Laboratory work 1 (video). Statistical estimators of the parameters of the population
Section "Fundamentals of mathematical statistics". Topic "Statistical estimators of the parameters of the population»
We offer you a mini-test.
Question 1. The statistical estimators of the parameters of the general population are ...
a) point and interval;
b) dependent and independent;
c) random and non-random;
d) paired and multiple.
Choose the correct answer.
Question 2. The point estimator of a parameter has the following properties...
a) unbiased;
b) consistency;
c) efficiency;
d) all of the above.
Choose the correct answer.
Question 3. The interval estimator of the parameter is....
a) sample average;
b) the sample dispersion;
c) confidence interval;
d) sample mean square deviation.
Choose the correct answer, in your opinion.
Let's check your answers:
Question 1 – the correct answer is under the number 1; question 2 – the correct answer is 4; question 3 – the correct answer is 3.
Point statistical estimators can be found using Microsoft Excel and statistical functions: to find the average we use AVERAGE, to find the sample variance we use VAR.S and to find the mean square deviation of the standard slope we can use STDEVA.
You can also use statistical functions to find the mode — this is the MODE.SNGL function. To find the median of the sample population we use the MEDIAN function and QUARTILE.INC is used to find the quartile.
Using statistical functions, we can find the kurtosis (KURT) and we can find the skewness (SKEW).
We can also find confidence intervals in Microsoft Excel. To do this, we can use the statistical functions – CONFIDENCE.NORM or CONFIDENCE.T, depending on what interval we have, and what data we know. If we know the standard deviation of the general population, we are going find the confidence interval using the CONFIDENCE.NORM function. However, if we do not know the standard deviation for the general population, then we will find the confidence interval for the average value using the statistical function CONFIDENCE.T. In addition, in Microsoft Excel, we can find all these estimates using the analysis package. To do this, we need to select Data, Data Analysis and select the Descriptive statistics procedure at the toolbar.
Let us consider the problem in Microsoft Excel. We have the data (see the video). The x value is the first column. Now we are going to calculate the statistical functions using the corresponding functions, in order to find later the average variance, deviation, mode, median, kurtosis, skewness, and confidence interval. We are also to find the corresponding estimates using the analysis package.
So, let's find the average value. To do this, we use the Statistical Functions and select the AVERAGE function. We can select the entire data column here, and the value found is the average value for our sample population.
Let us find the variance for our sample population. To do this, we use the Statistical category and select the VAR.S function. We are also to select the data range.
Then, we need to find the mean-square deviation. To do this, we use the functions, the Statistical category and select the STDEVA. We enter our data and get the value of the mean square deviation in the corresponding cell.
Mode. In the Statistical category, we find the MODE.SNGL function. We also enter the data range. There is no mode here, since all values occur once, so there is no mode here.
Next, we find the median. We use the functions, the Statistical category, and select the MEDIAN function. To do this, we enter the data range and get the value 16.
Let's count the kurtosis. We use the functions, the Statistical category and select the KUR function. We enter the data range and get the value.
Next, we are going to find the asymmetry. To do this, we use the functions, the Statistical category, select the SKEW function, enter the data range and get the corresponding value of the skewness of cell C8.
Let us calculate the confidence interval. Since the mean-square deviation of the general population is unknown, we are going to calculate Student's confidence interval for the average value. To do this, we use the functions, the Statistical category and select CONFIDENCE.T.
We choose the significance level of 0.05, corresponding to five percent of the error. We have calculated the standard deviation here, and we enter it. Accordingly, the sample size is the number of objects. In our case, the number of objects is 7, that is, we can manually enter this, and we can also enter it using the function that searches for the sample size – this is the COUNT function, the Statistical category. We enter the data range, and our value is obviously 7.
Now we are going to build a data table and I suggest using the data package for this purpose. We select Data, Data Analysis and select Descriptive Statistics here. The input interval is the range of the first column; we tick the Labels box, since we have the title of this column. Do not forget to tick the box where there is the final statistics and the reliability level. We select the output interval in this cell accordingly. The columns present the values. Thus, we leave it as it is. Now we have the table. If we look closely, we see that we have identical data for the medium, the variance, the kurtosis, and for the asymmetry. In addition to the minimum, the maximum is calculated as well. The sum is automatically calculated, the amount, and the last value here is the confidence interval.
The standard error is the following. The standard error indicates an error in the sample average.
Thus, we conclude that it is more appropriate to calculate point and interval estimates using the Analysis Package. Accordingly, you can evaluate the resulting table. Note what confidence interval for the average value we have. The average value is 17. Accordingly, we add to 17 what we have in the reliability level – 4.27, that is, the confidence interval is 17±4.27. This can be written in another way (17 – 4.27; 17 + 4.27). If we calculate this, we get (12.73; 21.27). This is the confidence interval for our task. In other words, the general average with 95% reliability is in the interval from 12.73 to 21.27.
I offer tasks for solving on your own (see the video).
I wish you success in solving the tasks! Thank you for your attention.