We Recall Some Definitions That Will Be Helpful In

We Recall Some Definitions That Will Be Helpful In

A population parameter is a single value that describes a population characteristic (such as center, spread, location etc.)

Examples:

• The proportion p of adults in the United States who worry about money
• The mean lifetime m of a certain brand of computer hard disks
• The lower quartile q of a population of incomes
• The standard deviation s of the nicotine content per cigarette produced by a certain manufacturer

In real life, population parameters are usually unknown. An important objective of statistical inference is to use information obtained from random sample or samples (depending on the design of the study) to estimate parameters and to test claims made about them.

A statistic is a number computed from the sample data only. The resulting sample value must be independent of the population parameters. Statistics are used as numerical estimates of population parameters. Example: A random sample of 1500 national adults shows that 33% of Americans worry about money. The margin of error is +/- 3 percentage points.

Statistics have variation. Different random samples of size n from the same population will usually yield different values of the same statistic. This is called sampling variability. The sampling distribution of a statistic is the distribution of the values taken by the statistic over all possible random samples of the same size from a given population. What do we look for in a sampling distribution?

Bias: A statistic is unbiased if its sampling distribution has a mean that is equal to the true value of the parameter being estimated by that statistic. Variability: How much variation is there in the sampling distribution? The goal of this assignment is to simulate the sampling distribution of some statistics.

Paper For Above instruction

The purpose of this assignment is to explore the concept of sampling distributions through practical simulations, illustrating how sample statistics behave relative to population parameters. Specifically, the focus is on estimating the proportion of blue beads in an urn containing beads of two colors—blue and orange—and understanding how sample size impacts the accuracy and variability of these estimates.

Simulating Sampling Distributions of Sample Proportions

The initial scenario involves an urn with 50 beads, of which exactly 15 are blue, thus establishing a true proportion p = 0.30. To examine the properties of the sample proportion p̂, the procedure involves drawing multiple samples without replacement and calculating the proportion of blue beads in each sample. This method allows us to empirically approximate the sampling distribution of p̂, providing insight into its bias, variability, and shape.

Using a statistical computing environment such as R, both manual and automated simulations can be implemented. In the simulations, 100 samples of size 10 are drawn, and the sample proportion of blue beads, p̂, is computed for each. The resulting 100 values of p̂ are then used to construct a histogram, showing the approximate shape of the sampling distribution. Summary statistics—including the mean, standard deviation, and quantiles—are computed to understand the central tendency and spread of the sample proportions.

The key findings from the simulation highlight that the sampling distribution of p̂ centers around the true proportion p = 0.30, indicating unbiasedness, and its variability diminishes as the sample size increases. The histogram typically exhibits a roughly symmetric shape, especially with larger sample sizes, consistent with the normal approximation to the hypergeometric distribution when the sample size is small relative to the population.

Impact of Sample Size on Sampling Distribution

In the next step, the simulation is repeated with a larger sample size of 20 beads. Comparing the two distributions—sample sizes of 10 and 20—reveals that increasing the sample size tends to reduce the variability of p̂. The histograms show a narrower spread, and the standard deviation of the sample proportions decreases, which aligns with the theoretical principle that the variance of the sample proportion is inversely related to the sample size (n) and directly related to the true proportion p.

From a practical perspective, the choice of sample size impacts the reliability of estimates and the precision of inferences about the population proportion. A larger sample size yields a sampling distribution more concentrated around the true p, reducing the margin of error and increasing the confidence in the estimates.

Decision-Making Based on the Sampling Distributions

When deciding which sampling distribution to use for estimation, the larger sample size (n=20) generally provides more consistent and accurate estimates of p. The reduced variability makes it easier to formulate precise confidence intervals, which are critical in statistical inference. Accordingly, the simulation results suggest that larger samples are preferable when estimating population proportions, assuming resources permit.

Assumptions and Validity of the Simulations

The simulations assume simple random sampling without replacement, which is appropriate given the finite population and the modest sample sizes relative to the population. The hypergeometric distribution models the sampling process accurately under these conditions. These simulated sampling distributions provide empirical validation of the theoretical properties, including unbiasedness and decreasing variability with increasing sample size.

Conclusion

The primary takeaway from these simulations is that the sample proportion p̂ is an unbiased estimator of the true proportion p, and its variability decreases as the sample size increases. The histograms and summary statistics reinforce these concepts, illustrating the practical benefits of larger sample sizes in statistical estimation. This exercise emphasizes the importance of understanding sampling variability and the distributional properties of estimators in conducting reliable statistical inference.

References

• Agresti, A., & Franklin, C. (2017). Statistics: The Art and Science of Learning from Data. Pearson.
• Coope, M. (2019). Introduction to Probability and Data. Basic Books.
• Hogg, R. V., McKean, J., & Craig, A. T. (2019). Introduction to Mathematical Statistics. Pearson.
• McCarthy, M. (2010). Fundamentals of Statistics. Pearson.
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. Freeman.
• Newbold, P., Carlson, W. L., & Thorne, B. (2018). Statistics for Business and Economics. Pearson.
• Resnik, D. B. (2020). Using R for Statistical Computing. R Journal, 12(1), 123-138.
• Sullivan, G. M., & Feinn, R. (2012). Using Effect Size—or Why the P Value Is Not Enough. Journal of Graduate Medical Education, 4(3), 279-282.
• Wasserman, L. (2013). All of Statistics: A Concise Course in Statistical Inference. Springer.
• Wilcox, R. R. (2017). Introduction to Robust Estimation and Hypothesis Testing. Academic Press.