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Statistics for Business: Random Variables and Probability Distributions

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  • How do you find the missing probability in a discrete probability distribution when all other probabilities are known?
    Subtract the sum of the known probabilities from 1.
  • Is the amount of water a person drinks in one day a discrete or continuous random variable?
    It is a continuous random variable because the amount can take any real value within a range.
  • What is the binomial probability formula used for?
    To find the probability of a specific number of successes in a fixed number of independent trials with the same probability of success.
  • How do you calculate the probability that exactly 2 widgets are defective if the defect rate is 10% and 8 widgets are selected?
    Use the binomial probability formula with n=8, p=0.1, and x=2.
  • What is the expected value in a lottery game where the probability of winning is 1 in 5000 and the prize is \$1000?
    Expected value = (Probability of winning × Prize) + (Probability of losing × Cost of ticket).
  • What distribution models the number of power outages in a city per month if the average is 2.3 outages?
    The Poisson distribution models the number of outages.
  • How is the maximum time determined in a uniform distribution with a 90% probability the event occurs within that time?
    Use the uniform distribution formula to find the time corresponding to the 90th percentile.
  • What is the total area under the standard normal curve to the left of z = -1.29 or to the right of z = 1.29?
    The total area is approximately 0.20, representing the combined tails of the distribution.
  • How do you find the probability that a standardized test score falls between z = -1.2 and z = 2.0?
    Calculate the area under the standard normal curve between z = -1.2 and z = 2.0 using a Z-table or calculator.
  • What is the probability that a plant's height is between 64 cm and 72 cm if heights are normally distributed with mean 50 cm and standard deviation 8 cm?
    Use the normal distribution to find the area between 64 and 72 cm, which corresponds to the shaded region in the graph.
  • What does the shaded area in the standard normal distribution graph between z = -1.29 and z = 1.29 represent?
    It represents the probability that a value falls within that z-score range, approximately 80%.
  • What is the continuity correction when approximating a binomial probability with a normal distribution?
    Adjust the discrete x-value by ±0.5 before converting to a z-score for the normal approximation.
  • How do you determine if an event is unusual using probability?
    An event is unusual if its probability is less than 0.05.
  • What is the mean and standard deviation in a normal distribution used for?
    The mean indicates the center, and the standard deviation measures the spread of the distribution.
  • How do you calculate the probability of at least 2 employees preferring remote work in a sample of 8 if 40% prefer remote work?
    Use the binomial distribution to find P(X ≥ 2) with n=8 and p=0.4.
  • What is the probability that the first box with damaged fruit is found in the first or second box if the damage rate is 1 in 50?
    Calculate the probability using the geometric distribution for the first or second trial.
  • What is the significance of the z-value in a standard normal distribution?
    The z-value represents the number of standard deviations a data point is from the mean.
  • How do you interpret a probability of 0.7745 for a test score between 73 and 88 with mean 82 and SD 6?
    It means there is a 77.45% chance a randomly chosen student scored between 73 and 88, which is not unusual.
  • What is the formula for the binomial probability P(X = x)?
    P(X = x) = \(\binom{n}{x} p^x (1-p)^{n-x}\)
  • What is the formula for the Poisson probability P(X = x)?
    P(X = x) = \(\frac{e^{-\lambda} \lambda^x}{x!}\)