Ch. 5 - Discrete Probability Distributions

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 5 - Discrete Probability Distributions

Problem 5.2.38a

Chapter 5, Problem 5.2.38a

Politics The County Clerk in Essex, New Jersey, was accused of cheating by not using randomness in assigning the order in which candidates’ names appeared on voting ballots. Among 41 different ballots, Democrats were assigned the desirable first line 40 times. Assume that Democrats and Republicans are assigned the first line using a method of random selection so that they are equally likely to get that first line.

a. Use the range rule of thumb to identify the limits separating values that are significantly low and those that are significantly high. Based on the results, is the result of 40 first lines for Democrats significantly high?

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Step 1: Understand the problem. The County Clerk is accused of not using randomness in assigning the order of candidates on ballots. We need to determine if the result of Democrats being assigned the first line 40 out of 41 times is significantly high using the range rule of thumb.

Step 2: Define the random process. If Democrats and Republicans are equally likely to be assigned the first line, the probability of Democrats being assigned the first line on any given ballot is 0.5. This follows a binomial distribution where n = 41 (number of ballots) and p = 0.5 (probability of success).

Step 3: Calculate the mean (μ) and standard deviation (σ) for the binomial distribution. The formulas are:

μ = n \times p

and

σ = \sqrt{n \times p \times (1 - p)}

Step 4: Apply the range rule of thumb to identify the limits for significantly low and significantly high values. The range rule of thumb states that values are significantly low if they are below

μ - 2 \times σ

and significantly high if they are above

μ + 2 \times σ

Step 5: Compare the observed value (40 first lines for Democrats) to the calculated limits. If 40 exceeds the upper limit, it is considered significantly high, suggesting that the assignment process may not be random.

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Key Concepts

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

Range Rule of Thumb

The Range Rule of Thumb is a statistical guideline that suggests using the range of a data set to estimate the limits of typical values. Specifically, it states that the limits for significantly low and high values can be determined by calculating the mean and standard deviation, where significantly low values are below the mean minus two standard deviations, and significantly high values are above the mean plus two standard deviations. This rule helps in identifying outliers in a data set.

Random Selection

Random selection is a fundamental concept in statistics that ensures each member of a population has an equal chance of being chosen for a sample. This method is crucial for eliminating bias in the selection process, thereby allowing for more reliable and valid conclusions. In the context of the question, it implies that both Democrats and Republicans should have an equal opportunity to appear first on the ballot, which is essential for fair elections.

Significance Testing

Significance testing is a statistical method used to determine if the observed results in a data set are due to chance or if they reflect a true effect. In this context, it involves assessing whether the occurrence of Democrats appearing first on the ballot 40 times out of 41 is statistically significant, meaning it is unlikely to have happened under the assumption of random selection. This helps in evaluating the fairness of the ballot assignment process.

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Step 2: Calculate Test Statistic

Related Practice

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.

a. Use the range rule of thumb to identify the limits separating values that are significantly low and those that are significantly high. Based on the results, is the result of 152 yellow peas either significantly low or significantly high?

150

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Textbook Question

Using Probabilities for Significant Events

a. Find the probability of getting exactly 1 match.

200

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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.

a. Find the mean and standard deviation for the numbers of peas with green pods in the groups of 16.

153

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Textbook Question

Lottery. In Exercises 15–20, refer to the accompanying table, which describes probabilities for the California Daily 4 lottery. The player selects four digits with repetition allowed, and the random variable x is the number of digits that match those in the same order that they are drawn (for a “straight” bet).

Using Probabilities for Significant Events

a. Find the probability of getting exactly 2 matches.

113

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Textbook Question

In Exercises 25–28, find the probabilities and answer the questions.

Internet Voting Based on a Consumer Reports survey, 39% of likely voters would be willing to vote by Internet instead of the in-person traditional method of voting. For each of the following, assume that 15 likely voters are randomly selected.

a. What is the probability that exactly 12 of those selected would do Internet voting?

167

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Textbook Question

Expected Value in North Carolina’s Pick 4 Game In North Carolina’s Pick 4 lottery game, you can pay \(1 to select a four-digit number from 0000 through 9999. If you select the same sequence of four digits that are drawn, you win and collect \)5000.

a. How many different selections are possible?

151

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