A Random Sample Of N=500 Observations From A Binomial Population Produced X=460 Successes. 1. Give The Best Point Estimate For The Binomial Proportion P? 2. Calculate The Confidence Interval For P?

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1. Point Estimate for the Binomial Proportion p

In statistical analysis, estimating the binomial proportion p is a fundamental task, especially when dealing with categorical data. Binomial proportions represent the probability of success in a sequence of independent trials, and accurately estimating this proportion is crucial for making informed decisions and drawing meaningful conclusions. In this specific scenario, we have a random sample of n = 500 observations drawn from a binomial population, resulting in x = 460 successes. Our primary objective is to determine the best point estimate for the binomial proportion p, denoted as p^{\hat{p}}. The point estimate serves as a single value that best represents the true population proportion based on the available sample data. To achieve this, we employ the sample proportion formula, which directly relates the number of successes to the total number of trials. The sample proportion, calculated as the number of successes divided by the total number of trials, provides an unbiased estimate of the population proportion. In this context, the formula is expressed as p^=xn{\hat{p} = \frac{x}{n}}, where x represents the number of successes and n represents the total number of observations in the sample. By substituting the given values, x = 460 and n = 500, into the formula, we can derive the point estimate for the binomial proportion p. This calculation provides a numerical value that serves as our best guess for the true proportion of successes in the underlying binomial population. Understanding the concept of a point estimate is paramount in statistical inference, as it forms the basis for constructing confidence intervals and conducting hypothesis tests. The point estimate provides a central value around which we can quantify the uncertainty associated with our estimation process. By utilizing the sample proportion as our point estimate, we leverage the information contained within the sample to make inferences about the larger population from which it was drawn. This process allows us to generalize our findings beyond the specific sample and gain insights into the broader characteristics of the binomial population under investigation. In the subsequent sections, we will delve deeper into the interpretation and implications of this point estimate, exploring its role in further statistical analysis and decision-making.

Calculation and Result:

To calculate the best point estimate for the binomial proportion p, we use the formula:

p^=xn=460500=0.92\hat{p} = \frac{x}{n} = \frac{460}{500} = 0.92

Therefore, the best point estimate for the binomial proportion p is 0.92. This value represents the proportion of successes observed in the sample and serves as our best single-value estimate for the true population proportion.

p^=0.92\hat{p} = 0.92

2. Confidence Interval for p

Introduction to Confidence Intervals

Confidence intervals play a pivotal role in statistical inference, providing a range of values within which the true population parameter is likely to lie. Unlike point estimates, which offer a single value, confidence intervals quantify the uncertainty associated with our estimation process. This is particularly important when dealing with sample data, as it acknowledges the inherent variability and limitations of drawing conclusions based on a subset of the population. The width of the confidence interval reflects the precision of our estimate, with narrower intervals indicating greater certainty and wider intervals suggesting more uncertainty. Constructing a confidence interval involves specifying a confidence level, which represents the probability that the interval will contain the true population parameter. Common confidence levels include 90%, 95%, and 99%, each offering a different balance between precision and confidence. A higher confidence level implies a wider interval, reflecting a greater degree of certainty but potentially sacrificing precision. Conversely, a lower confidence level results in a narrower interval, providing a more precise estimate but with a reduced probability of capturing the true parameter. The choice of confidence level depends on the specific context of the analysis and the desired trade-off between precision and confidence. In this section, we will focus on constructing a confidence interval for the binomial proportion p, utilizing the sample data provided. This interval will provide a range of plausible values for the true proportion of successes in the underlying population, taking into account the inherent variability in our sample data. Understanding the principles of confidence interval construction is essential for interpreting statistical results and making informed decisions based on sample data. By quantifying the uncertainty associated with our estimates, confidence intervals provide a more nuanced and comprehensive understanding of the population parameter of interest.

Formula for Confidence Interval

Constructing a confidence interval for the binomial proportion p involves utilizing the sample proportion p^{\hat{p}}, the sample size n, and a critical value z corresponding to the desired confidence level. The formula for the confidence interval is given by:

p^±zp^(1−p^)n\hat{p} \pm z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

where:

  • p^{\hat{p}} is the sample proportion,
  • z is the critical value from the standard normal distribution corresponding to the desired confidence level,
  • n is the sample size.

Calculations for 90% and 95% Confidence Intervals

a. For a 90% confidence interval:

The critical value z for a 90% confidence level is approximately 1.645. Using the previously calculated p^{\hat{p}} = 0.92 and n = 500, the 90% confidence interval is:

0.92±1.6450.92(1−0.92)5000.92 \pm 1.645 \sqrt{\frac{0.92(1-0.92)}{500}}

0.92±1.6450.92(0.08)5000. 92 \pm 1.645 \sqrt{\frac{0.92(0.08)}{500}}

0.92±1.6450.07365000. 92 \pm 1.645 \sqrt{\frac{0.0736}{500}}

0.92±1.6450.00014720. 92 \pm 1.645 \sqrt{0.0001472}

0.92±1.645imes0.012130. 92 \pm 1.645 imes 0.01213

0.92±0.02000. 92 \pm 0.0200

Therefore, the 90% confidence interval is approximately (0.900, 0.940).

b. For a 95% confidence interval:

The critical value z for a 95% confidence level is approximately 1.96. Using the same values for p^{\hat{p}} and n, the 95% confidence interval is:

0.92±1.960.92(1−0.92)5000.92 \pm 1.96 \sqrt{\frac{0.92(1-0.92)}{500}}

0.92±1.960.92(0.08)5000. 92 \pm 1.96 \sqrt{\frac{0.92(0.08)}{500}}

0.92±1.960.07365000. 92 \pm 1.96 \sqrt{\frac{0.0736}{500}}

0.92±1.960.00014720. 92 \pm 1.96 \sqrt{0.0001472}

0.92±1.96imes0.012130. 92 \pm 1.96 imes 0.01213

0.92±0.02380. 92 \pm 0.0238

Therefore, the 95% confidence interval is approximately (0.896, 0.944).

Interpretation of Confidence Intervals

The 90% confidence interval (0.900, 0.940) suggests that we are 90% confident that the true binomial proportion p lies within this range. Similarly, the 95% confidence interval (0.896, 0.944) indicates that we are 95% confident that the true proportion p falls within this interval. The wider 95% confidence interval reflects the higher level of confidence, providing a broader range of plausible values for p. These intervals provide valuable insights into the uncertainty associated with our estimate of the binomial proportion, allowing for more informed decision-making and statistical inference.

Conclusion

In conclusion, analyzing the random sample of n = 500 observations with x = 460 successes from a binomial population has provided us with valuable insights into the estimation of the binomial proportion p. The best point estimate for p was found to be 0.92, representing the proportion of successes observed in the sample. Furthermore, we constructed both 90% and 95% confidence intervals for p, which provide a range of plausible values for the true population proportion. The 90% confidence interval was calculated as (0.900, 0.940), while the 95% confidence interval was determined to be (0.896, 0.944). These intervals quantify the uncertainty associated with our estimate, with the wider 95% interval reflecting a higher level of confidence. By utilizing these statistical tools, we can make more informed decisions and draw more meaningful conclusions about the underlying binomial population. Understanding the concepts of point estimates and confidence intervals is crucial for statistical analysis and inference, enabling us to effectively interpret sample data and generalize findings to the broader population.