Problems Related To Distributions, Probability, And Statisti ✓ Solved

Problems related to distributions probability and statistical concepts

Problems related to distributions, probability, and statistical concepts

From the time of early studies by Sir Francis Galton in the late nineteenth century linking it with mental ability, the cranial capacity of the human skull has played an important role in arguments about IQ, racial differences, and evolution, sometimes with serious consequences. Suppose that the mean cranial capacity measurement for modern, adult males is 1171 cc (cubic centimeters) and that the standard deviation is 283 cc. Complete the following statements about the distribution of cranial capacity measurement for modern, adult males. (a) According to Chebyshev’s theorem, at least 36% of the measurements lie between _____ cc and _____ cc. (Round your answer to the nearest integer.) (b) According to Chebyshev’s theorem, at least _____% of the measurements lie between 605 cc and 1737 cc. (c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately _____% of the measurements lie between 605 cc and 1737 cc. (d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the measurements lie between _____ cc and _____ cc.

Find in the probability table the values to give a legitimate discrete probability distribution for the discrete random variable X, whose possible values are 1, 3, 4, 5, and 6.

An ordinary (fair) coin is tossed 3 times. Outcomes are triples of "heads" (h) and "tails" (t). For each outcome, let R be the number of tails in the outcome. For example, if the outcome is tht, then R = 2. Suppose the random variable X is defined as X = 4R - 2R² - 2. Calculate the probability distribution function pX(x). Fill in values of X for each outcome and compute the associated probabilities.

Let X be a random variable with a given probability distribution: Value x of X and P(X=x). Given the values, find the expectation E(X) and variance Var(X).

A machine that manufactures automobile pistons is estimated to produce a defective piston 3% of the time. Suppose a random sample of 90 pistons is taken. (a) Estimate the expected number of defective pistons. (b) Quantify the uncertainty by calculating the standard deviation of the number of defective pistons.

Anita’s fast food chain tracks the proportion of customers who dine in. If 48% order food to go, what is the probability that exactly 2 out of 4 randomly selected customers order to go?

Suppose 45% of all babies born in a hospital are boys. If 7 babies are randomly selected, what is the probability that fewer than 2 are boys? Provide intermediate calculations and round appropriately.

Loretta, age eighty, is interested in her systolic blood pressure compared to peers. The mean systolic BP for women over 75 is 134.3 mmHg, with a standard deviation of 5.6 mmHg. (a) Using Chebyshev’s theorem, find the minimum percentage of measurements between 125.9 mmHg and 142.7 mmHg. (b) The percentage between 123.1 mmHg and 145.5 mmHg. (c) Assuming a bell shape, approximate percentage between these values using the empirical rule. (d) For a bell-shaped distribution, estimate the range containing 99.7% of measurements.

Fill in the probability distribution values for a discrete random variable X with values 0, 1, 2, 3, and 4. Then, considering a scenario where Clara guesses the suit of cards randomly, estimate the expected number of correct guesses in 15 trials and the standard deviation, assuming no clairvoyance.

Paper For Above Instructions

This comprehensive analysis explores various statistical concepts, including distribution characteristics, probability calculations, and real-world applications related to human biological measures, manufacturing quality control, and game theory. Each problem exemplifies fundamental probability and statistics principles, offering insights into their practical applications.

Analysis of Cranial Capacity Distributions

The distribution of cranial capacities is often examined within a statistical framework to understand biological variability and inferential insights. Given a mean (μ) of 1171 cc and a standard deviation (σ) of 283 cc, Chebyshev’s theorem provides bounds applicable regardless of the distribution shape, while the empirical rule applies specifically to normal distributions.

(Part a) Chebyshev’s theorem states that at least (1 - 1/k²) × 100% of data lies within k standard deviations of the mean. To find the interval where at least 36% of observations reside, we set:

1 - 1/k² = 0.36 → 1/k² = 0.64 → k² = 1/0.64 ≈ 1.5625 → k ≈ 1.25.

Then, the interval is:

1171 - 1.25×283 ≈ 1171 - 353 ≈ 818 cc,

1171 + 353 ≈ 1524 cc.

Rounding to the nearest integer, the interval is approximately 818 cc to 1524 cc.

(Part b) Using the same approach for at least 80% (since 100% - 20% = 80%), k for 80% is:

1 - 1/k² = 0.8 → 1/k²= 0.2 → k²= 5 → k=√5 ≈ 2.236.

The interval:

1171 - 2.236×283 ≈ 1171 - 632.8 ≈ 538 cc,

1171 + 632.8 ≈ 1804.8 cc.

Thus, at least 80% of the cranial capacities lie between approximately 538 cc and 1805 cc.

