Practical Application Scenario 1 To Complete This Scenario U

Practical Application Scenario 1to Complete This Scenario Use Sas Ent

To complete this scenario, use SAS Enterprise Guide or the Confidence Interval Calculator and the Area Gas Prices document, provided in Updates and Handouts. The Star Tribune newspaper has decided to write an article about gasoline prices in the Twin Cities area (Minneapolis and St. Paul, Minnesota). The newspaper has designed a survey and taken a simple random sample of regular unleaded gas prices at 70 area stations. The results are in the Area Gas Prices file.

The newspaper's policy is that no data will be reported unless they are 95 percent confident that the numbers are correct. As the new manager of the data verification unit at the Star Tribune, you need to develop a 95 percent confidence interval for the average price of regular unleaded gasoline in the Twin Cities area. You already know how to compute sample parameters, like the sample mean and sample standard deviation; this scenario simply asks you to complete a confidence interval for the data provided and communicate the results to the Star Tribune's business manager.

Paper For Above instruction

The task involves calculating a 95% confidence interval for the mean price of regular unleaded gasoline in the Twin Cities, based on a sample of 70 stations. Using the data provided in the Area Gas Prices file and statistical tools like SAS Enterprise Guide or the Confidence Interval Calculator, the first step is to compute the sample mean (\(\bar{x}\)) and the sample standard deviation (s). Once these parameters are determined, the next step is to apply the formula for the confidence interval of the population mean when the population standard deviation is unknown, which typically is based on the t-distribution if the sample size is relatively small or the population standard deviation is unknown. For large samples, the Z-distribution can be used as an approximation. Assuming the sample size of 70 is sufficiently large, the formula simplifies to:

\[ CI = \bar{x} \pm Z_{\frac{\alpha}{2}} \times \frac{s}{\sqrt{n}} \]

where \( Z_{\frac{\alpha}{2}} \) is the critical Z-value for 95% confidence (approximately 1.96). After computing the interval, the results should be clearly communicated to the business manager, emphasizing the range within which the true average gasoline price likely falls with 95% confidence. This analysis ensures that the report adheres to the newspaper's policy of only reporting data with high confidence, supporting informed decision-making and accurate reporting.

Using the sample data, suppose the sample mean gasoline price is $3.15 per gallon and the sample standard deviation is $0.25. The standard error (SE) is calculated as:

\[ SE = \frac{0.25}{\sqrt{70}} \approx 0.03 \]

The margin of error (ME) is:

\[ ME = 1.96 \times 0.03 \approx 0.06 \]

Thus, the 95% confidence interval is:

\[ 3.15 \pm 0.06 \Rightarrow [3.09, 3.21] \]

This means the true average price of regular unleaded gasoline in the Twin Cities is estimated to lie between $3.09 and $3.21 with 95% confidence. The results can be formatted appropriately in the report to inform readers and stakeholders accurately.

In conclusion, the calculated confidence interval provides a statistically sound estimate of the average gasoline price in the Twin Cities area, allowing the Star Tribune to report the data transparently while satisfying its confidence policy. Accurate reporting based on this interval can influence consumer perceptions and policy discussions about fuel pricing in the region.

Practical Application Scenario 2

To complete this scenario, use the Sample Size Estimator file provided in Resources. The third shift at the Microsoft's Windows security unit in Redmond, Washington is considering replacing its coffee brewing units with new German-made brewers. As a team member, develop a 99 percent confidence interval for the average yield (cups per pound) for the new German coffee brewer. The margin of error should be no bigger than 0.3 cups, and based on experience with current coffee makers, the best estimate for the population standard deviation, sigma, is 1.2 cups.

Calculate the required sample size needed to achieve a margin of error no greater than 0.3 cups at 99% confidence. The formula for the sample size, n, when the population standard deviation is known, is:

\[ n = \left( \frac{Z_{\frac{\alpha}{2}} \times \sigma}{E} \right)^2 \]

where \( Z_{\frac{\alpha}{2}} \) is the Z-value for 99% confidence (approximately 2.576), \( \sigma = 1.2 \), and the desired margin of error \( E = 0.3 \). Substituting the values:

\[ n = \left( \frac{2.576 \times 1.2}{0.3} \right)^2 \approx \left( 10.33 \right)^2 = 106.66 \]

Rounding up, a sample size of 107 units is needed to ensure the margin of error does not exceed 0.3 cups with 99% confidence.

The above calculation indicates that sampling at least 107 coffee brewers will provide the desired precision for estimating the mean yield. After data collection, the sample mean can be used to construct the 99% confidence interval by applying:

\[ \bar{x} \pm Z_{0.005} \times \frac{\sigma}{\sqrt{n}} \]

where \(\bar{x}\) is the sample mean. For example, if the sample mean yield observed is 8.5 cups per pound, then the confidence interval becomes:

\[ 8.5 \pm 2.576 \times \frac{1.2}{\sqrt{107}} \approx 8.5 \pm 0.30 \Rightarrow [8.2, 8.8] \]

This interval indicates that, with 99% confidence, the true mean yield lies within this range. Communicating these findings helps stakeholders understand the expected performance of the new German coffee brewers and make informed purchasing decisions.

Conclusion

Both practical scenarios demonstrate how confidence intervals and sample size calculations are vital tools in real-world decision-making. The gasoline price analysis provides a statistically valid estimate that aids in transparent reporting, while the coffee brewer yield estimation ensures precise sampling plans to meet quality assurance standards. Proper application of these statistical techniques ultimately supports data-driven decisions, ensures accuracy, and enhances credibility in organizational reporting and procurement processes.

References

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