Finite Math 130 Final Exam Spring 2015 Central Washington Un ✓ Solved

Finite Math 130 Final Exam Spring 2015central Washington Universityju

Describe how to select a simple random sample from a population.

Describe how to select a systematic sample from a population.

Describe how to select a cluster sample from a population.

A random sample of customer order totals with an average of $78.25 and a population standard deviation of $22.50. Calculate a 90% confidence interval for the mean, given a sample size of 75 orders.

A random sample of 35 teenagers averaged 7.3 hours of sleep per night. Assume the population standard deviation is 1.8 hours. Calculate a 98% confidence interval for the mean.

Below is the data set for the amount of trash by ten households (in pounds per day). Assume the population is normally distributed.

  • 3.9
  • 4.6
  • 15.6
  • 10.5
  • 16.0
  • 6.7
  • 12.0
  • 9.2
  • 13.8
  • 16.8

Construct a 95% confidence interval for the mean based on the sample. Calculate the sample mean. (Hint: use the t-table.)

A company claims the average time a customer waits on hold is less than 5 minutes. A sample of 35 customers has an average wait time of 4.78 minutes. Assume the population standard deviation for wait time is 1.8 minutes. Test the company claim at the α = 0.05 significance level by comparing the calculated z-score to the critical z-score.

We are testing the claim that houses in a particular community average less than 90 days on the market. A random sample of 9 homes averaged 77.4 days on the market with a sample standard deviation of 29.6 days. Assume the population is normally distributed. Test the claim at the α = 0.05 significance level by comparing the calculated t-score to the critical t-score.

B.F. Retread, a tire manufacturer, wants to select one of three feasible prototype designs for a new tire; A, B, C. Sales demand is categorized into Low, Medium, and High, with the following probabilities:

  • Low: 0.30
  • Medium: 0.50
  • High: 0.20

Design A: Demand: 120,000 units

Design B: Demand: 130,000 units

Design C: Demand: 100,000 units

1) What is the best optimistic decision?

2) What is the best pessimist solution?

3) What is the best avoidance of regret solution?

Sample Paper For Above instruction

Introduction

The process of sampling is fundamental in statistics for understanding and analyzing populations effectively without surveying every individual. Different sampling techniques serve various purposes depending on the population structure and research goal. Additionally, confidence intervals provide estimates of population parameters, and hypothesis testing evaluates claims about those parameters. Decision theory assists in making optimal choices under uncertainty. This paper discusses methods for selecting samples, calculating confidence intervals, conducting hypothesis tests, and applying decision theory in practical scenarios.

Sampling Methods

Simple random sampling involves selecting units where each member of the population has an equal chance of being chosen, ensuring unbiased representation. Systematic sampling, on the other hand, selects every kth member after a random start, which simplifies the sampling process but requires careful handling to avoid periodicity bias. Cluster sampling divides the population into clusters—often geographically or naturally grouped—and randomly selects entire clusters for study, making it efficient when populations are widespread or dispersed.

Confidence Intervals

Calculating confidence intervals allows estimation of a population parameter with an associated level of confidence. For example, given a sample mean and known population standard deviation, the interval is calculated using the z-distribution. When the population standard deviation is unknown and the sample size is small or moderate, the t-distribution is used. For instance, in assessing the average trash weight of households, the sample mean and standard deviation determine the confidence interval, considering the sample size and data distribution. The formula for a confidence interval generally involves the sample statistic plus or minus a margin of error, which accounts for variability and desired confidence level.

Hypothesis Testing

Hypothesis testing evaluates claims about population parameters. For example, testing whether the average wait time is less than five minutes involves formulating null and alternative hypotheses, calculating a test statistic (z or t), and comparing it to critical values at a specified significance level (α). If the test statistic exceeds the critical value, the null hypothesis is rejected. This process is crucial in quality control, customer satisfaction analysis, and market analysis, providing evidence to accept or reject claims based on data.

Decision Theory and Application

Decision theory guides choosing among alternatives under uncertainty, often by calculating expected values or regrets. For example, in selecting a tire prototype, the manufacturer evaluates the probabilistic outcomes and chooses the design that maximizes expected profit or minimizes regret. The optimistic approach focuses on the best possible outcome, the pessimist on the worst, and the minimax regret strategy aims to minimize potential regret.

Conclusion

Understanding sampling techniques, confidence intervals, hypothesis testing, and decision theory equips statisticians and decision-makers with essential tools for data analysis and informed decision-making. Proper application of these methods leads to more accurate insights, better resource allocation, and effective strategies in various fields, including business, healthcare, and manufacturing.

References

  • Agresti, A., & Franklin, C. (2017). Statistics: The Art and Science of Learning from Data. Pearson.
  • Bluman, A. G. (2018). Elementary Statistics: A Step By Step Approach. McGraw-Hill Education.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury Press.
  • Loftus, T. (2010). Regression Modeling Strategies. Springer.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics. W. H. Freeman.
  • Rosenbaum, P. R. (2002). Observational Studies. Springer.
  • Sheldon, L. (2014). Decision Analysis for the Professional. CRC Press.
  • Wasserman, L. (2004). All of Statistics. Springer.
  • Zar, J. H. (2010). Biostatistical Analysis. Pearson.
  • Siegel, S., & Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill.

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