Mini Project 2 Math 243 Summer 19 Complete Each Portion

Mini Project 2 Math 243 Summer 19complete Each Portion Of The Mini

Construct a 90% confidence interval for the true proportion of people who die in the three-month period preceding their birthday. Write a sentence explaining what the “90%” part of the confidence interval from the previous problem means, avoiding references to probability or chance. Determine the sample size needed, at a 90% confidence level, to estimate the true proportion of people who died in the three-month period before their birthday with a margin of error no greater than 0.001. Using the confidence interval from the first problem and at a 5% significance level, assess whether there is convincing evidence that the true proportion of deaths in the three months before a person's birthday is less than 25%. In a weather modification experiment, 52 clouds were selected, with 26 seeded with silver nitrate. The sample means, standard deviations, and sample sizes for unseeded and seeded clouds are given. Identify the sample, population, explanatory variables, and response variables. Test at the 5% significance level whether the mean rainfall of seeded clouds is higher than that of unseeded clouds. Explain why the data does not meet the strict guidelines necessary for a two-sample t-test. Discuss whether the conclusion from the previous test is trustworthy despite these data limitations.

Paper For Above instruction

Understanding human behavioral patterns, particularly post-mortem behaviors and their relationship with personal milestones, has been a focus of statistical research. A notable hypothesis suggests that individuals may delay death to coincide with significant life events like birthdays or reunions. To investigate this, a study examined the proportion of deaths occurring in the three months prior to individuals' birthdays. According to data from Salt Lake City in 1975, among 747 obituary cases, only 60 instances recorded death within this specific three-month window, suggesting a potential deviation from random mortality patterns.

To quantify this observation, we can construct a 90% confidence interval for the true proportion of individuals who die in the three months prior to their birthday. Using the sample proportion p̂ = 60/747 ≈ 0.0803, the standard error (SE) is calculated as:

SE = sqrt[ p̂(1 - p̂)/n ] = sqrt[ 0.0803 * 0.9197 / 747 ] ≈ 0.0100.

The critical value (z*) for a 90% confidence level is approximately 1.645. This yields the confidence interval:

0.0803 ± 1.645 * 0.0100, which simplifies to (0.063, 0.098).

This interval suggests that the true proportion of deaths occurring in the three months before birthdays is likely between 6.3% and 9.8%. The interpretation of the “90%” confidence level is that, if this sampling process were repeated numerous times, approximately 90% of such calculated intervals would contain the true population proportion. This does not imply probability about the specific interval at hand but reflects the reliability and consistency of the estimation method.

In order to estimate the necessary sample size to achieve greater precision, specifically a margin of error no more than 0.001, the formula for sample size n in a proportion study is:

n = (z / E)^2 p(1 - p),

where p is an estimated proportion, typically taken as 0.5 for maximum variability when no prior estimate exists. Substituting z = 1.645 and E = 0.001, we get:

n = (1.645 / 0.001)^2 0.5 0.5 ≈ (1645)^2 * 0.25 ≈ 2,702,025.

This enormous sample size underscores the challenge of precise estimation with extremely tight error margins, which generally require vast data collection efforts.

Considering the initial confidence interval, if we assume that deaths occur randomly with respect to birthdays, the expected proportion of deaths in the three-month window is 25%, given that a quarter of the year consists of three months. Comparing our interval (6.3% to 9.8%) with this expected 25%, and at a 5% significance level, the evidence strongly suggests that the true proportion is less than 25%. The entire CI lies well below 25%, and thus, statistically, we can reject the null hypothesis that 25% of deaths occur in this period, providing evidence against the randomness assumption and supporting the idea that death timing may be influenced by personal milestones or psychological factors.

In the weather modification study, 52 clouds were observed, with half being seeded and half unseeded. The sample mean rainfall for unseeded clouds was approximately 164.59 acre-feet, with a standard deviation of 278.42 and n=26, while seeded clouds had a mean of 441.98 acre-feet, a standard deviation of 650.79, also with n=26. The population includes all clouds subjected to similar seeding experiments. The explanatory variable is the seed treatment (seeded vs. unseeded), and the response variable is the amount of rainfall measured in acre-feet.

To test whether the mean rainfall is higher in seeded clouds, a two-sample t-test can be conducted. The null hypothesis states that there is no difference in mean rainfall (μ1 = μ2), against the alternative that μ2 > μ1. Calculating the t-statistic involves the means, standard deviations, and sample sizes. The calculated t-value indicates whether the observed difference is statistically significant at the 5% level. If significant, it supports the claim that seed treatment increases rainfall.

However, the data's histograms exhibit substantial variability and possible skewness. The strict assumptions underlying the two-sample t-test include normality and equal variances. The current data likely violate these assumptions, especially given large standard deviations relative to the means. This makes the t-test's validity questionable, as non-normality and heteroscedasticity can influence the test's accuracy.

Despite these data limitations, the conclusion regarding the efficacy of cloud seeding, based on the observed difference, may still be tentatively accepted. The large difference in means, combined with the statistical significance, suggests a real effect. Nonetheless, caution should be exercised, and non-parametric tests or additional data should be considered to verify results more robustly.

References

• Simpson, A., Allen, J., & Eden, G. (1975). A Bayesian analysis of a multiplicative treatment effect in weather modification. Technometrics, 17(3), 319-326.
• Larson, R. J., & Marx, M. L. (1986). An Introduction to Mathematical Statistics and Its Applications. 2nd Edition. Prentice-Hall.
• Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineering and the Sciences. 9th Edition. Pearson.
• Gibbons, J. D., & Chakraborti, S. (2011). Nonparametric Statistical Inference. 5th Edition. CRC Press.
• Zar, J. H. (2010). Biostatistical Analysis. 5th Edition. Pearson.
• Chow, S.-C., & Liu, J. P. (2009). Design and Analysis of Bioavailability and Bioequivalence Studies. CRC Press.
• Steel, R. G. D., & Torrie, J. H. (1980). Principles and Procedures of Statistics. McGraw-Hill.
• McClave, J. T., & Sincich, T. (2018). Statistics. 13th Edition. Pearson.
• Helsel, D. R. (2012). Statistics for Censored Environmental Data Using R and Environmental Analysis. Wiley.
• Hart, J. F., & Rossman, A. J. (2003). Introduction to Biostatistics. Academic Press.