Assume That Both Populations Are Normally Distributed.a) Test Whether $\mu_1 \neq \mu_2$ At The $\alpha=0.01$ Level Of Significance For The Given Sample Data.b) Construct A \$99\%$[/tex\] Confidence Interval About

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Hypothesis Testing and Confidence Interval Construction for Normally Distributed Populations

In statistical analysis, hypothesis testing and confidence interval construction are two essential tools 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 confidence interval construction for the given sample data.

Null and Alternative Hypotheses

The null hypothesis is a statement of no effect or no difference, while the alternative hypothesis is a statement of an effect or a difference. In this case, we want to test whether the mean of population 1 is not equal to the mean of population 2. Therefore, the null and alternative hypotheses are:

  • H0: μ1 = μ2 (The means of population 1 and population 2 are equal)
  • H1: μ1 ≠ μ2 (The means of population 1 and population 2 are not equal)

Level of Significance

The level of significance, denoted by α, is the maximum probability of rejecting the null hypothesis when it is true. In this case, we are given α = 0.01, which means that there is only a 1% chance of rejecting the null hypothesis when it is true.

Test Statistic

The test statistic is a numerical value that is used to determine whether the null hypothesis should be rejected. For normally distributed populations, the test statistic is given by:

  • t = (xÌ„1 - xÌ„2) / sqrt((s1^2 / n1) + (s2^2 / n2))

where x̄1 and x̄2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

Critical Region

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

  • t < -tα/2 or t > tα/2

where tα/2 is the critical value from the t-distribution with n1 + n2 - 2 degrees of freedom.

Test Results

To perform the hypothesis test, we need to calculate the test statistic and determine whether it falls within the critical region. If it does, we reject the null hypothesis; otherwise, we fail to reject it.

Example

Suppose we have two samples of data:

  • Sample 1: xÌ„1 = 10, s1 = 2, n1 = 10
  • Sample 2: xÌ„2 = 12, s2 = 3, n2 = 15

We want to test whether the mean of population 1 is not equal to the mean of population 2 at the α = 0.01 level of significance.

First, we calculate the test statistic:

  • t = (10 - 12) / sqrt((2^2 / 10) + (3^2 / 15)) = -2 / sqrt(0.4 + 0.6) = -2 / sqrt(1) = -2

Next, we determine the critical value from the t-distribution with 10 + 15 - 2 = 23 degrees of freedom:

  • tα/2 = 2.069

Since the test statistic t = -2 does not fall within the critical region (-2.069, 2.069), we fail to reject the null hypothesis.

Conclusion

In this article, we performed hypothesis testing and confidence interval construction for normally distributed populations. We assumed that both populations are normally distributed and tested whether the mean of population 1 is not equal to the mean of population 2 at the α = 0.01 level of significance. We also constructed a 99% confidence interval about the difference between the means of the two populations.

Confidence Interval Formula

The confidence interval for the difference between the means of two normally distributed populations is given by:

  • CI = (xÌ„1 - xÌ„2) ± tα/2 * sqrt((s1^2 / n1) + (s2^2 / n2))

where x̄1 and x̄2 are the sample means, s1 and s2 are the sample standard deviations, n1 and n2 are the sample sizes, and tα/2 is the critical value from the t-distribution with n1 + n2 - 2 degrees of freedom.

Example

Suppose we have two samples of data:

  • Sample 1: xÌ„1 = 10, s1 = 2, n1 = 10
  • Sample 2: xÌ„2 = 12, s2 = 3, n2 = 15

We want to construct a 99% confidence interval about the difference between the means of the two populations.

First, we calculate the margin of error:

  • ME = tα/2 * sqrt((s1^2 / n1) + (s2^2 / n2)) = 2.695 * sqrt(0.4 + 0.6) = 2.695 * sqrt(1) = 2.695

Next, we calculate the confidence interval:

  • CI = (10 - 12) ± 2.695 = -2 ± 2.695 = (-4.695, 0.695)

Therefore, we are 99% confident that the difference between the means of the two populations lies between -4.695 and 0.695.

Conclusion

In this article, we performed hypothesis testing and confidence interval construction for normally distributed populations. We assumed that both populations are normally distributed and tested whether the mean of population 1 is not equal to the mean of population 2 at the α = 0.01 level of significance. We also constructed a 99% confidence interval about the difference between the means of the two populations.
Frequently Asked Questions (FAQs) on Hypothesis Testing and Confidence Interval Construction

In the previous article, we discussed hypothesis testing and confidence interval construction for normally distributed populations. In this article, we will address some frequently asked questions (FAQs) related to these topics.

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

A: A hypothesis test is used to determine whether a statement about a population parameter is true or false, while a confidence interval is used to estimate a population parameter with a certain level of accuracy.

Q: What is the purpose of a null hypothesis?

A: The purpose of a null hypothesis is to provide a statement of no effect or no difference, which serves as a basis for testing the alternative hypothesis.

Q: What is the level of significance (α) and how is it used in hypothesis testing?

A: The level of significance (α) is the maximum probability of rejecting the null hypothesis when it is true. It is used to determine the critical region for the test statistic and to decide whether to reject the null hypothesis.

Q: What is the test statistic and how is it calculated?

A: The test statistic is a numerical value that is used to determine whether the null hypothesis should be rejected. It is calculated using the sample means, sample standard deviations, and sample sizes.

Q: What is the critical region and how is it used in hypothesis testing?

A: The critical region is the range of values of the test statistic for which the null hypothesis is rejected. It is used to determine whether the test statistic falls within the critical region and whether to reject the null hypothesis.

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

A: A one-tailed test is used to test a hypothesis about a population parameter in one direction (e.g., whether the mean is greater than a certain value), while a two-tailed test is used to test a hypothesis about a population parameter in both directions (e.g., whether the mean is greater than or less than a certain value).

Q: How is the confidence interval constructed?

A: The confidence interval is constructed by calculating the margin of error and adding or subtracting it from the point estimate of the population parameter.

Q: What is the purpose of a confidence interval?

A: The purpose of a confidence interval is to provide a range of values within which the population parameter is likely to lie with a certain level of accuracy.

Q: How is the margin of error calculated?

A: The margin of error is calculated using the critical value from the t-distribution, the sample standard deviations, and the sample sizes.

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

A: A 95% confidence interval is constructed using a critical value from the t-distribution with 95% confidence, while a 99% confidence interval is constructed using a critical value from the t-distribution with 99% confidence.

In this article, we addressed some frequently asked questions (FAQs) related to hypothesis testing and confidence interval construction. We hope that this article has provided a better understanding of these topics and has helped to clarify any confusion.

For further information on hypothesis testing and confidence interval construction, we recommend the following resources:

  • Statistical textbooks: There are many excellent statistical textbooks available that provide a comprehensive introduction to hypothesis testing and confidence interval construction.
  • Online resources: There are many online resources available that provide tutorials, examples, and practice problems on hypothesis testing and confidence interval construction.
  • Statistical software: There are many statistical software packages available that can be used to perform hypothesis testing and confidence interval construction, such as R, Python, and SPSS.
  • Hogg, R. V., & Tanis, E. A. (2001). Probability and statistical inference. Prentice Hall.
  • Moore, D. S., & McCabe, G. P. (2005). Introduction to the practice of statistics. W.H. Freeman and Company.
  • Rosner, B. (2010). Fundamentals of biostatistics. Cengage Learning.