(Part c) Assuming a bell-shaped distribution, approximately 95% of measurements fall within two standard deviations: [μ - 2σ, μ + 2σ] which is:

[1171 - 2×283, 1171 + 2×283] = [1171 - 566, 1171 + 566] = [605 cc, 1737 cc].

Therefore, approximately 95% lie between 605 cc and 1737 cc, which aligns with empirical rule expectations.

(Part d) For 68%, which corresponds to one standard deviation:

[1171 - 283, 1171 + 283] = [888 cc, 1454 cc].

Probability Distributions for Discrete Variables

For the discrete random variable X with possible values 1, 3, 4, 5, and 6, the probability values need to be assigned such that their sum equals 1, satisfying the properties of a probability distribution. For example:

  • x=1, p=0.1
  • x=3, p=0.2
  • x=4, p=0.3
  • x=5, p=0.2
  • x=6, p=0.2

The actual probabilities depend on the context definitively provided, but assume these as illustrative for the method.

Probability Distribution of a Coin Toss Experiment

In three tosses of a fair coin, the number of tails R can be 0, 1, 2, or 3, with probabilities given by the binomial distribution:

P(R=r) = C(3,r) × (0.5)^r × (0.5)^{3-r}.

Corresponding X values are calculated as X = 4R - 2R² - 2 for each R:

  • R=0: X=4×0 - 2×0 - 2 = -2
  • R=1: X=4×1 - 2×1 - 2 = 4 - 2 - 2= 0
  • R=2: X=8 - 8 - 2= -2
  • R=3: X=12 - 18 - 2= -8

Probability for each X is calculated accordingly, using the binomial probabilities.

Estimation of Defective Items in Manufacturing

The expected number (mean) for defective pistons in a sample of 90, with defect rate p=0.03, is:

μ = n × p = 90 × 0.03 = 2.7 pistons.

Standard deviation:

σ = √(n × p × (1 - p)) = √(90 × 0.03 × 0.97) ≈ √(2.6259) ≈ 1.620.

These estimates assist in quality control decisions.

Customer Preferences and Probabilities

Given a proportion of 48% for dine-in customers, the probability exactly 2 out of 4 select dine-in follows a binomial distribution:

P = C(4,2) × 0.48^2 × 0.52^2, which calculates as approximately 0.265.

Probability of Baby Gender Composition

With probability 0.45 each baby is a boy, binomial distribution applies:

P(X

Rounded to two decimal places, the probability is approximately 0.26.

Blood Pressure Analysis

Using Chebyshev’s theorem:

Part a: at least (1 - 1/ (k)²) × 100%, with k = (142.7 - 134.3)/5.6 ≈ 1.52,

so at least 1 - 1/1.52² ≈ 56.7% of measurements lie between 125.9 mmHg and 142.7 mmHg.

Part b: for the wider interval, similar calculations yield at least approximately 94% coverage.

Assuming a normal distribution, the empirical rule indicates about 95% within two standard deviations: [124.1 mmHg, 144.5 mmHg].

For 99.7%, a range approximately three standard deviations wide: [118.5 mmHg, 150.1 mmHg].

Probability Distribution for Discrete Variable X

Assigning probabilities such that their sum equals 1, e.g.,

  • 0: P=0.1
  • 1: P=0.2
  • 2: P=0.3
  • 3: P=0.2
  • 4: P=0.2

enables computation of expected values and variances accordingly.

Card Guessing Scenario

Each guess has a 1/4 probability of being correct, since the suits are evenly distributed, and guesses are random. For 15 guesses, the expected number of correct guesses is 15 × 0.25 = 3. The standard deviation for the number of correct guesses is √(15×0.25×0.75) ≈ 1.5.

These calculations demonstrate basic probability models applied to games and guessing strategies.

Summary

Across these problems, core statistical methods such as Chebyshev’s theorem, empirical rule, probability distributions, and expectation and variance calculations are crucial for understanding data variability, making predictions, and interpreting real-world phenomena. Such tools are essential in fields ranging from biomedical research to manufacturing and social sciences.

References

  • Galton, F. (1883). Inquiries into Human Faculty and Its Development. Macmillan.
  • Gould, S. J. (1996). The Mismeasure of Man. W.W. Norton & Company.
  • DeGroot, M., & Schervish, M. (2012). Probability and Statistics. Pearson.
  • Freedman, D., Pisani, R., & Purves, R. (2007). Statistics. W.W. Norton & Company.
  • Ross, S. (2014). A First Course in Probability. Pearson.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics. W.H. Freeman.
  • Wasserman, L. (2004). All of Statistics. Springer.
  • Wilkinson, L., & Task Force on Statistical Inference. (1999). Statistical Methods in Psychology Journals. American Psychologist.
  • Agresti, A., & Franklin, C. (2013). Statistics: The Art and Science of Learning from Data. Pearson.
  • Devore, J. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.

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