Математика. Материалы курса

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Section Testing Statistical Hypotheses. Topic Comparison of Sampled Population Means with Unknown Population Variances.

A short test for you. Question 1. Statistical criteria are divided into … Possible answers: parametric and non-parametric, dependent and independent, conditional and unconditional, random and non-random. Choose the correct answer

Question 2. The statistical hypothesis of comparing the sampled population means (if the population variances are unknown) is… Possible answers: non-parametric, zero, parametric, alternative. Choose the correct answer.

Question 3. Statistical hypotheses of the values of parameters of a known distribution of a random variable are called … Answer options: non-random, random, non-parametric, parametric. Choose the correct answer.

Let us check your answers. Question 1. The correct answer is 1. Question 2. The correct answer is 3. Question 3. The correct answer is 4.

Let us consider the algorithm for independent sample populations. Here, the criterion is applicable for small independent sample populations if there is reason to consider the population variances equal.

Step one. It is necessary to formulate hypotheses and choose the level of significance. The null hypothesis is “Population means are equal”, the alternative hypothesis is “Population means are not equal.”

Step two. It is necessary to find the empirical value of the criterion. The formula is shown on the screen. Explanations are provided here.

Step three. You need to find the critical value of the Student’s criterion using the statistical function STUDENT. OBR. 2X. Here, note that the probability is the significance level. The level of significance can be any level, and it is chosen by the researcher. This can be 0.05 (most often). It can be 0.01, it can be 0.1, and etc. This value will be shown here. Please note that the degree of freedom is calculated using this formula, and the resulting number is put in the corresponding cell.

Step four. It is necessary to compare the empirical and critical value of the criterion. Here, we take into account that the criterion is two-sided. If the empirical value is less than the critical one, the null hypothesis is accepted. Otherwise, an alternative hypothesis is accepted.

Let us consider the following example. When measuring the pulse of 10 patients in the experimental group and 12 patients in the control group, the following results were obtained. The average is 70 and 68 beats per minute, respectively. The variance estimates are 9 and 4 beats per square minute, respectively. To determine whether the average pulse values in patients of both groups differ significantly for the significance level of alpha = 0.05.

The null hypothesis indicates that the average pulse values are equal. An alternative hypothesis suggests that the average pulse values are not equal. The empirical value of the criterion is found by the formula. The average value and estimates of the variance of the sample volume values are substituted in this formula. The empirical value is 1.87. Then, we find the critical value of the criterion. To do this, we use the statistical function STUDENT. OBR. 2X.

Here, the probability is 0.05 (this is the significance level), and the degree of freedom is the value k. This is the total number of patients (10 – in the first group, 12 – in the second one), we subtract 2 from it. The critical value is approximately 2.09.

Now we will compare the empirical and critical values. 1.87<2.09. At the significance level of 0.05, the null hypothesis is accepted. The conclusion is that the difference in average heart rate values is not statistically significant and may be due to random causes, and not to the impact of the manipulations.

Keep in mind that you can use the Analysis Package to solve tasks based on the corresponding criteria. To do this, find the Data in the toolbar, select Data Analysis and the two-sample t-test procedure with the same variances. In the interval of variable 1, we enter the range of sample values of the first population. In the interval of variable 2, we add a range of sample values of the second population. The hypothetical mean here is 0, which corresponds to the null hypothesis.

Remember to check the Tags if you have headres. Alpha is automatically selected as 0.05, i.e. the reasoning is 95% reliable, or you can add your own level of significance. The output interval is the cell where the results are displayed.

This is how the window where the corresponding values are entered looks like. Here you enter the range of the first sample set. Here is the range of values for the second sample set. Here we can do nothing, in this case it will automatically be considered that 0 was entered. Check the Tags if there are headers. And here, if we do not add anything, the significance level will automatically be 0.05, or you can change it, and you enter the output interval yourselves. Remember to click OK.