Test Statistic and Critical Region

Oct 13, 2025 by windailys.com 94 views

Introduction

In statistics, hypothesis testing and confidence intervals are two fundamental concepts used to make inferences about a population based on a sample of data. When dealing with two normal populations, it is often necessary to compare their means to determine if there is a significant difference between them. In this article, we will discuss how to test whether the means of two normal populations are equal or not, and how to construct a confidence interval to estimate the difference between the means.

Assumptions and Notations

Before we proceed, let's make some assumptions and notations:

We assume that both populations are normally distributed.
We denote the population means as $$\mu_1$$ and $$\mu_2$$.
We denote the sample means as $$\bar{x}_1$$ and $$\bar{x}_2$$.
We denote the sample standard deviations as $$s_1$$ and $$s_2$$.
We denote the sample sizes as $$n_1$$ and $$n_2$$.
We denote the level of significance as $$\alpha$$.

Hypothesis Testing

The null hypothesis is that the means of the two populations are equal, i.e., $$H_0: \mu_1 = \mu_2$$. The alternative hypothesis is that the means of the two populations are not equal, i.e., $$H_1: \mu_1 \neq \mu_2$$. We will use the two-sample t-test to test this hypothesis.

Two-Sample t-Test

The two-sample t-test is a statistical test used to compare the means of two independent samples. The test statistic is given by:

t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}

where $$\bar{x}_1$$ and $$\bar{x}_2$$ are the sample means, $$s_1^2$$ and $$s_2^2$$ are the sample variances, and $$n_1$$ and $$n_2$$ are the sample sizes.

Critical Region

The critical region is the region of the t-distribution where the null hypothesis is rejected. The critical region is determined by the level of significance $$\alpha$$ and the degrees of freedom $$n_1 + n_2 - 2$$.

Test Statistic and Critical Region

Suppose we have the following sample data:

$$x_1$$	$$x_2$$
10	15
12	18
11	16
13	19
14	20

The sample means are $$\bar{x}_1 = 12$$ and $$\bar{x}_2 = 17$$. The sample standard deviations are $$s_1 = 1.41$$ and $$s_2 = 1.73$$. The sample sizes are $$n_1 = 5$$ and $$n_2 = 5$$.

The test statistic is:

t = \frac{12 - 17}{\sqrt{\frac{1.41^2}{5} + \frac{1.73^2}{5}}} = -2.45

The degrees of freedom are $$n_1 + n_2 - 2 = 8$$.

The critical region is determined by the level of significance $$\alpha = 0.01$$ and the degrees of freedom $$n_1 + n_2 - 2 = 8$$. The critical value is $$t_{0.01,8} = -2.896$$.

Since the test statistic $$t = -2.45$$ is less than the critical value $$t_{0.01,8} = -2.896$$, we reject the null hypothesis.

Confidence Interval

A confidence interval is a range of values within which we expect the true population parameter to lie. In this case, we want to estimate the difference between the means of the two populations.

Formula for Confidence Interval

The formula for the confidence interval is:

\bar{x}_1 - \bar{x}_2 \pm t_{\alpha/2, n_1 + n_2 - 2} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}

where $$\bar{x}_1$$ and $$\bar{x}2$$ are the sample means, $$s_1^2$$ and $$s_2^2$$ are the sample variances, $$n_1$$ and $$n_2$$ are the sample sizes, and $$t{\alpha/2, n_1 + n_2 - 2}$$ is the critical value from the t-distribution.

Confidence Interval for the Given Sample Data

Suppose we have the same sample data as before:

$$x_1$$	$$x_2$$
10	15
12	18
11	16
13	19
14	20

The sample means are $$\bar{x}_1 = 12$$ and $$\bar{x}_2 = 17$$. The sample standard deviations are $$s_1 = 1.41$$ and $$s_2 = 1.73$$. The sample sizes are $$n_1 = 5$$ and $$n_2 = 5$$.

The confidence interval is:

12 - 17 \pm 2.896 \sqrt{\frac{1.41^2}{5} + \frac{1.73^2}{5}} = -5 \pm 3.45

The 99% confidence interval is $$(-8.45, -1.55)$$.

Conclusion

In this article, we discussed how to test whether the means of two normal populations are equal or not, and how to construct a confidence interval to estimate the difference between the means. We used the two-sample t-test to test the hypothesis and constructed a 99% confidence interval for the given sample data. The results showed that the null hypothesis was rejected, indicating that the means of the two populations are not equal. The 99% confidence interval was also constructed, which provided a range of values within which we expect the true population parameter to lie.

Q1: What is the purpose of hypothesis testing in statistics?

A1: The purpose of hypothesis testing in statistics is to make inferences about a population based on a sample of data. It involves testing a hypothesis about a population parameter, such as the mean or proportion, to determine if it is true or false.

Q2: What is the difference between a null hypothesis and an alternative hypothesis?

A2: A null hypothesis is a statement of no effect or no difference, while an alternative hypothesis is a statement of an effect or a difference. In the context of hypothesis testing, the null hypothesis is typically denoted as H0, while the alternative hypothesis is denoted as H1.

Q3: What is the significance level (α) in hypothesis testing?

A3: The significance level (α) is the maximum probability of rejecting the null hypothesis when it is true. It is a threshold value that determines the critical region of the test. Common values of α include 0.05 and 0.01.

Q4: What is the critical region in hypothesis testing?

A4: The critical region is the region of the test statistic where the null hypothesis is rejected. It is determined by the significance level (α) and the degrees of freedom of the test.

Q5: What is the difference between a one-tailed and a two-tailed test?

A5: A one-tailed test is a test where the alternative hypothesis is directional, i.e., it specifies the direction of the effect. A two-tailed test is a test where the alternative hypothesis is non-directional, i.e., it does not specify the direction of the effect.

Q6: What is the formula for the two-sample t-test?

A6: The formula for the two-sample t-test is:

t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}

where $$\bar{x}_1$$ and $$\bar{x}_2$$ are the sample means, $$s_1^2$$ and $$s_2^2$$ are the sample variances, and $$n_1$$ and $$n_2$$ are the sample sizes.

Q7: What is the formula for the confidence interval?

A7: The formula for the confidence interval is:

\bar{x}_1 - \bar{x}_2 \pm t_{\alpha/2, n_1 + n_2 - 2} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}

Q8: What is the difference between a 95% confidence interval and a 99% confidence interval?

A8: A 95% confidence interval is a range of values within which we expect the true population parameter to lie with 95% confidence. A 99% confidence interval is a range of values within which we expect the true population parameter to lie with 99% confidence.

Q9: How do I choose the significance level (α) for my hypothesis test?

A9: The choice of significance level (α) depends on the research question and the level of precision desired. Common values of α include 0.05 and 0.01.

Q10: What is the importance of hypothesis testing in real-world applications?

A10: Hypothesis testing is an essential tool in many real-world applications, including medicine, engineering, economics, and social sciences. It helps researchers and practitioners to make informed decisions based on data and to evaluate the effectiveness of interventions or treatments.

Conclusion

In this article, we have discussed frequently asked questions (FAQs) on hypothesis testing and confidence interval for two normal populations. We have covered topics such as the purpose of hypothesis testing, the difference between a null hypothesis and an alternative hypothesis, the significance level (α), and the critical region. We have also provided formulas for the two-sample t-test and the confidence interval, and discussed the importance of hypothesis testing in real-world applications.