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Chapter 8: Problem 86

Independent random samples of \(n_{1}=50\) and \(n_{2}=60\) observations were selected from populations 1 and 2 , respectively. The sample sizes and computed sample statistics are given in the table:

Short Answer

Expert verified
Answer: The four main steps to compare the sample statistics of two independent random samples are: 1. Identifying the given information from the table, such as sample sizes, means, and variances. 2. Calculating the pooled variance using the formula: \(s_{p}^2 = \frac{(n_{1}-1)s_{1}^2 + (n_{2}-1)s_{2}^2}{n_{1}+n_{2}-2}\). 3. Calculating the standard error of the difference between means using the formula: \(SE_{\bar{x}_{1}-\bar{x}_{2}} = \sqrt{\frac{s_{1}^2}{n_{1}} + \frac{s_{2}^2}{n_{2}}}\). 4. Performing a hypothesis test for equal means using a two-sample t-test, including stating the null and alternative hypothesis, computing the test statistic, and comparing the t-value to the critical t-value.

Step by step solution

01

Identifying the given information

First, we need to identify all the given information from the table. The table provides us with sample sizes \(n_{1}=50\) and \(n_{2}=60\) and the sample statistics; however, we don't have the table here. We will assume that the table includes the means (\(\bar{x}_{1}\) and \(\bar{x}_{2}\)) and the variances (\(s_{1}^2\) and \(s_{2}^2\)) for both populations.
02

Calculating pooled variance

Now, let's compute the pooled variance. Pooled variance is an estimate of the common variance between two populations, calculated by pooling their variances and sample sizes. The formula for pooled variance is: \(pooled\, variance = s_{p}^2 = \frac{(n_{1}-1)s_{1}^2 + (n_{2}-1)s_{2}^2}{n_{1}+n_{2}-2}\) Note that you will need the variances and sample sizes from the table to compute the pooled variance.
03

Calculating standard error of the difference between means

Next, we will calculate the standard error of the difference between means. This will help us determine whether the means of the two populations are significantly different. The formula for the standard error of the difference between means is: \(SE_{\bar{x}_{1}-\bar{x}_{2}} = \sqrt{\frac{s_{1}^2}{n_{1}} + \frac{s_{2}^2}{n_{2}}}\) Plug the variances and sample sizes from the table into the formula to find the standard error.
04

Performing a hypothesis test for equal means

If we want to perform a hypothesis test to compare the means of the two populations, we can use a two-sample t-test. 1. State the null hypothesis \(H_{0}\): The population means are equal, i.e., \(\mu_{1} = \mu_{2}\). 2. State the alternative hypothesis \(H_{A}\): The population means are not equal, i.e., \(\mu_{1} \ne \mu_{2}\). Next, compute the test statistic: \(t = \frac{(\bar{x}_{1} - \bar{x}_{2}) - (\mu_{1} - \mu_{2})}{SE_{\bar{x}_{1}-\bar{x}_{2}}}\) Here, \((\mu_{1} - \mu_{2})=0\) since we assume population means are equal under the null hypothesis. Use the means and standard error calculated in the previous steps to compute the t-value. Finally, compare the calculated t-value to the critical t-value with the \((n_{1} + n_{2} - 2)\) degrees of freedom from the t-distribution table, determine the p-value, and make a decision about the null hypothesis. Keep in mind that we don't have the table or the data here, but these are the steps you would follow to perform the required analysis.

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Key Concepts

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

Pooled Variance
Pooled variance is a key concept in the two-sample t-test, particularly when comparing means from two independent samples. It represents a way of estimating the common variance between two distinct populations by combining their individual variances and sample sizes. The idea behind pooled variance is that it assumes the two populations have the same variance.

To compute the pooled variance, use the formula:
  • First, multiply each sample variance by its respective degrees of freedom (sample size minus one).
  • Add these products together.
  • Then, divide the result by the combined degrees of freedom of both samples (i.e., total of both sample sizes minus two).
From a practical perspective, the formula simplifies to:\[ s_{p}^2 = \frac{(n_{1}-1)s_{1}^2 + (n_{2}-1)s_{2}^2}{n_{1}+n_{2}-2} \]Using this formula ensures that the pooled variance provides a more precise estimate of the variance common to both groups, thus enhancing the reliability of subsequent analyses, such as the standard error calculation.
Standard Error of the Difference Between Means
The standard error of the difference between means is a crucial component in hypothesis testing, particularly in determining if two sample means can be considered significantly different. This metric reflects how much the difference in sample means is expected to vary simply due to sampling variability.

To find the standard error of the difference between means, you apply the following formula:\[ SE_{\bar{x}_{1}-\bar{x}_{2}} = \sqrt{\frac{s_{1}^2}{n_{1}} + \frac{s_{2}^2}{n_{2}}} \]Here鈥檚 how to interpret and calculate it:
  • Take each sample variance and divide it by the corresponding sample size.
  • Add these two results together.
  • Finally, calculate the square root of the sum obtained.
This provides a standard error value that helps in understanding the range within which the true difference between population means lies. The smaller the standard error, the more precise is our estimate of the difference between means, which aids in making informed decisions in hypothesis testing.
Null and Alternative Hypotheses
In the context of a two-sample t-test, formulating the null and alternative hypotheses is an essential step. These hypotheses are foundational because they guide the interpretation of statistical tests.

The **null hypothesis** (\(H_0\)) posits that there is no significant difference between the population means, suggesting that any observed difference in the sample means is due to random sampling error. Mathematically, this is represented as:\[H_0: \mu_1 = \mu_2 \]On the other hand, the **alternative hypothesis** (\(H_A\)) represents the counterclaim鈥攖hat the difference in sample means reflects a true difference in population means.

There are two forms that the alternative hypothesis can take:
  • If you are only interested in detecting any difference, you would use a two-tailed test, stated as: \(H_A: \mu_1 eq \mu_2\).
  • If your study is focused on directionality (one mean being greater or lesser), a one-tailed test might be applicable.
Ultimately, the hypotheses guide your analytical narrative. Rejecting or failing to reject the null hypothesis depends on the test statistic value compared against critical values and/or p-values, helping determine the statistical significance of your test results.

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