Ch. 5 - Discrete Probability Distributions

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 5 - Discrete Probability Distributions

Problem 5.2.29b

Chapter 5, Problem 5.2.29b

In Exercises 29 and 30, assume that different groups of couples use the XSORT method of gender selection and each couple gives birth to one baby. The XSORT method is designed to increase the likelihood that a baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5.

Gender Selection Assume that the groups consist of 36 couples.

b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high.

Verified step by step guidance

Step 1: Understand the problem. The XSORT method is assumed to have no effect, so the probability of a girl (success) is p = 0.5. The number of trials (couples) is n = 36. We are tasked with using the range rule of thumb to find the values that separate significantly low and significantly high results.

Step 2: Recall the range rule of thumb. It states that values are significantly low if they are below μ - 2σ and significantly high if they are above μ + 2σ, where μ is the mean and σ is the standard deviation of the binomial distribution.

Step 3: Calculate the mean (μ) of the binomial distribution. The formula for the mean is μ = n × p. Substitute n = 36 and p = 0.5 into the formula.

Step 4: Calculate the standard deviation (σ) of the binomial distribution. The formula for the standard deviation is σ = √(n × p × (1 - p)). Substitute n = 36, p = 0.5, and (1 - p) = 0.5 into the formula.

Step 5: Use the range rule of thumb to find the significantly low and high values. Compute μ - 2σ for the significantly low threshold and μ + 2σ for the significantly high threshold. These values will define the range of typical results.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, the probability of having a girl is given as 0.5, indicating that there is an equal chance of having a boy or a girl. Understanding probability is essential for analyzing outcomes in gender selection and interpreting the results of the XSORT method.

Range Rule of Thumb

The range rule of thumb is a statistical guideline that suggests using the range of data to identify significant deviations from the mean. Specifically, it states that values beyond two standard deviations from the mean can be considered significantly low or high. This concept is crucial for determining the thresholds for what constitutes unusual results in the context of the birth outcomes of the couples.

Standard Deviation

Standard deviation is a statistic that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates a wider spread. In this scenario, calculating the standard deviation of the number of girls born will help in applying the range rule of thumb to identify significantly low or high results.

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Textbook Question

Binomial Probability Formula. In Exercises 13 and 14, answer the questions designed to help understand the rationale for the binomial probability formula.

Guessing Answers Standard tests, such as the SAT, ACT, or Medical College Admission Test (MCAT), typically use multiple choice questions, each with five possible answers (a, b, c, d, e), one of which is correct. Assume that you guess the answers to the first three questions.

b. Beginning with WWC, make a complete list of the different possible arrangements of two wrong answers and one correct answer, and then find the probability for each entry in the list.

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Textbook Question

In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.

Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.

b. Find the probability that in a single day, there are 16 births.

Textbook Question

In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.

b. In a 118-year period, how many years are expected to have no hurricanes?

Textbook Question

b. In a 118-year period, how many years are expected to have 10 hurricanes?

Textbook Question

In Exercises 31 and 32, assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.75 probability that a pea has green pods (as in one of Mendel’s famous experiments).

Hybrids Assume that offspring peas are randomly selected in groups of 16.

b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high.

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Textbook Question

One of Mendel’s famous experiments with peas resulted in 580 offspring, and 152 of them were yellow peas. Mendel claimed that under the same conditions, 25% of offspring peas would be yellow. Assume that Mendel’s claim of 25% is true, and assume that a sample consists of 580 offspring peas.

b. Find the probability of exactly 152 yellow peas.

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