Statistical Inference II J Lee Assignment 1

Hw1 Stat206pdfstatistical Inference Ii J Lee Assignment 1problem 1

Suppose the day after the Drexel-Northeastern basketball game, a poll of 1000 Drexel students was conducted and it was determined that 850 out of the 1000 watched the game (live or on television). Assume that this was a simple random sample and that the Drexel undergraduate population is 20,000. (a) Generate an unbiased estimate of the true proportion of Drexel undergraduate students who watched the game. (b) What is your estimated standard error for the proportion estimate in (a)? (c) Give a 95% confidence interval for the true proportion of Drexel undergraduate students who watched the game. Problem 2. (Exercise 18 in Chapter 7 of Rice) From independent surveys of two populations, 90% confidence intervals for the population means are conducted. What is the probability that neither interval contains the respective population mean? That both do? Problem 3. (Exercise 23 in Chapter 7 of Rice) (a) Show that the standard error of an estimated proportion is largest when p = 1/2. (b) Use this result and Corollary B of Section 7.3.2 (also, on Page 17 of the lecture notes) to conclude that the quantity 1/2 √{N− n}/{N(n− 1)} is a conservative estimate of the standard error of p̂ no matter what the value of p may be. (c) Use the central limit theorem to conclude that the interval p̂ ± √{N− n}/{N(n− 1)} contains p with probability at least .95.

Paper For Above instruction

Statistical inference plays a vital role in understanding and interpreting data from samples to make informed conclusions about entire populations. In this analysis, we examine the proportion of Drexel undergraduate students who watched a recent basketball game, applying fundamental principles of estimation, standard error calculation, and confidence interval construction to provide accurate and reliable statistical insights.

In the initial scenario, a simple random sample of 1000 students was surveyed, with 850 confirming they watched the game, either live or on television. To estimate the true proportion of the entire Drexel undergraduate population who watched the game, the first step involves calculating the sample proportion, denoted as p̂. This unbiased estimator is derived directly from the sample data and is computed as the ratio of students who watched the game to the total sample size: p̂ = x/n = 850/1000 = 0.85. Given the properties of simple random sampling, p̂ serves as an unbiased estimate of the population proportion p, the true parameter reflecting the entire student body's viewing habits.

Next, understanding the variability or uncertainty associated with this estimate requires calculating the standard error (SE). The standard error of a sample proportion is given by the formula:

SE = √[p̂(1 - p̂)/n]

Substituting the values obtained:

SE = √[0.85 * 0.15 / 1000] = √[0.1275 / 1000] ≈ √0.0001275 ≈ 0.0113.

This value quantifies the typical deviation expected between the sample proportion and the true population proportion, considering the sample size and observed data.

To construct a 95% confidence interval for the true proportion p, we utilize the normal approximation to the binomial distribution, supported by the Central Limit Theorem when the sample size is sufficiently large. The interval is calculated as:

p̂ ± z × SE, where z is the critical value corresponding to the desired confidence level. For a 95% confidence interval, z* ≈ 1.96.

Thus, the interval becomes:

0.85 ± 1.96 × 0.0113 = (0.85 - 0.0222, 0.85 + 0.0222) ≈ (0.8278, 0.8722).

This interval suggests that, with 95% confidence, between approximately 82.78% and 87.22% of all Drexel undergraduates watched the game.

Transitioning to related concepts, when dealing with multiple independent intervals for different populations, the probability that neither contains the true means is derived by multiplying their individual non-inclusion probabilities. Conversely, the probability that both intervals contain the respective true means involves multiplying their inclusion probabilities; for 90% confidence intervals, each has a 10% chance of not containing the true mean, so the probability that neither contains it is 0.1 × 0.1 = 1%, and that both do is 0.9 × 0.9 = 81%.

Considering the maximum variability of the estimated proportion p̂, this occurs at p=0.5, where the product p̂(1 - p̂) reaches its maximum of 0.25. This property allows us to determine a conservative (worst-case) standard error estimate by plugging in p = 0.5 into the standard error formula or its bounds. Specifically, the quantity 1/2 × √(N−n)/(N(n−1)) provides an upper bound, ensuring our confidence interval maintains at least 95% coverage probability regardless of the actual value of p, owing to the properties of the binomial distribution and the Central Limit Theorem.

By applying the central limit theorem, we approximate the sampling distribution of p̂ as normal for large samples. Consequently, the interval p̂ ± √(N− n)/(N(n−1)) is a 95% confidence interval for p, guaranteeing that the true proportion is contained within this interval with at least 95% probability. This conservative approach ensures reliable estimation even under worst-case variability scenarios, reinforcing the robustness of statistical inference methods in practice.

References

  • Rice, J. (2007). Mathematical Statistics and Data Analysis. 3rd Edition. Cengage Learning.
  • Freedman, D., Pisani, R., & Purves, R. (2007). Statistics. W. W. Norton & Company.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.
  • Agresti, A., & Franklin, C. (2009). Statistics: The Art and Science of Learning from Data. Pearson.
  • Kirk, R. E. (2013). Experimental Design: Procedures for the Behavioral, Educational, and Agricultural Sciences. CRC Press.
  • Johnson, R. A., & Wichern, D. W. (2007). Applied Multivariate Statistical Analysis. Pearson.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
  • Moore, D. S., Notz, W. I., & Flinger, M. A. (2013). The Basic Practice of Statistics. W. H. Freeman.
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  • Lehmann, E. L., & Casella, G. (1998). Theory of Point Estimation. Springer.

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