Exercise 7.1: Number Of Faculty

Exercise 7 1q 10number Of Faculty The Numbers Of Faculty At 32 Rand

The assignment involves analyzing data related to various statistical topics, such as estimating means and proportions, constructing confidence intervals, and performing hypothesis tests. Specifically, you are asked to use sample data to estimate population parameters with a specified confidence level, interpret confidence intervals, and determine the significance of differences or associations between groups or variables using appropriate statistical methods.

Applying these principles, you will perform calculations including point estimates, confidence intervals (ranging from 90% to 99%), and hypothesis tests at different significance levels. The data sets include information on faculty numbers, job counts of retired men, daycare tuition costs, gasoline taxes, university dance company sizes, students' belief in haunted places, voter opinions on hunger spending, movie viewing habits, salaries of government employees, student expenditures, park acreage, cookie calorie content, physician demographics, doctoral students' salaries, teacher salaries, home prices, student ages, test scores, obstacle course times, mistakes during practice, and financial aid distributions. For each problem, interpret your results in context, and determine whether your data provides sufficient evidence to support claims or hypotheses.

Sample Paper For Above instruction

Estimating Population Means and Proportions: Applications and Interpretations

Introduction

Statistics play a vital role in understanding and interpreting data collected from various fields such as education, health, economics, and social sciences. Estimating population parameters, such as means and proportions, enables decision-makers to infer insights from sample data with a certain level of confidence. This paper demonstrates the application of statistical estimation and hypothesis testing using real or hypothetical data to illustrate how these methods inform conclusions in practical scenarios.

Estimating the Mean Number of Faculty at Colleges

A study was conducted on 32 randomly selected state-controlled colleges and universities with enrollments under 12,000 students. The number of faculty members at each institution was recorded. To estimate the average number of faculty across all such colleges with 92% confidence, we calculate the sample mean and standard deviation from the sample data. Assuming the sample mean is 45 faculty members, and sample standard deviation is 12, the standard error is computed as SE = SD / √n = 12 / √32 ≈ 2.12. Using the t-distribution for 31 degrees of freedom, the critical value at 92% confidence is approximately 1.75. The margin of error is thus ME = critical value × SE ≈ 1.75 × 2.12 ≈ 3.71. Therefore, the 92% confidence interval for the mean number of faculty is approximately 45 ± 3.71, or from 41.29 to 48.71 faculty members. This interval indicates that, with 92% confidence, the true average faculty size in all such colleges lies within this range.

Assessing Retirement Job Counts

A sociologist sampled 50 retired men, finding an average of 7.2 jobs during their lifetime, with a known population standard deviation of 2.1. To estimate the typical number of jobs among all retired men, we employ a confidence interval for the mean. Because the population standard deviation is known, we use the z-distribution. The standard error is SE = 2.1 / √50 ≈ 0.297. For a 95% confidence level, the z-value is approximately 1.96; for 99%, it is about 2.576. The interval at 95% is 7.2 ± 1.96 × 0.297 ≈ 7.2 ± 0.58, i.e., from 6.62 to 7.78. At 99%, it is 7.2 ± 2.576 × 0.297 ≈ 7.2 ± 0.765, from 6.435 to 7.965. The narrower 95% interval provides a more precise estimate, but the wider 99% interval offers greater confidence that the true mean lies within the range.

Estimating Day Care Tuition Fees

A sample of 50 children attending day care centers reported an average annual tuition of $3,987 with a known population standard deviation of $630. The 90% confidence interval for the mean cost is calculated similarly. The standard error is SE = 630 / √50 ≈ 89.2. The z-value for 90% confidence is approximately 1.645. The margin of error is ME = 1.645 × 89.2 ≈ 146.8. The confidence interval is thus $3,987 ± 146.8, or from approximately $3,840.2 to $4,133.8. This suggests that a day care center aiming to keep tuition rates low might consider charging around $3,840 to $4,134, balancing affordability with coverage of costs.

Estimating Gasoline Taxes

A sample of 12 states yields their gasoline tax rates in cents per gallon. The goal is to estimate the true average tax with 90% confidence. Suppose the sample mean is 44.2 cents, and the sample standard deviation is 4.5 cents. The standard error is SE = 4.5 / √12 ≈ 1.30. The critical z-value at 90% is 1.645, leading to a margin of error of 1.645 × 1.30 ≈ 2.14. The confidence interval is approximately 44.2 ± 2.14, from 42.06 to 46.34 cents. Since the interval includes the national average of 44.7 cents, we cannot conclude a significant difference at this confidence level.

Estimating the Size of University Dance Companies

Data from several small universities regarding dance company membership shows a sample mean size of 25 students with a standard deviation of 5 students, based on a random sample of similar institutions. For a 99% confidence interval with a sufficiently large sample size, the critical z-value is 2.576. The standard error is SE = 5 / √n (assuming n=30), approximately 0.91. Then, the margin of error is 2.576 × 0.91 ≈ 2.35. The estimated true mean size of dance companies is 25 ± 2.35, or between approximately 22.65 and 27.35 students, informing stakeholders about typical group sizes.

Conclusion

In summary, statistical estimation involves using sample data to infer about populations with a specified confidence level. Constructing confidence intervals provides a range within which the true parameter value lies, aiding in decision-making. Hypothesis testing further enables evaluating claims or differences between groups. These tools are essential in fields such as education, economics, health sciences, and public policy, guiding evidence-based decisions grounded in data analysis.

References

  • Agresti, A., & Finlay, B. (2009). Statistical methods for the social sciences. Pearson Education.
  • Casella, G., & Berger, R. L. (2002). Statistical inference (2nd ed.). Duxbury Press.
  • Creswell, J. W. (2014). Research design: qualitative, quantitative, and mixed methods approaches. Sage Publications.
  • Dean, C. E., & Voss, D. (2003). Design and analysis of experiments. Springer.
  • Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver and Boyd.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the practice of statistics. W. H. Freeman.
  • Newcombe, R. G. (1998). Two-sided confidence intervals for the binomial proportion. The American Statistician, 52(2), 121-126.
  • Zar, J. H. (2010). Biostatistical analysis. Pearson Education.
  • Lohr, S. L. (1999). Sampling: Design and analysis. Duxbury Press.
  • Wasserstein, R. L., & Lazar, N. A. (2016). The ASA statement on p-values: context, process, and purpose. The American Statistician, 70(2), 129-133.

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