The first step is to state the null hypothesis and an alternative hypothesis. We draw a random sample from Population \(1\) and label the sample statistics it yields with the subscript \(1\). Hypotheses concerning the relative sizes of the means of two populations are tested using the same critical value and \(p\)-value procedures that were used in the case of a single population. That is, you proceed with the p-value approach or critical value approach in the same exact way. Did you have an idea for improving this content? The samples must be independent, and each sample must be large: \(n_1\geq 30\) and \(n_2\geq 30\). If the confidence interval includes 0 we can say that there is no significant . The samples must be independent, and each sample must be large: To compare customer satisfaction levels of two competing cable television companies, \(174\) customers of Company \(1\) and \(355\) customers of Company \(2\) were randomly selected and were asked to rate their cable companies on a five-point scale, with \(1\) being least satisfied and \(5\) most satisfied. The data provide sufficient evidence, at the \(1\%\) level of significance, to conclude that the mean customer satisfaction for Company \(1\) is higher than that for Company \(2\). Refer to Example \(\PageIndex{1}\) concerning the mean satisfaction levels of customers of two competing cable television companies. The drinks should be given in random order. (In the relatively rare case that both population standard deviations \(\sigma _1\) and \(\sigma _2\) are known they would be used instead of the sample standard deviations.). Samples from two distinct populations are independent if each one is drawn without reference to the other, and has no connection with the other. Are these large samples or a normal population? . It is supposed that a new machine will pack faster on the average than the machine currently used. Assume the population variances are approximately equal and hotel rates in any given city are normally distributed. To use the methods we developed previously, we need to check the conditions. When developing an interval estimate for the difference between two population means with sample sizes of n1 and n2, n1 and n2 can be of different sizes. where and are the means of the two samples, is the hypothesized difference between the population means (0 if testing for equal means), 1 and 2 are the standard deviations of the two populations, and n 1 and n 2 are the sizes of the two samples. The following steps are used to conduct a 2-sample t-test for pooled variances in Minitab. Putting all this together gives us the following formula for the two-sample T-interval. Create a relative frequency polygon that displays the distribution of each population on the same graph. When dealing with large samples, we can use S2 to estimate 2. If so, then the following formula for a confidence interval for \(\mu _1-\mu _2\) is valid. Is this an independent sample or paired sample? If this rule of thumb is satisfied, we can assume the variances are equal. Start studying for CFA exams right away. At the beginning of each tutoring session, the children watched a short video with a religious message that ended with a promotional message for the church. How many degrees of freedom are associated with the critical value? For instance, they might want to know whether the average returns for two subsidiaries of a given company exhibit a significant difference. 95% CI for mu sophomore - mu juniors: (-0.45, 0.173), T-Test mu sophomore = mu juniors (Vs no =): T = -0.92. Example research questions: How much difference is there in average weight loss for those who diet compared to those who exercise to lose weight? We randomly select 20 couples and compare the time the husbands and wives spend watching TV. The confidence interval for the difference between two means contains all the values of (- ) (the difference between the two population means) which would not be rejected in the two-sided hypothesis test of H 0: = against H a: , i.e. follows a t-distribution with \(n_1+n_2-2\) degrees of freedom. The \(99\%\) confidence level means that \(\alpha =1-0.99=0.01\) so that \(z_{\alpha /2}=z_{0.005}\). Conducting a Hypothesis Test for the Difference in Means When two populations are related, you can compare them by analyzing the difference between their means. The following options can be given: The results of such a test may then inform decisions regarding resource allocation or the rewarding of directors. Trace metals in drinking water affect the flavor and an unusually high concentration can pose a health hazard. The results, (machine.txt), in seconds, are shown in the tables. B. larger of the two sample means. Note! This assumption is called the assumption of homogeneity of variance. ), \[Z=\frac{(\bar{x_1}-\bar{x_2})-D_0}{\sqrt{\frac{s_{1}^{2}}{n_1}+\frac{s_{2}^{2}}{n_2}}} \nonumber \]. Sort by: Top Voted Questions Tips & Thanks Want to join the conversation? To apply the formula for the confidence interval, proceed exactly as was done in Chapter 7. 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The symbols \(s_{1}^{2}\) and \(s_{2}^{2}\) denote the squares of \(s_1\) and \(s_2\). (In most problems in this section, we provided the degrees of freedom for you.). Remember, the default for the 2-sample t-test in Minitab is the non-pooled one. The following are examples to illustrate the two types of samples. Using the table or software, the value is 1.8331. \(t^*=\dfrac{\bar{x}_1-\bar{x_2}-0}{\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}}\), will have a t-distribution with degrees of freedom, \(df=\dfrac{(n_1-1)(n_2-1)}{(n_2-1)C^2+(1-C)^2(n_1-1)}\). In words, we estimate that the average customer satisfaction level for Company \(1\) is \(0.27\) points higher on this five-point scale than it is for Company \(2\). To find the interval, we need all of the pieces. Remember although the Normal Probability Plot for the differences showed no violation, we should still proceed with caution. Suppose we wish to compare the means of two distinct populations. As above, the null hypothesis tends to be that there is no difference between the means of the two populations; or, more formally, that the difference is zero (so, for example, that there is no difference between the average heights of two populations of . Wed love your input. The summary statistics are: The standard deviations are 0.520 and 0.3093 respectively; both the sample sizes are small, and the standard deviations are quite different from each other. The first three steps are identical to those in Example \(\PageIndex{2}\). The test statistic used is: $$ Z=\frac { { \bar { x } }_{ 1 }-{ \bar { x } }_{ 2 } }{ \sqrt { \left( \frac { { \sigma }_{ 1 }^{ 2 } }{ { n }_{ 1 } } +\frac { { \sigma }_{ 2 }^{ 2 } }{ { n }_{ 2 } } \right) } } $$. 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