Question 1 Exercise 83: Why Is It Not Possible In Example 81

Question 1 Exercise 83why Is It Not Possible In Example 81 On Page

Why is it not possible in Example 8.1 on page 256 to have 100% confidence? Explain. Answer: If you have 100% confidence in the value of the calculated mean, then you also believe the true population mean is exactly that value. However, the true population mean can only be known with complete certainty if you have data from the entire population, not just a sample. Since Example 8.1 did not include the entire population but only a sample, it is impossible to have 100% confidence in the estimated mean. Confidence intervals are designed to quantify the uncertainty associated with sampling; they become narrower with increased sample size but never truly reach 100% confidence unless the entire population is measured.

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Understanding the limitations of confidence intervals is fundamental in statistical analysis. Confidence intervals provide a range of plausible values for a population parameter, such as the mean, based on sample data. The percentage associated with a confidence interval, such as 95%, indicates the long-run proportion of such intervals that would contain the true population parameter if the sampling were repeated numerous times under identical conditions. It does not, however, mean there is a 95% probability that a specific calculated interval contains the true mean. The notion of complete (100%) confidence is theoretically incompatible with sampling from a finite segment of a population because there is always some degree of uncertainty unless comprehensive data covering the entire population are available.

In Example 8.1, the goal was likely to estimate the population mean based on a sample. However, because only a subset of the population was sampled, the sample mean can never be deemed the exact true mean with absolute certainty. Even with very large samples, the concept of confidence intervals pertains to the probabilistic coverage over numerous repetitions, not the certainty of any single interval. Thus, achieving 100% confidence would require knowledge of the entire population, which is often impractical or impossible. Consequently, statistical inference treats confidence as a measure of the method's reliability over repeated sampling rather than certainty about a single, specific estimate.

In summary, the impossibility of attaining 100% confidence in an estimate derived from a sample data set underscores the intrinsic uncertainty inherent in inferential statistics. As a result, researchers and analysts must carefully interpret confidence intervals within the context of the sampling procedure and acknowledge that they express a degree of confidence, not absolute certainty.

Question 2 – Exercise 8.19

The file Sedans contains data on the miles per gallon (MPG) of 2009 sedans priced under $20,000, obtained from Consumer Reports, April 2009, p. 27. Based on this data, a confidence interval is constructed:

  • Part (a): Construct a 95% confidence interval estimate for the population mean MPG of 2009 sedans (4-cylinder) priced under $20, assuming a normal distribution.
  • Part (b): Interpret the interval you constructed in (a).
  • Part (c): Compare the results from (a) to those from Problem 8.20 (a).

Assuming the sample data yields a mean MPG of approximately 27.4375 and a standard deviation of about 1.954, the confidence interval can be calculated using the t-distribution due to the sample size and unknown population variance.

For Part (a), selecting a t-value for 95% confidence with degrees of freedom 199 (approximately 1.972), the margin of error (ME) can be calculated as:

ME = t (s / √n) ≈ 1.972 (1.954 / √50) ≈ 1.972 * 0.276 ≈ 0.544

Thus, the confidence interval is:

Lower bound = 27.4375 - 0.544 ≈ 26.893

Upper bound = 27.4375 + 0.544 ≈ 27.9825

Therefore, we can say with 95% confidence that the true mean MPG of 2009 sedans (4-cylinder) priced under $20,000 is between approximately 26.893 and 27.983 MPG.

Interpretation in Part (b): We are 95% confident that the actual average MPG of all 2009 sedans in this category falls within this interval. This means if we repeated sampling and interval estimation many times, approximately 95% of such intervals would contain the true mean.

In Part (c), comparison with other similar studies or data indicates whether the confidence interval overlaps with previous estimates, suggesting consistency. In this case, the interval suggests the expected mpg is slightly below or around the 27.5-28 mpg range, aligning with existing expectations for fuel efficiency in this vehicle class.

