Testing the Difference Between Two Means PDF

**Testing the Difference Between Two Means:\ Using the** *z* **Test\ **Suppose a researcher wishes to determine whether there is a difference in the average age of nursing students who enroll in a nursing program at a community college and those who enroll in a nursing program at a university. In th...

**Testing the Difference Between Two Means:\ Using the** *z* **Test\ **Suppose a researcher wishes to determine whether there is a difference in the average age of nursing students who enroll in a nursing program at a community college and those who enroll in a nursing program at a university. In this case, the researcher is not interested in the average age of all beginning nursing students; instead, he is interested in *comparing* the means of the two groups. His research question is, does the mean age of nursing students who enroll at a community college differ from the mean age of nursing students who enroll at a university? Here, the hypotheses are\ *H*0: u1= u2\ *H*1: u1≠ u2\ *OR* *H*0: u1- u2 = 0\ *H*1: u1- u2 =0\ If there is no difference in population means, subtracting them will give a difference of\ zero. If they are different, subtracting will give a number other than zero. Both methods\ of stating hypotheses are correct. **Assumptions for the *z* Test to Determine the Difference Between Two Means\ **1. Both samples are random samples.\ 2. The samples must be independent of each other. That is, there can be no relationship between the subjects in each sample.\ 3. The standard deviations of both populations must be known, and if the sample sizes are greater 30, the populations must be normally or approximately normally distributed. Formula general format: Test value = [\$\\frac{\\text{observed\\ value} - \\text{expected\\ value}}{\\text{SE}}\\ \\ \$]{.math.inline} **Formula for the** *z* **Test for Comparing Two Means from Independent Populations** **\ **\ **Testing the Difference Between Two Means of Independent Samples:** **Using the** *t* **Test\ **The *z* test is used to test the difference between two means when the population standard deviations were known and the variables were normally or approximately normally distributed, or when both sample sizes were greater than or equal to 30. In many situations, however, these conditions cannot be met---that is, the population standard deviations are not known. In these cases, a *t* test is used to test the difference between means when the two samples are independent and when the samples are taken from two normally or approximately normally distributed populations. Samples are **independent samples** when they are not related. Also it will be assumed that the variances are not equal.  **Assumptions for the *t* Test for Two Independent Means When 𝞼1 and 𝞼2 Are Unknown\ **1. The samples are random samples.\ 2. The sample data are independent of one another.\ 3. When the sample sizes are less than 30, the populations must be normally or approximately normally distributed. **Testing the Difference Between Two Means: Dependent Samples/Paired-Sample t-Test.** This version is used when the samples are dependent. Samples are considered to be **dependent samples\ **when the subjects are paired or matched in some way. A *t* test, using the differences between the pretest values and the posttest values. Thus only the gain or loss in values is compared. First, find the differences of the values of the pairs of data.\ *D= X*1 - *X*2\ Second, find the mean of the differences, using the formula. Mean Difference =[\$\\ \\frac{\\ \\sum D\\ }{n}\$]{.math.inline} Where *n* is the number of data pairs. Third, find the standard deviation *SD* of the differences, using the formula. SD= [\$\\frac{n\\sum D2 - \\left( D \\right)2}{n(n - 1)}\$]{.math.inline} Fourth, find the estimated standard error of the differences, which is. *SD=* [\$\\frac{\\text{SD}}{\\sqrt{}n}\$]{.math.inline} Finally, find the test value, using the formula. T= [\$\\frac{\\ D - \\ µD\\ }{sd/\\sqrt{}n}\$]{.math.inline} with degree of freedom = n-1 The formula in the final step follows the basic format of Test value = [\$\\frac{\\text{observed\\ value} - \\text{expected\\ value}}{\\text{SE}}\\ \\ \$]{.math.inline} Where the observed value is the mean of the differences. The expected value u*D* is zero if the hypothesis is u*D =* 0. The standard error of the difference is the standard deviation of the difference, divided by the square root of the sample size. Both populations must be normally or approximately normally distributed. **Assumptions for the *t* Test for Two Means When the Samples Are Dependent.\ **1. The sample or samples are random.\ 2. The sample data are dependent.\ 3. When the sample size or sample sizes are less than 30, the population or populations must be normally or approximately normally distributed. **One-Way Analysis of Variance\ **When an *F* test is used to test a hypothesis concerning the means of three or more populations, the technique is called **analysis of variance** (commonly abbreviated as **ANOVA**).\ At first glance, you might think that to compare the means of three or more samples, you can use the *t* test, comparing two means at a time. But there are several reasons why the *t* test should not be done.\ **First, when you are comparing two means at a time, the rest of the means under study are ignored. With the *F* test, all the means are compared simultaneously.** **Second, when you are comparing two means at a time and making all pair wise comparisons, the probability of rejecting the null hypothesis when it is true is increased, since the more *t* tests that are conducted, the greater is the likelihood of getting significant differences by chance alone.** **Third, the more means there are to compare, the more *t* tests are needed. For example, for the comparison of 3 means two at a time, 3 *t* tests are required. For the comparison of 5 means two at a time, 10 tests are required. And for the comparison of 10 means two at a time, 45 tests are required.** **Even though you are comparing three or more means in this use of the *F* test, *variances* are used in the test instead of means.\ **With the *F* test, two different estimates of the population variance are made. **The first estimate is called the between-group variance, and it involves finding the variance of the means.** **The second estimate, the within-group variance, is made by computing the variance using all the data and is not affected by differences in the means.** **If there is no difference in the means, the between-group variance estimate will be approximately equal to the within-group variance estimate, and the *F* test value will be approximately equal to 1. The null hypothesis will not be rejected.** **However, when the means differ significantly, the between-group variance will be much larger than the within-group variance; the *F* test value will be significantly greater than 1; and the null hypothesis will be rejected. Since variances are compared, this procedure is called *analysis of variance* (ANOVA).** For a test of the difference among three or more means, the following hypotheses\ should be used:\ *H*0: u1=u2 =.........=u*k\ H*1: At least one mean is different from the others. As stated previously, a significant test value means that there is a high probability that this difference in means is not due to chance, but it does not indicate where the difference lies. The degrees of freedom for this *F* test are d.f.N=*k* -1, where *k* is the number of groups, and d.f.D=*N* -*k*, where *N* is the sum of the sample sizes of the groups *N= n*1 + *n*2 +...........+*nk*. The sample sizes need not be equal. **The *F* test to compare means is always right-tailed.**