two-tailed test?

A: A one-tailed test is used to determine whether a population parameter is greater than or less than a specified value, while a two-tailed test is used to determine whether a population parameter is equal to a specified value.

Q: What is the formula for a confidence interval, and how is it used?

A: The formula for a confidence interval is given by:

$$\bar{x}_1 - \bar{x}_2 \pm t_{\alpha/2} \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$$

where $\bar{x}_1$ and $\bar{x}_2$ are the sample means, $t_{\alpha/2}$ is the critical value from the t-distribution with $n_1 + n_2 - 2$ degrees of freedom, and $n_1$ and $n_2$ are the sample sizes.

Q: What is the difference between a confidence interval and a prediction interval?

A: A confidence interval is used to estimate the value of a population parameter, while a prediction interval is used to predict the value of a future observation.

Q: What are some common mistakes to avoid when performing hypothesis testing and confidence interval construction?

A: Some common mistakes to avoid when performing hypothesis testing and confidence interval construction include:

• Not checking the assumptions of the test or interval
• Not using the correct formula or software
• Not interpreting the results correctly
• Not considering the sample size and variability

Conclusion

In this article, we answered some frequently asked questions about hypothesis testing and confidence interval construction. We hope that this article has been helpful in clarifying some of the common misconceptions and mistakes that can occur when performing hypothesis testing and confidence interval construction.

Additional Resources

For more information on hypothesis testing and confidence interval construction, please see the following resources:

• [1] "Hypothesis Testing and Confidence Interval Construction" by [Author]
• [2] "Statistical Analysis with R" by [Author]
• [3] "Hypothesis Testing and Confidence Interval Construction" by [Author]

Note: The above resources are fictional and for demonstration purposes only.