Find The 99 Percent Confidence Interval For The Mean Bill

Find the 99 percent confidence interval for the mean bill of all lunch orders

A random sample of 19 lunch orders at Noodles and Company showed a mean bill of $12.83 with a standard deviation of $6.71. Calculate the 99% confidence interval for the mean bill of all lunch orders. Round your answers to 4 decimal places.

Paper For Above instruction

The task involves constructing a 99% confidence interval for the average lunch bill at Noodles and Company based on a sample. The data collected includes a sample size of 19 orders, with a sample mean of $12.83 and a sample standard deviation of $6.71. This statistical estimation aims to infer the range within which the true population mean likely falls with 99% certainty.

Given the sample data, the appropriate statistical method is to use the Student's t-distribution because the sample size is small (n

CI = x̄ ± tα/2, df * (s / √n)

Where:

  • x̄ = 12.83 (sample mean)
  • s = 6.71 (sample standard deviation)
  • n = 19 (sample size)
  • df = n - 1 = 18 (degrees of freedom)
  • tα/2, df = t-value for 99% confidence with 18 degrees of freedom.

Using a t-distribution table or calculator, the t-value for a 99% confidence level and 18 degrees of freedom is approximately 2.878. Now, calculate the standard error (SE):

SE = s / √n = 6.71 / √19 ≈ 6.71 / 4.3589 ≈ 1.5392

Next, determine the margin of error (ME):

ME = t SE = 2.878 1.5392 ≈ 4.4293

The confidence interval bounds are then:

  • Lower bound: 12.83 - 4.4293 ≈ 8.4007
  • Upper bound: 12.83 + 4.4293 ≈ 17.2593

Therefore, the 99% confidence interval for the mean lunch bill is approximately from $8.4007 to $17.2593, rounded to four decimal places.

References

  • Blitzstein, J., & Hwang, J. (2014). Introduction to Probability. CRC Press.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W.H. Freeman.
  • Newbold, P., Carlson, W., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
  • Devore, J. L. (2011). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Lehmann, E. L., & Romano, J. P. (2005). Testing Statistical Hypotheses. Springer.
  • Ross, S. M. (2014). Introduction to Probability and Statistics for Engineers and Scientists. Academic Press.
  • Agresti, A., & Franklin, C. (2013). Statistics: The Art and Science of Learning from Data. Pearson.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
  • Statistics.com. (2020). Confidence Intervals. Retrieved from https://statistics.com
  • UCLA Statistical Consulting Group. (2019). Constructing Confidence Intervals. https://stats.idre.ucla.edu

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