Business Statistics Lesson 11: Two Sample Test of Hypothesis

In this lesson, we will compare independent and dependent samples. We'll discuss the Z test, testing proportions, and using the t-test.

Hypothesis Testing Process The five-step process of hypothesis testing involves stating the null and alternate hypotheses, selecting a level of significance, identifying the test statistic (T or Z), formulating the decision rule based on critical values, and making a decision to reject or not reject the null hypothesis. The choice between using T or Z depends on whether we know the population standard deviation.

Two Sample Test Comparison In two sample tests, we compare two different samples to see if there's a difference. We use specific formulas for comparing sample means when knowing population standard deviation. For example, in a grocery store scenario where self-checkout speed is compared with standard checkout speed using known mean and standard deviation data.

To compare the means of two different samples, we compute the Z value using a specific formula and then compare it to a critical value. If the computed Z value is greater than the critical value, we reject the null hypothesis. In this case, with at least 1% chance of being wrong, it appears that one method is faster than another based on our sample.

In this chapter, we explore how to conduct a two-sample test of proportions. We consider scenarios such as comparing the proportion of male students passing a statistics class final with that of female students and examining differences in the proportion of workers calling in sick at different plants. The focus is on yes/no comparisons like pass/fail or afraid/not afraid.

In this example, we apply the five-step hypothesis testing procedure to compare proportions between younger and older women. We set up the null hypothesis that there is no difference in proportion and aim to prove an alternative with a significance level of 0.05. Using Z statistic computation, we find a computed Z value of -2.21 which leads us to reject the null hypothesis in favor of proving a difference.

Computing T for Two Samples When working with two samples, we need to pool the sample standard deviations and then compute the T value. Excel can easily help us in this process. The data analysis tool in Excel is a quick and efficient way to perform these computations.

Comparing Assembly Methods A lawnmower manufacturer has two different assembly methods, and we are measuring the time it takes to assemble using each method. We want to determine if there's a difference between these two methods.

The hypothesis testing procedure involves stating the null and alternate hypotheses, where the two-tailed test indicates a difference in procedures. A significance level of 0.1 is used, and the t-test is chosen due to unknown population standard deviations.

In a two-tailed test with a significance level of 0.1, the critical values are -1.833 and 1.833. These values were obtained by calculating the degrees of freedom (n1 + n2 - 2) and finding the level of significance for two-tailed degrees of freedom.

Computing Means Compute the means of sample one and sample two by dividing the sum of values by their respective sample sizes.

Standard Deviation Calculation Calculate the standard deviation for each sample using a formula that involves squared differences between each value and its mean, then divide by n minus 1. Once done, compute the pooled sample standard deviation using a specific formula shown on this slide.

T-Score Computation After obtaining the pooled sample standard deviations, manually calculate T-score using X bar 1 minus X bar 2 divided by square root of pooled variances times certain fractions based on both samples' sizes. In this case, we end up with a T score of negative .662 which leads to rejecting null hypothesis due to it not being more negative or positive than 1.833.

When comparing the means of two samples, it's important to distinguish between dependent and independent samples. Independent samples are not related to each other, while dependent samples involve before-and-after studies on the same sample (e.g., before and after taking a stats class). An example is appraising houses by two different companies to determine if there's a difference in their appraisals.

The process involves stating the null and alternate hypotheses, determining the significance level, finding the critical value using degrees of freedom, computing T involving paired dependent samples, comparing T with critical value to make a decision. Excel data analysis test also provides similar functionality.