91Ó°ÊÓ

Consider the hypothesis test \(H_{0}: \mu_{1}=\mu_{2}\) against \(H_{1}: \mu_{1} \neq \mu_{2} .\) Suppose that sample sizes \(n_{1}=10\) and \(n_{2}=10,\) that \(\bar{x}_{1}=7.8\) and \(\bar{x}_{2}=5.6,\) and that \(s_{1}^{2}=4\) and \(s_{2}^{2}=9 .\) Assume that \(\sigma_{1}^{2}=\sigma_{2}^{2}\) and that the data are drawn from normal distributions. Use \(\alpha=0.05\) a. Test the hypothesis and find the \(P\) -value. b. Explain how the test could be conducted with a confidence interval. c. What is the power of the test in part (a) if \(\mu_{1}\) is 3 units greater than \(\mu_{2} ?\) d. Assume that sample sizes are equal. What sample size should be used to obtain \(\beta=0.05\) if \(\mu_{1}\) is 3 units greater than \(\mu_{2}\) ? Assume that \(\alpha=0.05\).

Step by step solution

01

Formulate the Hypotheses

We need to test the null hypothesis \( H_0: \mu_1 = \mu_2 \) against the alternative hypothesis \( H_1: \mu_1 eq \mu_2 \). This is a two-tailed test.

02

Calculate Pooled Variance

Since \( \sigma_1^2 = \sigma_2^2 \), the pooled variance \( s_p^2 \) is calculated as follows: \[ s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} = \frac{(10 - 1)\cdot 4 + (10 - 1)\cdot 9}{18} = \frac{36 + 81}{18} = 6.5 \]

03

Compute the Test Statistic

The test statistic \( t \) is calculated using the formula: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_p^2 \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}} = \frac{7.8 - 5.6}{\sqrt{6.5 \left( \frac{1}{10} + \frac{1}{10} \right)}} = \frac{2.2}{1.14} \approx 1.93 \]

04

Determine the Critical Value and P-value

For a two-tailed test with \( \alpha = 0.05 \) and \( df = 18 \), find the critical t-value from the t-distribution table. The critical t-value is approximately 2.101. Since \(|1.93| < 2.101\), we fail to reject \( H_0 \). The P-value for \( t = 1.93 \) is greater than 0.05, supporting this decision.

05

Test Using Confidence Interval

Construct a 95% confidence interval for \( \mu_1 - \mu_2 \) using the formula: \[ (\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \cdot \sqrt{s_p^2 \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \] Where \( t_{\alpha/2} = 2.101 \). Calculate: \[ 2.2 \pm 2.101 \times 1.14 \approx 2.2 \pm 2.393 \] This interval is \([-0.193, 4.593]\), which includes 0, so we fail to reject \( H_0 \).

06

Calculate Power of the Test

Calculate the noncentrality parameter for \( \delta = \frac{3}{1.14} \approx 2.63 \). Find power using non-central t-distribution tables or software, considering \( \beta = 0.05 \), which represents the probability of a Type II Error.

07

Determine Required Sample Size for Desired Power

Use the formula for sample size required: \[ n = \left( \frac{(z_{\alpha/2} + z_{\beta}) \times \sigma}{\mu_1 - \mu_2} \right)^2 \] Given \( \alpha = \beta = 0.05 \), \( z_{0.025} = 1.96 \), \( z_{0.05} = 1.645 \), \( \sigma^2 = 6.5 \), Solve for \( n \) for \( \mu_1 - \mu_2 = 3 \). \[ n \approx \left( \frac{(1.96 + 1.645) \times \sqrt{6.5}}{3} \right)^2 = 25.7 \] Round up to \( n = 26 \) for each sample.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pooled Variance

Pooled variance is a method used in hypothesis testing to estimate the common variance of two independent samples. It is especially useful when we assume that the variances of the two groups are equal, a condition often referred to as "homogeneity of variance". Given two samples with their respective sizes and variances, pooled variance combines them by taking into account the degrees of freedom from both samples.

This formula for pooled variance is:

• Let \( n_1 \) and \( n_2 \) be the sample sizes, and \( s_1^2 \) and \( s_2^2 \) be the sample variances.
• The formula is \[ s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} \]
• This averages the variances of the two samples, weighted by their respective degrees of freedom.

In hypothesis testing, specifically for tests comparing means, pooled variance simplifies subsequent calculations for statistical testing.

Test Statistic

The test statistic is a standardized value used in hypothesis testing to determine the likelihood of a specific observation under the null hypothesis. In the context of a two-sample t-test, the test statistic helps to decide whether there is a significant difference between the sample means.

For the two-sample t-test, the formula to calculate the test statistic \( t \) is:

• \[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_p^2 \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}} \]
• where \( \bar{x}_1 \) and \( \bar{x}_2 \) are the sample means, and \( n_1 \) and \( n_2 \) are the sample sizes.
• The denominator is the standard error, incorporating the pooled variance \( s_p^2 \).

The computed t-value can then be used to compare against a critical value from the t-distribution or to find a p-value, helping in making a decision about the null hypothesis.

Confidence Interval

A confidence interval provides a range of values which is likely to contain the population parameter, such as the difference between two means. It is centered around the sample statistic and offers an estimate of the reliability of the statistical inference.

In practice, a 95% confidence interval for the difference between two means (\( \mu_1 - \mu_2 \)) in a two-sample t-test is calculated with:

• \[ (\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \cdot \sqrt{s_p^2 \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \]
• The \( t_{\alpha/2} \) is the critical value of the t-distribution for the chosen level of confidence.

If the confidence interval includes 0, this indicates that there is no significant difference between the means at the chosen level of confidence, thus failing to reject the null hypothesis.

Two-Tailed Test

A two-tailed test is a statistical test where the critical area for rejection of the null hypothesis is on both ends of the distribution. This implies that there is concern for both positive and negative differences from the null hypothesis in comparing the means.

In hypothesis testing, a two-tailed test is used when:

• The alternative hypothesis is of the form \( H_1: \mu_1 eq \mu_2 \), where the difference can be in either direction.
• This requires considering the extreme values on either side of the distribution that are unlikely if the null hypothesis is true.
• At a 95% confidence level, the alpha \( \alpha \) is split into two (each tail has 2.5%), crucial for calculating critical values in a t-distribution.

Thus, a two-tailed test provides a broader scope for detecting significance, accommodating hypotheses where either direction of effect is possible.