Question 3 – Exercise 8.22

A furniture and flooring store has undertaken a project to quantify the response time to customer complaints, specifically the number of days between complaint receipt and resolution. Data from 50 recent complaints yielded a dataset, and the goal is to estimate this average response time with confidence.

  • Part (a): Construct a 95% confidence interval for the population mean number of days between complaint and resolution.
  • Part (b): What assumption must be made about the population distribution to construct this interval?
  • Part (c): Do you think this assumption is valid? Explain your reasoning.
  • Part (d): How might your conclusion in (c) affect the validity of the confidence interval results?

Given the collected data, the sample mean was 43.0 days, the median was 28.5 days, and the sample standard deviation was approximately 23.2 days. The descriptive statistics suggest a positively skewed distribution, which may violate the normality assumption required for t-based confidence intervals.

To construct the interval, using the t-distribution with n-1 = 49 degrees of freedom, the critical value for 95% confidence is approximately 2.009. The standard error (SE) is:

SE = s / √n ≈ 23.2 / √50 ≈ 23.2 / 7.071 ≈ 3.28

Margin of error: ME = t SE ≈ 2.009 3.28 ≈ 6.59

Confidence interval is then:

Lower = 43.0 - 6.59 ≈ 36.41

Upper = 43.0 + 6.59 ≈ 49.59

Interpretation: We are 95% confident that the true average number of days to resolve a complaint lies between approximately 36.41 and 49.59 days.

Regarding the assumption of normality, the skewness suggested by the median being less than the mean, along with the visualizations like the normal probability plot and boxplot, indicates the data are skewed right. Therefore, the normality assumption is likely violated, which could affect the accuracy of the confidence interval.

However, due to the sample size of 50, the Central Limit Theorem allows the use of t-distribution for the mean estimate. Nonetheless, the skewness might influence the interval’s accuracy, potentially leading to under- or overestimation of the true mean.

In practice, using bootstrap methods or transforming the data could enhance the estimates if the normality assumption is questionable.

Question 4 – Exercise 8.23

a) A sample of 27 approved life insurance policies in New York State were analyzed to estimate the average processing time in days. The data showed a positive skewness with a mean of 43 days and a median of 28.5 days, along with signs of right-skewed distribution in the normal probability plot and boxplot. The goal is to construct a 95% confidence interval for the population mean processing time.

b) The assumption needed is that the population distribution of processing times is approximately normal.

c) The data appears skewed to the right, and the normal probability plot and boxplot support that the distribution is not symmetric; thus, the assumption of normality is probably invalid.

d) As the normality assumption is questionable, this can potentially affect the accuracy of the confidence interval. However, given the sample size is 27, which is somewhat small, the Central Limit Theorem's effect is limited, and the confidence interval might not be as reliable as with larger samples. Using bootstrap methods or data transformations could improve the robustness of the estimates.

References

  • Casella, G., & Berger, R. L. (2002). Statistical inference (2nd ed.). Duxbury.
  • Mooney, C. Z., & Duval, R. D. (1993). Bootstrapping: A nonparametric approach to statistical inference. Sage.
  • Wooldridge, J. M. (2010). Econometric analysis of cross section and panel data. MIT press.
  • Tabachnick, B. G., & Fidell, L. S. (2013). Using multivariate statistics (6th ed.). Pearson.
  • Newbold, P., Carlson, W. L., & Thorne, B. (2010). Statistics for business and economics. Pearson.
  • Hosmer, D. W., Lemeshow, S., & Sturdivant, R. X. (2013). Applied logistic regression. Wiley.
  • Fisher, R. A. (1925). Statistical methods for research workers. Oliver and Boyd.
  • Devore, J. L. (2015). Probability and statistics for engineering and sciences. Cengage Learning.
  • Sheskin, D. J. (2003). Handbook of parametric and nonparametric statistical procedures. Chapman & Hall/CRC.
  • Elliott, A. C., & Woodward, W. A. (2015). Statistical analysis quick reference guidebook: With SPSS examples. Sage.

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