Midterm 2 Review Updated With Current Text And Answers

Midterm 2 Review Updated To Current Text With Answersdsci 245a Ma

This assignment involves analyzing battery life data, probability distributions, hypothesis testing, and statistical calculations based on the provided information. It requires constructing contingency tables, calculating probabilities, understanding normal, binomial, Poisson, and hypergeometric distributions, and applying statistical formulas to interpret battery performance and related scenarios.

Paper For Above instruction

The analysis of battery life and reliability is a fundamental aspect of quality control and statistical inference in manufacturing and engineering. The provided data offers a comprehensive scenario within which various statistical methods can be applied to evaluate battery performance, production differences, and probability distributions. This paper discusses the statistical analysis based on the given data, including contingency table construction, probability calculations, distribution assessments, and hypothesis testing, emphasizing their application and interpretation.

Contingency Table and Initial Probabilities

The manufacturer sampled batteries during an 8-hour shift, taking two batteries each hour, resulting in 16 batteries. Of these, 4 batteries in the second half of the shift and 1 in the first half lasted less than 400 hours. Using this data, a contingency table based on relative frequencies is constructed to analyze the association between shift timing and battery longevity.

The probability that a randomly selected battery was produced during the first half is P(first half) = 0.5. Likewise, the probability that a battery lasts less than 400 hours is P(less than 400) = 0.3125.

The probability of selecting a battery during the first half that lasts less than 400 hours is P(first half AND less than 400) = 0.0625. The conditional probability that a battery lasts less than 400 hours, given it was produced in the first half, is P(less than 400 | first half) = 0.0625 / 0.5 = 0.125.

Similarly, the probability that a battery was produced during the first half given it lasted less than 400 hours is P(first half | less than 400) = 0.0625 / 0.3125 ≈ 0.2. These calculations suggest that battery life is dependent on production timing, as the probabilities do not multiply to the joint probability, indicating dependence.

Normal Distribution of Battery Life

Assuming battery life is normally distributed with a mean (μ) of 500 hours and standard deviation (σ) of 150 hours, the analysis proceeds to estimate various probabilities:

• The proportion of batteries lasting exactly 500 hours is zero, since the normal distribution is continuous.
• The probability that a battery lasts less than 350 hours is calculated as P(X
• The percentage of batteries lasting at least 300 hours is P(X ≥ 300) = NormalCDF(300, ∞, 500, 150) ≈ 0.798. This indicates most batteries last longer than 300 hours.
• The probability a battery lasts between 200 and 900 hours is P(200
• The battery life exceeded by only 2% of all batteries is determined through the inverse normal function: P(X > x) = 0.02, thus invNorm(0.98, 500, 150) ≈ 808 hours.

Binomial and Poisson Probabilities

Considering a proportion of batteries lasting less than 300 hours as 10% (p = 0.1), the probability during an 8-hour shift of sampling exactly 3 such batteries (out of 16 samples) is determined by the binomial probability mass function:

P(X=3) = BinomialPDF(n=16, p=0.1, x=3) ≈ 0.20.

The probability of at least one battery lasting less than 300 hours is P(X ≥ 1) = 1 – BinomialPDF(16, 0.1, 0) ≈ 0.8147.

For battery failure modeling over time, the Poisson distribution is used. If the average failure rate is 0.01 failures per hour, the expected number of failures in 50 hours is E = 0.01 * 50 = 0.5. The probability of no failures in this period is P(X=0) = e^-0.5 ≈ 0.6065, and the probability of 2 or more failures is P(X ≥ 2) = 1 – P(0) – P(1) ≈ 0.0758.

Distribution of Battery Life Lengths

Given a rounded probability distribution with specific life spans, calculating the probability that a battery lasts at least 700 hours involves adding the probabilities for 700 and 800 hours, summing to approximately 0.1875. The probability that the battery lasts less than 1100 hours is 1 minus the probability it lasts exactly 1100 hours, or 1 – 0.0625 = 0.9375.

The expected value of the distribution (mean life span) can be estimated by summing the product of midpoints and probabilities, resulting in an expected life close to 506 hours. The risk or variance is derived from summing squared deviations weighted by probabilities, with the standard deviation calculated to be approximately 204.54 hours.

Battery Lifespan Analysis with Minimum and Maximum Constraints

With a minimum of 200 hours and a maximum of 600 hours, the probability of a battery lasting less than 300 hours is calculated as P(200

When the most likely (mode) is at 500 hours, the probability that a battery lasts less than 300 hours slightly increases to about 0.0833, given the more peaked distribution.

Failure Rate and Predictive Probabilities

Assuming an average of 3 failures per 300 hours (failure rate λ = 0.01 per hour), the expected number of failures in any given period is 0.01 multiplied by hours. The probability of fewer than 2 failures in 200 hours is P(X

Quality Assurance and Batch Testing

For batch testing, the probability that no more than 20% of 5 randomly sampled batteries fail in less than 700 hours is calculated using the hypergeometric distribution. With 200 defective batteries in a batch of 1000, the probability of accepting the batch (acceptance threshold with at most 1 failure in the sample) is approximately 0.7375, resulting in a rejection probability of about 0.2625.

Additional Real-World Applications

Applying various distributions such as normal, exponential, uniform, and binomial, these statistical methods enable engineers, quality control specialists, and data analysts to predict battery performance, assess reliability, and make decisions about manufacturing processes. Proper understanding of these models enhances predictive maintenance, reduces costs, and improves product quality during production and usage cycles.

References

• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (9th ed.). Cengage Learning.
• Ross, S. M. (2014). Introduction to Probability Models (11th ed.). Academic Press.
• Moore, D. S., & McCabe, G. P. (2009). Introduction to the Practice of Statistics (7th ed.). W. H. Freeman and Company.
• Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
• Freund, J. E., & Walpole, R. E. (1987). Mathematical Statistics with Applications. Prentice-Hall.
• Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate Discrete Distributions. Wiley-Interscience.
• Rea, L. M., & Parker, R. A. (2014). Designing and Conducting Survey Research: A Comprehensive Guide. Jossey-Bass.
• McClave, J. T., & Sincich, T. (2012). Statistics. Pearson Education.
• Rubin, D. B. (2004). Multiple Imputation for Nonresponse in Surveys. Wiley.
• Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman & Hall.