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Chapter 6: Problem 48

Checking guidelines In a village having more than 100 adults, eight are randomly selected in order to form a committee of residents. \(40 \%\) of adults in the village are connected with agriculture. a. Verify that the guidelines have been satisfied about the relative sizes of the population and the sample, thus allowing the use of a binomial probability distribution for the number of selected adults connected with agriculture. b. Check whether the guideline was satisfied for this binomial distribution to be reasonably approximated by a normal distribution.

Short Answer

Expert verified
The sample satisfies binomial distribution criteria, but not normal approximation as np and n(1-p) are both less than 5.

Step by step solution

01

Conditions for Binomial Distribution

To determine if a distribution can be considered binomial, two main guidelines need to be checked: the 10% condition and binary outcomes. In this question, the population is above 100 adults, and only 8 are sampled. Therefore, 8 is less than or equal to 10% of the population (n 鈮� 0.1N), satisfying the 10% condition. The outcome (being connected to agriculture or not) is binary.
02

Parameters of the Binomial Distribution

The binomial distribution is defined by the parameters n (number of trials) and p (probability of success on each trial). Here, n = 8 (number of adults selected) and p = 0.40 (probability of an adult being connected to agriculture).
03

Checking Conditions for Normal Approximation

The guideline for using a normal approximation for a binomial distribution is that both np and n(1-p) should be greater than 5. Calculate np: np = 8 * 0.40 = 3.2. Calculate n(1-p): n(1-p) = 8 * (1 - 0.40) = 8 * 0.60 = 4.8. Both 3.2 and 4.8 are not greater than 5, so the normal approximation is not justified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

10% condition
The 10% condition is essential when working with a binomial distribution, especially when dealing with samples from larger populations. This guideline ensures that the sample size does not exceed 10% of the overall population. By keeping the sample size relatively small, we maintain the independence of each trial in the sampling process.

For example, in our exercise, the population consists of more than 100 adults and only 8 adults are selected to form a committee. Thus, we have that 8 is less than or equal to 10% of the population. This confirms that the 10% condition is satisfied, which is a green light to proceed with using a binomial probability distribution.
  • Guarantees independence of trials
  • Prevents skewing of results
  • Helps in keeping calculations valid
Binary outcomes
In a binomial distribution context, binary outcomes refer to the scenario where each trial results in one of only two possible outcomes. This duality is crucial for defining a binomial experiment.

These outcomes can be anything like "success or failure", "yes or no", "on or off". In our original exercise, the binary outcomes are clear: an adult either is connected to agriculture or is not. This aligns perfectly with the nature of binary outcomes and further supports using a binomial distribution for problem-solving.
  • Two possible outcomes per trial
  • Defines the framework of a binomial experiment
  • Simplifies probability calculations
Normal approximation
Normal approximation is a technique used when a binomial distribution is difficult to work with directly, especially with large sample sizes. It allows you to approximate the distribution with a normal curve, making calculations more manageable.

The rule of thumb for applying normal approximation is that both np and n(1-p) should be greater than 5. In our provided scenario with np = 3.2 and n(1-p) = 4.8, neither is greater than 5. Hence, the guideline suggests that using normal approximation is not suitable for this case. This is an important check because incorrect application of normal approximation can lead to inaccurate results.
  • Simplifies calculation with large samples
  • Requires np and n(1-p) > 5 for validity
  • Important for accurate results
Probability of success
The probability of success in a binomial setting is the likelihood that a particular outcome will occur in any given trial. It is a fundamental component that governs the behavior of the distribution.

Denoted by the symbol \( p \), this probability can range from 0 to 1. In the village committee task, the probability of selecting an adult connected to agriculture is 0.40, or 40%. Knowing \( p \) helps us in calculating the expected number of successes and other statistical measures in the binomial distribution.
  • Key parameter in binomial distribution
  • Expressed as a decimal between 0 and 1
  • Used to calculate expected outcomes

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Most popular questions from this chapter

For each of the following situations, explain whether the binomial distribution applies for \(X\). a. You are bidding on four items available on eBay. You think that you will win the first bid with probability \(25 \%\) and the second through fourth bids with probability \(30 \%\). Let \(X\) denote the number of winning bids out of the four items you bid on. b. You are bidding on four items available on eBay. Each bid is for \(\$ 70\), and you think there is a \(25 \%\) chance of winning a bid, with bids being independent events. Let \(X\) be the total amount of money you pay for your winning bids.

For an approximately normally distributed random variable \(X\) with a mean of 200 and a standard deviation of 36 , a. Find the \(z\) -score corresponding to the lower quartile and upper quartile of the standard normal distribution. b. Find and interpret the lower quartile and upper quartile of \(X\). c. Find the interquartile range (IQR) of \(X\). d. An observation is a potential outlier if it is more than \(1.5 \times\) IQR below \(\mathrm{Q} 1\) or above \(\mathrm{Q} 3\). Find the values of \(X\) that would be considered potential outliers.

Profit and the weather From past experience, a wheat farmer living in Manitoba, Canada, finds that his annual profit (in Canadian dollars) is 80,000 if the summer weather is typical, 50,000 if the weather is unusually dry, and 20,000 if there is a severe storm that destroys much of his crop. Weather bureau records indicate that the probability is 0.70 of typical weather, 0.20 of unusually dry weather, and 0.10 of a severe storm. Let \(X\) denote the farmer's profit next year. a. Construct a table with the probability distribution of \(X\) b. What is the probability that the profit is 50,000 or less? c. Find the mean of the probability distribution of \(X\). Interpret. d. Suppose the farmer buys insurance for 3000 that pays him 20,000 in the event of a severe storm that destroys much of the crop and pays nothing otherwise. Find the probability distribution of his profit. Find the mean and summarize the effect of buying this insurance.

Dental Insurance You plan to purchase dental insurance for your three remaining years in school. The insurance makes a one-time payment of \(\$ 1,000\) in case of a major dental repair (such as an implant) or \(\$ 100\) in case of a minor repair (such as a cavity). If you don't need dental repair over the next 3 years, the insurance expires and you receive no payout. You estimate the chances of requiring a major repair over the next 3 years as \(5 \%,\) a minor repair as \(60 \%\) and no repair as \(35 \%\). a. Why is \(X=\) payout of dental insurance a random variable? b. Is \(X\) discrete or continuous? What are its possible values? c. Give the probability distribution of \(X\).

The juror pool for the upcoming murder trial of a celebrity actor contains the names of 100,000 individuals in the population who may be called for jury duty. The proportion of the available jurors on the population list who are Hispanic is 0.40 . A jury of size 12 is selected at random from the population list of available jurors. Let \(X=\) the number of Hispanics selected to be jurors for this jury. a. Is it reasonable to assume that \(X\) has a binomial distribution? If so, identify the values of \(n\) and \(p\). If not, explain why not. b. Find the probability that no Hispanic is selected. c. If no Hispanic is selected out of a sample of size 12 , does this cast doubt on whether the sampling was truly random? Explain.
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