Written Homework 5i: I Have Attempted This Assignment Honest

Written Homework 5i Have Attempted This Assignment Honestly And The

Analyze the statistical and probabilistic aspects of various real-world scenarios, including evaluating improvements in accuracy based on sample data, calculating probabilities related to biking times, estimating sample sizes for survey accuracy, and constructing confidence intervals for population proportions. The assignment involves applying concepts from geometric and binomial distributions, sampling distribution models, and confidence interval construction to interpret data and support claims with statistical reasoning. Additionally, it requires understanding why some probabilities cannot be determined in certain cases and estimating potential outcomes using normal approximations and skewed distributions.

Paper For Above instruction

Statistical analysis plays a pivotal role in interpreting real-world data and making informed decisions based on probabilistic evidence. The assignment presents several scenarios where statistical concepts, including probability distributions, sample size estimation, and confidence intervals, are utilized to evaluate claims and predict future outcomes. This paper discusses each scenario in detail, illustrating how statistical reasoning can be employed to understand and analyze behavioral patterns, measurement uncertainties, and population parameters.

Evaluating the Archer’s Accuracy: Binomial Distribution and Significance

The scenario involving the archer's performance after purchasing a new bow exemplifies how binomial distribution can assess whether observed results suggest an actual change in skill or are attributable to chance. Historically, the archer hits the bull’s-eye with a probability of 80% (p=0.8). Under the null hypothesis that the bow has not improved her accuracy, the probability of hitting the bull’s-eye in any shot remains 0.8.

In her first three shots, she hits all three bull’s-eyes. The probability of this event, assuming her true accuracy remains 80%, is (0.8)^3 ≈ 0.512. While this might seem promising, it is plausible under the null hypothesis. However, after additional attempts totaling six bull’s-eyes out of six shots, the observed success rate is 100%. The probability of achieving six consecutive hits with p=0.8 is (0.8)^6 ≈ 0.262, still within a plausible range but increasingly unlikely if her true accuracy is truly 80%.

Further, considering 45 bull’s-eyes out of 50 shots, the sample proportion is 0.9, exceeding her presumed 80% accuracy. To test if this sample provides significant evidence of improvement, a binomial test or a normal approximation to the binomial distribution can be used. The expected number of bull’s-eyes under the null is 40 (0.8×50). The standard deviation is sqrt(50×0.8×0.2) ≈ 2.83. The observed value of 45 bull’s-eyes is 5 above the expected value, which is approximately 1.77 standard deviations away (z ≈ 1.77). Using standard normal tables, this is marginally significant but not conclusive. Therefore, statistically, the evidence that her accuracy has increased based solely on these observations is weak—more data would be needed for a definitive conclusion.

Probability Analysis of Biking Times Using Normal Approximation

The second scenario involves the bike commuting times, which are skewed left with a mean of 24 minutes and standard deviation of 5 minutes. The skewness complicates the use of normal approximation for individual probabilities, but the central limit theorem (CLT) justifies its application for averages over large samples. For a sample of 100 days, the mean of the sampling distribution of the average biking time is still 24 minutes, with a standard error of 5/√100 = 0.5 minutes.

In part (a), the question asks why the probability of a single ride being less than 20 minutes cannot be precisely determined using the normal approximation. Since the actual distribution is skewed and not perfectly normal, applying the normal model to individual observations can lead to inaccuracies, especially in the tails. Therefore, without knowledge of the exact distribution, the probability of a single ride under 20 minutes is uncertain.

Part (b) estimates the probability that the next four days all exceed 30 minutes. The probability that a single ride exceeds 30 minutes can be calculated assuming normality: z = (30 - 24)/5 = 1.2, with P(X > 30) ≈ 0.1151. The probability that all four daily rides are over 30 minutes is (0.1151)^4 ≈ 0.000177. This very low probability indicates that it is unlikely all four days will have durations exceeding 30 minutes.

Part (c) considers whether it is likely that the average of the next 10 days is less than 20 minutes. The sampling distribution of the mean has mean 24 and standard deviation 0.5. To find the probability that the average is less than 20 minutes, compute z = (20 - 24)/ (5/√10) ≈ -2.53. The probability of a sample mean less than 20 minutes is about 0.0057. Therefore, it is quite unlikely, about a 0.57% chance, that the average ride over the next 10 days will be less than 20 minutes.

Estimating Sample Sizes for Population Proportion

The third scenario involves estimating the proportion of Seattleites who enjoy the outdoors, presumed at 75%. To determine the number of respondents needed to estimate this proportion with a specified margin of error and confidence level, the standard formula for sample size calculation is employed:

For part (a), with a margin of error (ME) of 8% (0.08) and a confidence level of 90%, the critical z-value is approximately 1.645. The formula:

n = (z^2 p (1-p)) / ME^2

Substituting values: n = (1.645^2 0.75 0.25) / 0.08^2 ≈ (2.706 * 0.1875) / 0.0064 ≈ 0.507 / 0.0064 ≈ 79.2. Thus, at least 80 individuals need to be surveyed.

In part (b), to reduce the margin to 5% (0.05), recalculate:

n = (1.645^2 0.75 0.25) / 0.05^2 ≈ 2.706 * 0.1875 / 0.0025 ≈ 0.507 / 0.0025 ≈ 203.0. Therefore, approximately 203 respondents are necessary.

Part (c) aims for a 95% confidence level (z ≈ 1.96) with an even smaller margin of 2% (0.02). Reconciliation yields:

n = (1.96^2 0.75 0.25) / 0.02^2 ≈ 3.8416 * 0.1875 / 0.0004 ≈ 0.720 / 0.0004 ≈ 1800.0 respondents.

Constructing confidence intervals for each of these sample sizes involves taking sample proportions and calculating the margin of error with the formula:

CI = p̂ ± z * √(p̂(1 - p̂)/n)

For example, with the initial sample size of 80, assuming a sample proportion of 0.75, the margin of error is approximately 0.08 at 90% confidence, resulting in a confidence interval of roughly 67% to 83%. This interval indicates with 90% certainty that between 67% and 83% of Seattleites enjoy the outdoors. Similarly, increased sample sizes lead to narrower confidence intervals, providing more precise estimates.

Conclusion

The analysis demonstrates the importance of selecting appropriate statistical tools based on the nature of the data. Binomial distributions effectively assess success probabilities in quality improvement, while normal approximations allow reasonable estimates of probabilities for averages and aggregated data when sample sizes are sufficiently large. Accurate sample size determination supports reliable survey estimates, crucial for policy-making and research. Understanding these concepts empowers analysts to interpret data accurately, estimate uncertainties, and make informed predictions about populations and behaviors.

References

  • Agresti, A. (2018). Statistical methods for the social sciences. Pearson.
  • Conway, J. (2019). Statistics for business and economics. McGraw-Hill Education.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the practice of statistics. W.H. Freeman.
  • Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for business and economics. Pearson.
  • Siegel, S., & Castellan, N. J. (1988). Nonparametric statistics for the behavioral sciences. McGraw-Hill.
  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability and statistics for engineers and scientists. Pearson.
  • Zar, J. H. (2010). Biostatistical analysis. Pearson.
  • Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Routledge.
  • Mehta, N., & Patel, N. R. (2018). Sample size calculations in statistics. Journal of the Royal Statistical Society, 180(3), 607-624.
  • Lenth, R. V. (2016). Some practical guidance for effective use of Power and Sample size calculations. Journal of Statistical Software, 55(1), 1–19.

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