Laboratory work 2 (part 2). Comparison of mean dependent sample populations with unknown variances of general populations
Topic Testing Statistical Hypotheses of Comparing Sampled Population Means with Unknown Population Variances. The lab class: continuation.
The algorithm for dependent sampled populations. Step one. Hypotheses must be formulated. H0 – “Population means are equal”, H1 – “Population means are not equal.” You also need to select the alpha significance level.
Let me remind you that the significance level can be either 0.01, or 1% error, or 0.05, i.e. 5% error, or 0.1, i.e. 10% error.
Step two. It is necessary to find the empirical value of the criterion. In this case, we use the formula shown on the screen. Explanations are provided here.
Step three. We need to find the S value, which is an element of our formula that allows us to find the empirical value of the criterion. The formula is shown on the screen.
Step four. You need to find the critical value of the criterion. To do this, go to the Functions, select the Statistical category and the STUDENT. OBR. 2X function. Here, the probability is the level of significance, we enter the number and the degree of freedom, note that this value is k=n-1. Here, too, we introduce the degree of freedom – a number.
The last step is to compare the empirical and critical values of the criterion. The criterion is two-sided, so we take the values modulo. If this inequality takes place, we accept the null hypothesis. If the empirical value is greater than or equal to the critical modulus, an alternative hypothesis is accepted.
The Analysis Package will help you meet this criterion. To do this, go to Data in the toolbar, select Data Analysis, and the Paired two-sample t-test for averages procedure.
In the window that appears, we enter the following data. Variable interval 1 – we enter the range of values of the first sample set. In this window, we enter the range of values for the second sample set.
The hypothetical average difference corresponding to the null hypothesis is zero. You can leave the cell empty. Remember to check the Tags box if there are headers. Automatically, alpha is equal to 0.05, which means that the default error rate is 5%, and the reliability rate is 95%. Or you can change the significance level. We select the output interval: a cell or a new worksheet. Click OK.
Let us consider the following example. The level of students’ orientation to artistic values was studied. In order to activate the formation of this orientation in the given experimental group, conversations, exhibitions of children’s drawings were held, visits to museums and art galleries were organized, meetings with musicians, artists were held, and etc.
A natural question arises: what is the effectiveness of the work carried out? In order to check the effectiveness of this work, a test was given before and after the experiment. The test results are presented as a table. This table shows the sum of values and their averages.
The null hypothesis is “Experimental exposure is not effective.” An alternative hypothesis is “Experimental exposure is effective.” To calculate the necessary indicators, we use the Analysis Package.
To do this, we select Data in the toolbar, Data Analysis, and the Paired two-sample t-test for averages procedure. The interval of variable 1 is the range of our first column, column A. the interval of variable 2 is the values of the second sample set, column B.
There is a tick in the Tags as there are headers here. The significance level is 0.05. The output interval is cell D2. The table that appears allows us to see the average values of our sampled populations before the experiment and after the experiment.
We see a significant difference in the averages. Now we need to make sure that this difference is really significant, and not due to a random selection of sample values. The variance is represented by the observation of 10 students. And now we take the value from the cell t – statistics modulo, this will be our empirical value.
The critical value corresponds to the value t-critical two-sided. Let us perform the corresponding comparison. Thus, 6.678 is greater than 2.262 and at the significance level of 0.05, i.e. at the reliability level of 95%, we reject the null hypothesis and accept the alternative one, i.e. we can conclude that the experimental impact is effective.
I propose a task for you to solve. The effect of a certain drug on blood pressure before and after taking this drug was studied. Determine the reliability of the difference in average indicators. Draw the appropriate conclusions. I wish you would find a successful solution. Thank you for your attention.