M2 Analysis Assignment Number Of Pages: 2 (Double Spaced)

M2 Analysis Assignment Number of Pages: 2 (Double Spaced) Number of sources: 1

We will be using Excel to simulate a normal distribution with a mean of 300 and a standard deviation of 9. Please note there are five parts to this question.

1. Generate a random sample of 14 data points from a normally distributed population (using the =invnorm command in Excel). Create a histogram where the first bin starts at 275.25 and the width of each bin is 4.5. The last bin should be at 320.25. Save this histogram as an image file (.png, .jpg, or .bmp) and attach it.

2. Generate a random sample of 60 data points from the same distribution. Create the histogram with the same bin parameters, save it as an image file, and attach it.

3. Generate a sample of 320 data points. Create the histogram as before, save, and attach the image.

4. Generate a sample of 1700 data points. Create the histogram, save, and attach the image.

5. Analyze and explain in detail what you notice as the sample size increases, especially regarding the distribution of the sample. Address the shape, spread, and any convergence toward the theoretical distribution. You may record an audio explanation and attach it.

Additionally,

6. Using Excel, simulate the number of left-handed people in samples, assuming 10% of the population is left-handed. Generate samples of 6, 14, 65, and 3000 data points, count the number of left-handed individuals, and create a pie chart showing the proportion of left- vs. right-handed people. Save and attach each pie chart, noting the percentage of left-handed people in each sample.

7. Analyze and explain what you've observed regarding the proportion of left-handed individuals as sample size increases. Compare these sample proportions to the expected population proportion of 10%, and discuss any variations or patterns observed.

Paper For Above instruction

Simulation of normal distribution and analysis of sample size effects are fundamental exercises in understanding statistical principles and variability. Using Excel's =invnorm command, we can generate random samples from a normal distribution with specified parameters—mean of 300 and standard deviation of 9—and observe how the distribution's shape converges with increasing sample sizes.

Initially, with a small sample size of 14 data points, the histogram tends to display significant randomness, with the frequencies spread unevenly across bins. The distribution appears somewhat irregular and does not closely resemble the ideal normal curve. This is characteristic of small samples because they are more susceptible to sampling variability. The histograms often show clusters or gaps, and the shape might deviate notably from the theoretical distribution (Ott & Longnecker, 2010).

As the sample size increases to 60, the histogram begins to smooth out, and the shape appears more symmetric and bell-shaped, aligning more closely with the normal distribution. The frequencies across bins become less erratic, and the natural pattern of the distribution emerges more clearly. This phenomenon illustrates the Law of Large Numbers, which states that with larger samples, the sample distribution tends to mirror the population distribution more accurately (Casella & Berger, 2002). Nonetheless, some variability remains, though less pronounced than with smaller samples.

Further increasing the sample size to 320 enhances the approximation to the theoretical normal curve. The histogram's shape becomes more symmetric, and the spread of data within the expected range becomes more consistent with the normal distribution's properties. The central limit theorem supports this observation, emphasizing that the sampling distribution of the mean tends toward normality as sample sizes grow, even if the underlying data are not perfectly normal (Rice, 2007).

At 1700 data points, the histogram almost perfectly resembles the theoretical normal distribution with the specified mean and standard deviation. The frequencies across bins align closely with what is predicted by the probability density function, and the distribution appears smooth and symmetric. Variability due to randomness diminishes significantly with such a large sample size, demonstrating the power of large samples in statistical estimation and inference (Freeman, 2014).

This progression from small to large samples exemplifies key statistical principles. Small samples paint a noisy picture of the true distribution, but as the sample size increases, the empirical distribution converges toward the theoretical one. This convergence underscores the reliability of larger samples in statistical modeling, enabling more precise estimates and inferences. The exercise vividly illustrates the Law of Large Numbers and the Central Limit Theorem, which are foundational to statistical theory and practice (Wackerly, Mendenhall, & Scheaffer, 2008).

Regarding the exercise on estimating left-handed individuals, the simulated sample proportions fluctuate around 10% due to sampling variability, especially with smaller samples. For instance, in the sample of 6, it’s possible to observe either zero or one left-handed person, leading to a wide percentage swing. The 14-sample might still vary notably but tends to hover closer to the true proportion. With larger samples like 65 and 3000, the sample proportion stabilizes around 10%, demonstrating the Law of Large Numbers (Walpole, Myers, Myers, & Ye, 2012).

This exercise highlights the importance of sample size in statistical estimation. Smaller samples are subject to higher variability, which can lead to inaccurate representations of the population parameter. Larger samples provide more precise estimates, reducing variability and increasing confidence in the results. This phenomenon is crucial in designing experiments and surveys, emphasizing the need for adequate sample sizes to draw reliable conclusions (Moore, McCabe, & Craig, 2012).

References

• Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
• Freeman, J. (2014). Stats: Data and Models (3rd ed.). Pearson.
• Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis (6th ed.). Brooks/Cole.
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics (7th ed.). W. H. Freeman and Company.
• Rice, J. A. (2007). Mathematical Statistics and Data Analysis (3rd ed.). Cengage Learning.
• Wackerly, D. D., Mendenhall, W., & Scheaffer, R. L. (2008). Mathematical Statistics with Applications (7th ed.). Thomson Brooks/Cole.
• Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineering and the Sciences (9th ed.). Pearson.