For P(x65), And P(62 ✓ Solved

1) For , give a) P (x 65), and c) P (62 65) = c) P (62 165lbs average (sample of 16): b) >162.5lbs average (sample of 36): c) P(158

1) For , give a) P (x 65), and c) P (62

In this assignment, we will tackle various probability and statistics questions, focusing on probability calculations, confidence intervals, and distributions. We will address problems involving normal distributions, binomial distributions, Poisson distributions, and the calculation of confidence intervals for population means and proportions.

Normal Distribution Calculations

1. Probability Calculations

Given the normal distribution with mean (μ) and standard deviation (σ), we need to compute:

  • a) P (x Use the Z-score formula: Z = (X - μ) / σ, where μ is the mean and σ is the standard deviation. For a specific calculation, we need the values of μ and σ.
  • b) P (x > 65): Also apply the Z-score formula and use Z-tables or software to find the probability.
  • c) P (62 Calculate the Z-scores for both values and find the associated probabilities, then compute P(62

Finding Z Values

2. Corresponding Z Values

Given a set of values, determine the corresponding Z-scores:

  • a) For x=54: Compute Z = (54 - μ) / σ.
  • b) For x=61.5: Compute Z = (61.5 - μ) / σ.
  • c) For x=69: Compute Z = (69 - μ) / σ.

Binomial Distribution Problems

3. Probability of Successes in Binomial Trials

For a binomial distribution, where you have a number of attempts (n) and success probability (p), compute:

  • a) P (10 Use binomial probabilities for the given range.
  • b) P (8 Calculate similarly.

Battery Lifespan Analysis

4. The lifespan of a battery follows a normal distribution with a mean of 1248 days and a standard deviation of 185 days. To find the proportion of batteries replaced under a guarantee of 1080 days, compute:

  • P(X
  • To ensure that no more than 10% is replaced, find the guarantee duration where P(X

Success Probabilities in Binomial Distributions

5. Given p = 0.6:

  • Three or fewer successes out of five attempts: Calculate using the binomial formula.
  • Between three and six successes out of seven attempts: Calculate using cumulative distribution functions.

Sample Size Analysis

In various scenarios, the sample size impacts the probability and confidence intervals. For a likelihood of p=0.3:

  • Four or more successes out of six attempts: Compute the complementary probabilities.
  • Likelihood of exactly two successes: Direct calculation using binomial probability.

Cinema Attendance Probability

For a cinema audience composed of 60% men and 40% women:

  • a) Probability of six women in a sample of six: Use binomial probability P(X=6).
  • b) Probability of more than three men: Calculate cumulative probabilities.
  • c) Probability of fewer than three women: Complementary calculations.

Quality Control Systems

10. In a quality control system:

  • Determine the probability of one or more defects in three samples and the final decision based on the quality checks.

Train Arrival Probabilities

12. Assess the likelihood of five trains arriving in ten minutes (Poisson distribution with λ=8), and calculate:

  • a) Probability of exactly five trains: Use the Poisson formula.
  • b) Probability of three or fewer trains: Sum up the probabilities for 0, 1, 2, and 3 trains.

Carpet Weaving Errors

For the probability of finding between 3 and 5 errors in a two-yard stretch, apply Poisson distribution where the mean is 3.

Population Weight Analysis

Regarding the average weight and variance:

  • Compute probabilities for sample means over specific weight thresholds.
  • Use the central limit theorem for samples of varying sizes.

Confidence Intervals

Construct 90% and 95% confidence intervals for the specified datasets while considering the population variance.

References

  • Anderson, D. R., Sweeney, D. J., & Williams, T. A. (2018). Statistics for Business and Economics. Cengage Learning.
  • Bernoulli, J. (1713). Exposition of a New Theory on Probability.
  • Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for The Behavioral Sciences. Cengage Learning.
  • Triola, M. F. (2018). Elementary Statistics. Pearson.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Wiley.
  • Barrow, A. (2019). Applied Statistics. McGraw Hill.
  • McClave, J. T., & Sincich, T. (2017). Statistics. Pearson.
  • Moore, D. S., & McCabe, G. P. (2017). Introduction to the Practice of Statistics. W. H. Freeman.
  • Gibbons, J. D., & Chakraborti, S. (2011). Nonparametric Statistical Inference. CRC Press.
  • Keller, G. (2018). Statistics. Cengage Learning.

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