Part 1 Multiple Choice: The Gallup Poll Interviewed 1423 Ran

Part 1 Multiple Choice: The Gallup Poll interviewed 1423 Randomly Selec

Analyze the provided statistical scenario involving Gallup Poll data, probability distributions, descriptive statistics, probability calculations, and interpretation of statistical events. The task involves identifying population parameters, classifying statistical measures, describing distributions, calculating probabilities, interpreting Bayesian probability, and analyzing distribution shapes and properties across multiple contexts. Additionally, it includes solving problems related to gambling profits, pregnancy durations, misleading resumes, and sampling probabilities, requiring clear statistical reasoning, formula application, and proper data interpretation.

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The data collected by the Gallup Poll, which involved interviewing 1,423 randomly selected American citizens, serves as a foundational example of inferential statistics. When 44% of respondents identified violence as the most bothersome TV content, this percentage functions as a sample statistic—a measure derived from the sample data used to estimate a population parameter. The distinction between a parameter and a statistic is crucial: whereas a parameter describes a characteristic of an entire population, a statistic describes a sample and is used to make inferences about the population (Moore et al., 2013).

The concept of a distribution is fundamental in understanding how variables behave. The distribution of a variable indicates the range of values it can take and the frequency of these values. For example, when considering the number of calls for reservations to Regional Airways, the data follow a binomial distribution because each call independently results in either a reservation or not, with a fixed probability of success (p = 0.85). Binomial distributions are discrete and characterized by parameters n (number of trials) and p (probability of success), which in this case, with n = 100 and p = 0.85, describe the expected number of reservation calls (Devore, 2011).

In describing the normal distribution of city gas mileage, the parameters are the mean and standard deviation. For instance, a mean of 21.2 MPG and a standard deviation of 5.4 MPG describe the spread and center of the data. Calculations such as finding the probability of a vehicle achieving 32 MPG or higher involve standardizing the observed value using the z-score formula, then consulting standard normal tables (Wackerly et al., 2008). For example, the probability that a randomly selected vehicle has a city MPG of 32 or more is computed as P(Z ≥ (32 - 21.2)/5.4).

Sample mean calculations and their distributions are grounded in the Central Limit Theorem, which states that the sampling distribution of the sample mean approaches a normal distribution with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size (n). For example, with n=36, the standard error is σ/√n = 5.4/6. Then, the probability that the mean MPG exceeds 23 uses the standard normal distribution with this standard error (Freeman et al., 2010).

When analyzing restaurant ratings and prices, percent calculations provide insights into data distribution. Determining what percentage of restaurants received a ‘Good’ rating involves dividing the count by the total sample size. Similarly, analyzing the probability of a restaurant being ‘Excellent’ given a certain meal price involves conditional probability concepts, calculated as P(E | Price), which is P(E and Price)/P(Price) (Casella & Berger, 2002).

The probability of selecting left-handed people from a sample is modeled using the binomial distribution since each individual can be classified as left-handed or not, with a fixed probability p=0.10. Calculating the probability of exactly 5 left-handed individuals out of 20 employs the binomial probability formula P(X=k) = C(n, k) p^k (1-p)^{n-k} (Ross, 2010). The expected number is n·p, which for n=20 and p=0.10, equals 2.

In analyzing employment data, the probabilities of combined events such as being married and college graduated utilize rules of joint probability and conditional probability. For example, P(M and C) is found using P(M|C)·P(C). Independence between events M and C would be established if P(M and C) equals P(M)·P(C); otherwise, they are dependent. The calculations demonstrate the relationships among these probabilities and the use of Venn diagrams for visualization (Kreyszig, 2011).

The normal distribution describes many natural phenomena, characterized by symmetry and a bell-shaped curve. The probability calculations related to the distribution of bottle contents involve finding the z-score for the mean volume and then deciphering the area under the curve. Similarly, for the marble problem, basic probability rules determine the chance of drawing specific color combinations without replacement, requiring multiplication and conditional probability (Lehmann & Romano, 2005).

Calculations of normal curve areas between specific z-values are performed using standard normal tables or software, providing probabilities like the area between -1.6 and 2.9. These areas help describe the likelihood that a variable falls within certain intervals, key to hypotheses testing and confidence interval construction (Wackerly et al., 2008). Overall, understanding the shape and properties of the normal distribution aids in interpreting real-world data, as it frequently approximates many natural and social phenomena.

The problem-solving section encompasses applied probability and statistics, requiring the application of formulas for binomial and normal distributions. Wendy’s gambling profits involve summing probabilities from discrete distributions, while pregnancy durations utilize the normal distribution with specified parameters. For example, finding the probability that a pregnancy lasts less than 200 days requires standardizing and using the cumulative distribution function.

Similarly, the analysis of resumes among executives employs binomial probability calculations for the exact number of misleading resumes, the probability of at least one misleading resume, and the expected count. The calculations involve combinatorial coefficients and the binomial probability formula, which are essential in quality control, risk assessment, and decision-making processes in applied statistics (Moore et al., 2013).

In conclusion, this extensive set of statistical problems demonstrates the application of core concepts such as probability distributions, descriptive statistics, inference, and hypothesis testing. Mastery of these concepts enables analysts to interpret data accurately, make informed predictions, and support decision-making based on statistical evidence. The integration of theory, calculation, and interpretation forms the backbone of statistical literacy essential in diverse fields from social sciences to business.

References

  • Casella, G., & Berger, R. L. (2002). Statistical inference (2nd ed.). Duxbury.
  • Devore, J. L. (2011). Probability and statistics for engineering and the sciences (8th ed.). Brooks Cole.
  • Freeman, H., et al. (2010). Statistics: Informed decisions using data (2nd ed.). Pearson.
  • Kreyszig, E. (2011). Advanced engineering mathematics. John Wiley & Sons.
  • Lehmann, E. L., & Romano, J. P. (2005). Testing statistical hypotheses (3rd ed.). Springer.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2013). Introduction to the practice of statistics. W. H. Freeman.
  • Ross, S. M. (2010). Introduction to probability models. Academic Press.
  • Wackerly, D. D., Mendenhall, W., & Scheaffer, R. L. (2008). Mathematical statistics with applications (7th ed.). Cengage Learning.

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