Chapter 5 Minitab Expression: This Activity You Will Compare

Chapter 5 Minitab Expressin This Activity You Will Compare Population

In this activity, you will compare population parameters calculated from the entire dataset with statistics calculated from samples taken from the dataset. Using the textbook dataset, find the mean, median, range, and standard deviation for public school final enrollment (C16), SAT grand total average score (C21), and public school expenditures (C18). Notice that these are population parameters. To demonstrate the sampling process, I will work with the variable Violent Crimes (C22). I will examine the descriptive statistics first (Statistics, Describe, Descriptive Statistics).

You must do each variable individually. Here is my output: Remember you can use the statistics option to change the output. Let’s get an idea of the distribution for Violent Crimes. Here is a dot plot of Violent Crimes. There are some counties with high violent crimes (Guilford, Mecklenburg, Wake).

To create a sample of 20%, use the command Data, Sample from Columns: Now I have a sample of 20% (20 rows since this dataset has 100 rows) in C79. I can use the sample to determine the means for the same variables used in the first question. For the data in C79, here are the summary statistics. The mean is a bit higher, the median is lower, and the standard error (SE) is higher (why?).

Parameters: Let’s compare this mean (396.5) with the mean from the full data set which was 339.66. The standard error of the mean for the full dataset was 69.64. So, the mean from the 20% sample falls within one standard error of the mean for the full dataset. Notice how the variability of a variable affects the sample’s accuracy. Remember that accuracy depends on sample size and confidence.

We can use the empirical rule to compare the mean of the full data set to the mean of the sample. Use the standard error to determine the approximate 95% confidence interval (± 2 standard errors). Compare the width of the interval for the full dataset and the sample. For my example: 95% confidence interval for mean from full dataset: (200.38, 478.94). 95% confidence interval for 20% sample: (136.3, 656.7). Notice how the intervals differ. Which interval is more accurate? More precise? Note: Each time you draw a sample of 20 rows, the summary statistics will be different since a different set of counties is selected.

Paper For Above instruction

The process of statistical analysis often involves understanding the differences between population parameters and sample statistics. This activity offers a comprehensive illustration of these concepts through the analysis of a dataset covering various educational and social metrics across counties. By examining the population parameters of certain variables and comparing them with sample statistics derived from a subset of the data, students can grasp the implications of sampling variability, accuracy, and precision.

In the first part of the analysis, key population parameters for variables such as public school final enrollment (C16), SAT scores (C21), and public school expenditures (C18) are calculated. These parameters include measures such as the mean, median, range, and standard deviation, which provide foundational insights into the distribution and central tendency of the data. For instance, the mean enrollment and expenditure offer a sense of typical values across all counties, while the measures of spread, like standard deviation and range, indicate the variability present.

Focusing on Violent Crimes (C22), the dataset reveals a skewed distribution with a few counties exhibiting notably high violent crime rates, such as Guilford, Mecklenburg, and Wake. Visual tools like dot plots help in visually assessing the distribution, highlighting outliers and the overall spread. To better understand sampling variability, a subset constituting 20% of the data (20 counties) is randomly selected. The statistics derived from this sample—mean, median, standard error—are then compared to those of the full dataset.

The comparison demonstrates that the sample mean for Violent Crimes is higher than the population mean, which underscores one of the fundamental principles of sampling variability: sample statistics fluctuate around true population parameters. The standard error quantifies this variability and helps in constructing confidence intervals. For example, the confidence interval derived from the full dataset (200.38, 478.94) is narrower than that from the 20% sample (136.3, 656.7), illustrating the effect of sample size on accuracy and precision.

This exercise emphasizes the importance of sample size in statistical accuracy. Larger samples tend to produce estimates closer to population parameters, reducing variability. However, smaller samples, while quicker and easier to obtain, may lead to wider confidence intervals and less reliable estimates. The empirical rule aids in understanding the expected bounds of these estimates, with approximately 95% of sample means falling within ±2 standard errors of the true mean.

Repeated sampling further introduces variability, causing different sample statistics each time, which reinforces the importance of using appropriate sample sizes and confidence levels in statistical inference. Overall, this activity underscores that understanding the relationship between population parameters and sample statistics is crucial in making valid inferences in research and data analysis.

References

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