Математика. Материалы курса

 

 

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

A short test for you. Question 1. Statistical hypotheses are divided into the following types… The answers are given. Choose the correct answer.

Question 2. The statistical hypothesis of comparing the sampled population means if the population variances are known 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: independent, parametric, non-parametric, random. 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 2.

Let us consider the algorithm for testing the hypothesis of equality of means with known variances.

Step one. You should select the alpha significance level. The most commonly used values are 0.01 or 1%, 0.05, which corresponds to 5%, and 0.1, which corresponds to 10 percent error.

Next, you need to formulate the hypotheses. H0 is a null hypothesis “Population means are equal.”

H1 can be of three types. The most frequently used “Population means are not equal.”

Step two. It is necessary to find the empirical value of the criterion. The formula is on the screen. Here are the sample means for normally distributed variables x1 and x2, respectively, and D1 and D2 are known population variances for x1 and x2, respectively.

Step three. You need to find the critical value of the criterion. To do this, go to the Wizard. Select Statistical and NORMSINV.

Here, we enter the value p/2+0.5 in the Probability window. Here is an example where the reliability is 0.95, i.e. the significance level is 0.05.

The last step of the algorithm is to compare the empirical and critical values of the criterion. If the empirical value is less than the critical one, the null hypothesis is accepted at the alpha significance level; otherwise we can accept the alternative hypothesis.

I would like to draw your attention to the fact that, in mathematical statistics, the rule for inferring an alternative hypothesis is stricter than it is supposed to be when solving problems in biology or medicine using the appropriate criteria for means.

The inference rule is as follows: if with a reliability of 0.95, i.e. with a significance level of 0.05, it turns out that the empirical value is greater than or equal to the critical z, it is recommended to increase the reliability, i.e. if we take, e.g., 0.99, i.e. alpha will be 0.01, and, under these conditions, we should check the criterion again.

If the same inequality is obtained in this case, we can say with 95% reliability (or with a 5% error) that the H1 hypothesis, i.e. an alternative one, can be accepted.

Let us consider the following example. As a result of measuring the height of 70 randomly selected first-year students and 80 second-year students, the following results were obtained.

The average height is known, i.e. the population mean. At the same time, estimates of the height variance were known, i.e. the population variances were known. It is necessary to find out whether it is possible to assert a significant difference in the average height values of the first- and second-year students, representing the population at the level of significance alpha=0.05, based on the results obtained.

The null hypothesis, “The difference in the average height values of the first- and second-year students is not statistically significant.” An alternative hypothesis is statistically significant.

Let us find the empirical value of the criterion by substituting the average value and the population variance in the formula. The empirical value is 13.23.

Next, we find the critical value of the criterion, which is calculated using the statistical function of the NORMSINV. Here, it is approximately equal to 1.96. Let us compare the empirical and critical values.

Since 13.23>1.96, we reject the null hypothesis at the significance level of 0.05 and can conclude that the experimentally observed difference in the average height of the first- and second-year students in the population may be significant.

It was possible to use the Analysis Package to calculate the values of this criterion. In the Data toolbar, you could select Data Analysis and the Two-Sample z-Test for the means.

We would use the Analysis Package if the data on the first- and the second- year students were given. In this case, the interval of variable 1 would be references to cells with the values of the first selection. The interval of variable two is a reference to the values of the second selection. The hypothetical average is 0.

It means that H0 initially sounds like “Population means are equal”, thus, their difference is zero. The variance of variable 1 is known, as well as the variance of variable 2. Remember to check the Tags if your data have headers. Alpha will automatically be selected as 0.05, i.e. our reasoning is 95% reliable, or you can choose your own significance level. For the output interval, select the cell where our results will be displayed.

I have some tasks for you. Task 1. Note that the variances of 5 mm² and 7 mm² are given here, i.e., in fact, the errors corresponding to the first and second machines are given. These are population variances for the first and second sampled populations.

Task 2. In this task, note that two populations of 100 objects are given, and when you enter data, take into account the repeatability of each value. I wish you would find a successful solution. Thank you for your attention.