Using Excel To Construct A 95% Confidence Interval ✓ Solved

Using Excel to Construct a 95% Confidence Interval

Please use the attached Excel template to complete the assignment. The first tab in the spreadsheet includes an example. The inputs you see in the template are based on the example. You need to enter the data given in the assignment and complete it.

A random sample of 150 people from Manchester, New Hampshire, have been given cholesterol tests, and 60 of these people had levels over the "safe" count of 200. Using Excel, construct a 95% confidence interval for the population proportion of people in New Hampshire with cholesterol levels over 200. This question requires a confidence interval for a proportion, the interval that is likely to capture the proportion of all population members that satisfy a specified property.

The formula you will use is: p estimate ± z-critical value × sqrt(p estimate (1-p estimate)/n). Note that, for population proportion, we use the z-multiple, not the t-multiple.

Paper For Above Instructions

In this assignment, we are tasked with constructing a 95% confidence interval to estimate the population proportion of individuals in Manchester, New Hampshire, with cholesterol levels exceeding 200. This analysis hinges on the principles of inferential statistics, specifically focusing on population proportions.

Overview of Confidence Intervals

A confidence interval is a range of values, derived from a dataset, that is likely to contain the value of an unknown population parameter. In this case, we want to estimate the proportion of all individuals in New Hampshire with cholesterol levels over 200. A 95% confidence interval suggests that, if we were to take many samples and create confidence intervals from each, approximately 95% of those intervals would contain the true population proportion.

Understanding the Sample

From the assignment, we know that a sample of 150 individuals was randomly selected, and 60 of these individuals had cholesterol levels above 200. This gives us the necessary data to proceed with the calculations.

The sample proportion (p̂) is calculated as follows:

p̂ = X/n

Where X is the number of successes (people with cholesterol levels over 200), and n is the total number of observations in the sample.

In our situation:

X = 60

n = 150

Thus, p̂ = 60/150 = 0.4.

Calculating the 95% Confidence Interval

To construct the confidence interval, we also need to determine the z-critical value for a 95% confidence level. Typically, this value can be found in z-tables or standard normal distribution tables. For a 95% confidence level, the z-critical value (z*) is 1.96.

We will now plug the values into the confidence interval formula:

Confidence Interval = p̂ ± z* × sqrt(p̂(1 - p̂)/n)

Let's calculate the standard error (SE):

SE = sqrt(p̂(1 - p̂)/n) = sqrt(0.4(1 - 0.4)/150) = sqrt(0.4 × 0.6 / 150) = sqrt(0.0024) ≈ 0.049.

Now we can calculate the margin of error (ME):

ME = z* × SE = 1.96 × 0.049 ≈ 0.096.

Finally, we can construct the confidence interval:

Lower limit = p̂ - ME = 0.4 - 0.096 = 0.304.

Upper limit = p̂ + ME = 0.4 + 0.096 = 0.496.

Thus, the 95% confidence interval for the population proportion of people in New Hampshire with cholesterol levels over 200 is approximately (0.304, 0.496).

Using Excel to Construct the Confidence Interval

To efficiently compute these values, we can utilize Excel. The steps are simple:

  1. Input the sample size (n = 150) and the number of successes (X = 60) into the spreadsheet.
  2. Calculate the sample proportion using the formula =X/n.
  3. Calculate the standard error using the formula =SQRT((p*(1-p))/n).
  4. Calculate the margin of error using the formula =z SE.
  5. Finally, determine the confidence interval limits using the calculated values.

Conclusion

In summary, we calculated a 95% confidence interval for the proportion of individuals in Manchester, New Hampshire, with cholesterol levels above 200 by utilizing statistical principles and Excel. The findings indicate that we can be 95% confident that the true proportion of the population falls between approximately 30.4% and 49.6%.

References

  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2016). Introduction to the Practice of Statistics. W.H. Freeman.
  • Daniel, W. W., & CROSS, C. L. (2018). Biostatistics: A Foundation for Analysis in the Health Sciences. Wiley.
  • Agresti, A., & Coull, B. A. (1998). Approximate is Better than Exact for Interval Estimation of Binomial Proportions. The American Statistician, 52(2), 119-126.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.
  • Rosner, B. (2011). Fundamentals of Biostatistics. Cengage Learning.
  • Walbourn, L. (2016). Application of Confidence Intervals in Health Research. Oxford University Press.
  • Wackerly, D. D., Mendenhall, W., & Beaver, R. J. (2008). Mathematical Statistics with Applications. Cengage Learning.
  • Lehmann, E. L., & Casella, G. (1998). Theory of Point Estimation. Springer.
  • Haan, M. D., & Rens, D. (2015). Intermediate Statistics. Wiley.
  • Wood, S. N. (2017). Generalized Additive Models: An Introduction with R. Chapman and Hall/CRC.

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