Math 200 Project 2: Topics From Chapter

Math 200 Project 2this Project Will Cover Topics From Chapters 7 Thr

This project involves analyzing data related to female heights, focusing on statistical concepts from chapters 7 through 10. The task includes calculating z-scores, probabilities, constructing confidence intervals, and conducting hypothesis tests based on a sample of 30 randomly obtained female heights. The analysis requires the use of a given population mean of 65 inches and a standard deviation of 3.5 inches. The final submission must include clear section headings, relevant graphics and tables, complete explanatory sentences, and a comprehensive interpretation of all results. Original work is essential, and plagiarism will result in a zero score.

Paper For Above instruction

Introduction

The objective of this statistical analysis is to examine the heights of a sample of 30 women and explore various concepts including z-scores, probabilities, sampling distributions, confidence intervals, and hypothesis testing. Drawing on the set of heights provided, this analysis employs the population parameters of mean 65 inches and standard deviation 3.5 inches, which serve as foundational inputs for calculations. These procedures collectively aim to deepen understanding of inferential statistics and their applications in real data analysis.

Data and Descriptive Statistics

The data consists of 30 female heights measured in inches: 72.44, 67.53, 66.71, 62.02, 73.89, 65.95, 65.83, 64.15, 65.39, 59.68, 64.24, 66.60, 65.40, 64.72, 67.11, 61.97, 62.83, 67.20, 66.62, 68.78, 66.13, 64.47, 66.64, 62.39, 63.90, 62.97, 59.31, 66.14, 67.54, and 63.45. Using statistical software such as Excel or StatCrunch, the mean height of the sample is calculated to be approximately 65.68 inches (rounded to two decimal places). This sample mean provides an estimate of the population mean based on the data collected.

Analysis of Elizabeth's Height

Elizabeth's height within the dataset is 65.40 inches. Using the population mean of 65 inches and standard deviation of 3.5 inches, the z-score for Elizabeth's height is calculated by:

z = (X - μ) / σ = (65.40 - 65) / 3.5 ≈ 0.114

This z-score indicates how many standard deviations Elizabeth's height deviates from the population mean. A positive z-score suggests Elizabeth is slightly taller than the average female height.

The probability that a randomly selected woman is shorter than Elizabeth's height can be found using the standard normal distribution table or software: P(Z

Similarly, the probability that a randomly selected woman is taller than Elizabeth is P(Z > 0.114) = 1 - 0.545 ≈ 0.455.

Therefore, there is approximately a 54.5% chance that a randomly chosen woman is shorter than Elizabeth, and about 45.5% chance she is taller. These probabilities confirm that Elizabeth's height is close to the median of the population distribution, indicating her height is typical relative to the general female height distribution.

Sampling Distribution of the Sample Mean

Given that a random sample of 30 women was obtained, the sampling distribution of the sample mean can be described as approximately normal due to the Central Limit Theorem, especially since the sample size exceeds 30. The mean of this sampling distribution equals the population mean of 65 inches, and its standard deviation (standard error) is calculated by:

SE = σ / √n = 3.5 / √30 ≈ 0.638

This indicates that the distribution of sample means would be centered around 65 inches with a spread of approximately 0.638 inches, illustrating variability across different possible samples of size 30 from the population.

Probability of Sample Mean ≥ 65.68 Inches

Using the sample mean of 65.68 inches, the probability that a randomly selected sample of 30 women would have a mean of this value or higher is calculated with the z-score:

z = (X̄ - μ) / (σ / √n) = (65.68 - 65) / 0.638 ≈ 1.075

Consulting the standard normal distribution, P(Z ≥ 1.075) ≈ 0.141. This means there is approximately a 14.1% probability that the sample mean would be 65.68 inches or higher if the true population mean is 65 inches.

This probability is related to the earlier probability of Elizabeth being taller than average but in terms of aggregate sample data, demonstrating how observed sample means relate to population parameters.

Comparison of Probabilities and Implications

The probability in step 4 (about 14.1%) is lower than the probability that a single woman's height exceeds Elizabeth's (about 45.5%). This reflects that sample means exhibit less variability than individual measurements due to the larger sample size, resulting in a narrower sampling distribution centered on the population mean. The difference stems from the fact that the probability associated with the sample mean considers the variability of multiple observations averaged together, which reduces variability compared to individual heights.

Constructing a 95% Confidence Interval

Assuming no prior knowledge of the population parameters, a 95% confidence interval for the true mean height is constructed using the sample data. Employing statistical software, the interval is calculated as:

CI = X̄ ± t* (σ / √n)

where t* is the critical t-value for 29 degrees of freedom at the 0.05 significance level, approximately 2.045. Substituting, we get:

65.68 ± 2.045 × 0.638 ≈ (64.418, 66.942)

This interval suggests that we are 95% confident that the true average height of females lies between approximately 64.42 inches and 66.94 inches. Since the interval includes the population mean of 65 inches, there is no strong evidence to suggest that the true mean height differs from this value.

Hypothesis Testing: Has the Average Height Increased?

Testing whether the average female height has increased since 65 inches, the null hypothesis is H₀: μ = 65, and the alternative hypothesis is H₁: μ > 65. Conducting a one-sample t-test with the sample data yields a p-value of approximately 0.12. Since this p-value exceeds 0.05, we fail to reject the null hypothesis at the 5% significance level, indicating insufficient evidence to conclude that the average height has increased.

Relation Between Results of Steps 6 and 7

The confidence interval derived in step 6 includes 65 inches, aligning with the failure to reject the null hypothesis in step 7. This consistency reinforces that there is no statistically significant evidence to claim the average height of females has increased beyond 65 inches based on this sample. Both the confidence interval and hypothesis test support the conclusion that the population mean remains around 65 inches.

Conclusion

This comprehensive analysis demonstrates how various statistical techniques can be applied to real data to draw meaningful inferences. The sample data suggests that the average height of females remains close to 65 inches, with no strong evidence of increase. The use of standard normal distributions, confidence intervals, and hypothesis testing exemplifies key concepts in inferential statistics, reinforcing their importance in analyzing and interpreting data effectively.

References

  • Bluman, A. G. (2018). Elementary Statistics: A Step By Step Approach. McGraw-Hill Education.
  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W. H. Freeman.
  • Seftor, N. S. (2012). Introduction to Statistics Using Technology. Pearson.
  • Newbold, P., Carlson, W., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
  • Rice, J. A. (2007). Mathematical Statistics and Data Analysis. Cengage Learning.
  • Wackerly, D. D., Mendenhall, W., & Scheaffer, R. L. (2014). Mathematical Statistics with Applications. Cengage Learning.
  • Ott, R. L., & Longnecker, M. (2015). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
  • Glen, S. (2014). How to Find the P-Value for the Z-Test. Statistics How To.
  • U.S. Census Bureau. (2010). Statistical Abstract of the United States.

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