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Template Confidence Interval for Population Proportion Notes

Number of successes in sample: 421

Sample size, n: 500

Confidence level, c: 0.95

Sample proportion, p: 0.842

Critical z value, z: 1. Uses the NORMSINV function in excel; NORMSINV((1+c)/2)

Standard error, SE: sqrt(p*(1-p)/n)

Margin of error, MOE: z * SE

Lower confidence limit: p - MOE

Upper confidence limit: p + MOE

Enter data highlighted in yellow. Rows 5 through 10 are calculated using data from rows 2 through 4.

Paper For Above Instructions

In statistics, a confidence interval is a range of values that is likely to contain the population parameter with a specified level of confidence. In this case, we are focusing on the population proportion, which is the ratio of the number of successes to the total sample size. The data provided includes information on a proportion based on a sample of 500 individuals, where 421 have exhibited the characteristic of interest.

Understanding Confidence Intervals

The confidence level (c) set at 0.95 implies that we can be 95% confident that the population proportion lies within the calculated range. This is a commonly used confidence level in statistical analysis. The sample proportion (p) is calculated as the number of successes divided by the sample size, giving us:

p = 421 / 500 = 0.842.

Critical Z Value

To calculate the confidence interval, we first determine the critical z value associated with our confidence level. For a 95% confidence interval, we need to find the z value that corresponds to the upper 2.5% of the normal distribution (since the interval is two-tailed). This can be found using the NORMSINV function in Excel:

z = NORMSINV((1+c)/2) = NORMSINV(0.975) ≈ 1.96.

Standard Error Calculation

The standard error (SE) measures the variability of the sample proportion. It is calculated using the formula:

SE = sqrt(p * (1 - p) / n).

Substituting the values, we find:

SE = sqrt(0.842 (1 - 0.842) / 500) = sqrt(0.842 0.158 / 500) ≈ 0.022.

Margin of Error

The margin of error (MOE) provides the range around the sample proportion that accounts for sampling variability. It is computed as the product of the critical z value and the standard error:

MOE = z SE = 1.96 0.022 ≈ 0.043.

Calculating the Confidence Interval

The confidence interval is then calculated by subtracting and adding the margin of error from the sample proportion:

  • Lower confidence limit = p - MOE = 0.842 - 0.043 ≈ 0.799
  • Upper confidence limit = p + MOE = 0.842 + 0.043 ≈ 0.885

Thus, we can conclude that we are 95% confident that the true population proportion lies between approximately 0.799 and 0.885.

Conclusion

Confidence intervals for population proportions are a crucial tool in inferential statistics, allowing researchers to estimate population parameters based on sample data. The confidence interval calculated above (0.799, 0.885) provides valuable insight into the proportion of the population exhibiting the characteristic of interest based on the sample analyzed. Further research could involve testing this confidence interval against other variables to assess its robustness and applicability to different populations.

References

  • OpenStax. (n.d.). Introductory Business Statistics. Retrieved from https://openstax.org/books/introductory-business-statistics
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2016). Introduction to the Practice of Statistics (8th ed.). W.H. Freeman and Company.
  • Newman, D., & Schaeffer, B. (2016). Fundamentals of Statistics. Pearson.
  • Wackerly, D., Mendenhall, W., & Scheaffer, L. D. (2014). Mathematical Statistics with Applications (7th ed.). Cengage Learning.
  • Gerald, H. (2019). Applied Statistics. Wiley.
  • Mendenhall, W., Beaver, R. J., & Beaver, B. M. (2016). Introduction to Probability and Statistics (14th ed.). Cengage Learning.
  • Field, A. P. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). SAGE Publications.
  • Ghasemi, A., & Zahediasl, S. (2012). Normality Tests for Statistical Analysis: A Guide for Non-statisticians. International Journal of Endocrinology and Metabolism, 10(2), 486-489.
  • Pallant, J. (2020). SPSS Survival Manual (7th ed.). Allen & Unwin.
  • Altman, D. G., & Bland, J. M. (2011). How to Obtain Confidence Intervals for a Proportion. BMJ, 343, d2020.

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