Identify Population Parameters From Sample Statistics ✓ Solved

Identify Population Parameters From Sample Statisticsprompt The Owner

Identify Population Parameters From Sample Statisticsprompt The Owner

The owners of a small mail order business want to estimate their average annual sales. They randomly select 100 sales from their first quarter to analyze. For the samples, the average amount spent was $235.12 with a standard error of $12.41. Use what you have learned about sampling distributions to answer the following questions.

Response Parameters

1. What conditions, or assumptions, should be verified before using the sample values to estimate the population mean?

Before using the sample mean to estimate the population mean, several key assumptions should be verified. Primarily, the sample should be random and representative of the population to avoid bias. The sampling distribution of the sample mean should be approximately normal, which typically requires that the sample size be sufficiently large, usually n ≥ 30, according to the Central Limit Theorem. Additionally, the data should be independent, meaning that individual sales are not influenced by each other. If these conditions are satisfied, then the sample mean can be used as a reliable estimate of the population mean with valid inference.

2. If these conditions are satisfied, what is the probability that you would get a sample with a sample mean of $230.00 or less?

Given the sample mean (x̄) = $235.12 and the standard error (SE) = $12.41, we treat this as a normal distribution. To find the probability of obtaining a sample mean of $230 or less, we calculate the z-score:

z = (x̄ - μ) / SE

We have x̄ = $230, but the true population mean μ is unknown. However, assuming the sample mean is an unbiased estimator of μ, we can proceed to find the probability corresponding to x̄ = $230:

z = ($230 - $235.12) / $12.41 ≈ -0.419

Using standard normal distribution tables or a calculator, the probability that z ≤ -0.419 is approximately 0.337. Therefore, there is about a 33.7% chance of observing a sample mean of $230 or less if the true population mean is around $235.12.

3. If they expect to have 50,000 sales this year, what is their expected value of the total sales for the year?

The expected total sales for the year can be estimated by multiplying the average sales per transaction by the total number of transactions:

Total expected sales = average sales × number of sales

= $235.12 × 50,000 ≈ $11,756,000

This estimate provides the business owners with an anticipated total revenue for the year based on the sample data.

4. What do the following two values represent, in terms of the sampling distribution?

“X + Z₀.05 • σ/√n = $214.77” and “X + Z₀.95 • σ/√n = $255.47”

These values are the lower and upper bounds of the 90% confidence interval for the population mean, based on the sample data. The term “X” represents the sample mean ($235.12), while Z₀.05 and Z₀.95 are the z-scores for the 5th and 95th percentiles, corresponding to the bounds of the interval. The term σ/√n is the standard error of the mean. Thus, these calculations indicate that, with 90% confidence, the true average sale amount lies between $214.77 and $255.47.

In terms of the sampling distribution, these values indicate the range in which the true population mean is likely to fall, considering the variability in the sample data and the chosen confidence level. The narrower the interval, the more precise the estimate; the wider the interval reflects more uncertainty.

5. What do these values mean in terms of the expected total sales for the year?

The lower and upper bounds of the confidence interval for the average sale amount translate to a range for the total annual sales. Using these bounds, the expected total sales can be estimated as:

- Minimum total sales estimate: $214.77 × 50,000 = $10,738,500

- Maximum total sales estimate: $255.47 × 50,000 = $12,773,500

These bounds give the business owners a range within which the total sales for the year are likely to fall, with 90% confidence. Such an interval can be instrumental for planning, budgeting, and strategic decision-making, offering a probabilistic understanding of their revenue expectations based on the sample data.

References

  • Agresti, A., & Franklin, C. (2017). Statistics: The Art and Science of Learning from Data. Pearson.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W. H. Freeman.
  • Weiss, N. A. (2012). Introductory Statistics. Pearson.
  • Ott, R. L., & Longnecker, M. (2015). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
  • Lehman, A. (2013). Understanding and Applying Basic Statistical Methods. John Wiley & Sons.
  • Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
  • Freund, J. E. (2010). Modern Elementary Statistics. Prentice Hall.
  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Scholkopf, B., & Burges, C. (2018). Statistics and Data Analysis in Business and Economics. Springer.
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