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Chapter 8: Problem 57

In a certain political poll, each person polled has a \(90 \%\) probability of telling his or her real preference. Suppose that \(55 \%\) of the population really prefer candidate Goode, and \(45 \%\) prefer candidate Slick. First find the probability that a person polled will say that he or she prefers Goode. Then find the approximate probability that, if 1,000 people are polled, more than \(52 \%\) will say they prefer Goode.

Short Answer

Expert verified
The probability that a person will say they prefer Goode is 54%. There is approximately an 89.8% probability that more than 52% of the 1,000 people polled will say they prefer Goode.

Step by step solution

01

Calculate the Probability of Saying They Prefer Goode

First, we will derive the probability of a person saying they prefer Goode. We know that 90% of the time, people will tell their real preference. Let G be the event of a person really preferring Goode, S the event of a person really preferring Slick, and R the event of saying they prefer Goode. Then the probability that a person polled will say they prefer Goode can be found using the law of total probability: \(P(R) = P(R|G)P(G) + P(R|S)P(S)\) We know the following probabilities: - \(P(G) = 0.55\) - \(P(S) = 0.45\) - \(P(R|G) = 0.90\) (90% probability of telling true preference given they prefer Goode) - \(P(R|S) = 0.10\) (100% - 90% probability of telling true preference given they prefer Slick) Now we can calculate the probability of a person saying they prefer Goode: \(P(R) = (0.90)(0.55) + (0.10)(0.45) = 0.495 + 0.045 = 0.54\) So, the probability that a person will say they prefer Goode is 54%.
02

Approximate the Probability of More Than 52% Saying They Prefer Goode

Now, we will find the approximate probability that more than 52% of 1,000 people polled will say they prefer Goode. We will use the Central Limit Theorem to model the situation using a normal distribution. Given that we're dealing with a large sample size, this should give us a reasonable approximation. First, we need to find the mean and standard deviation for the sampling distribution of the sample proportion (\(\hat{p}\)): - Mean (\(\mu_{\hat{p}}\)) = \(P(R) = 0.54\) - Standard deviation (\(\sigma_{\hat{p}}\)) = \(\sqrt{\frac{P(R)(1 - P(R))}{n}} = \sqrt{\frac{(0.54)(1 - 0.54)}{1000}} \approx 0.0157\) Now, we can standardize the proportion we're interested in, 52%, or 0.52: \(z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}} = \frac{0.52 - 0.54}{0.0157} \approx -1.27\) Using a standard normal distribution table or a calculator, we can find the probability of obtaining a z-score less than -1.27: \(P(Z < -1.27) \approx 0.102\) Since we want the probability of more than 52% saying they prefer Goode, we need to find the complement: \(P(Z > -1.27) = 1 - P(Z < -1.27) \approx 1 - 0.102 = 0.898\) So, there is approximately an 89.8% probability that more than 52% of the 1,000 people polled will say they prefer Goode.

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Key Concepts

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

Central Limit Theorem
The Central Limit Theorem (CLT) is a fundamental principle of probability theory that plays a pivotal role in the field of statistics. It states that as the size of a sample drawn from a population with a finite level of variance increases, the sample means will be approximately normally distributed, regardless of the shape of the original population's distribution.

This theorem is incredibly useful when dealing with large datasets because it allows us to make predictions and inferences about population parameters that would be challenging to calculate otherwise. In the context of polling, where the number of individuals is large (like a sample size of 1000 people), the CLT justifies the use of the normal distribution to approximate the sampling distribution of the sample proportion. This simplifies the process of calculating probabilities and confidence intervals for population proportions.

When we say that the sampling distribution becomes 'approximately normal', it means that standard statistical tables (like the Z-table used in the problem's solution) can be employed to find probabilities related to the sample mean, which is highly convenient and efficient.
Law of Total Probability
The law of total probability is a fundamental rule that provides a way to break down complex probability problems into more manageable parts. It tells us that the total probability of an outcome is the sum of the probabilities of that outcome occurring across different, disjoint scenarios or paths that cover the entire sample space.

For example, in the problem we have two groups of voters - those that prefer Goode and those that prefer Slick. The question is not just about whether people prefer Goode, but whether they'll say they prefer Goode. We must consider both the probability of someone really preferring Goode and telling the truth and the probability of someone preferring Slick but saying they prefer Goode.

By applying the law of total probability, we can accurately account for all possible scenarios that lead to the event of interest (people stating their preference for Goode) and thus calculate the overall probability of this event. This provides a clearer understanding of the total likelihood of an outcome in a complex probability model, which is crucial for accurate and meaningful inferences in polling.
Sampling Distribution
A sampling distribution is the probability distribution of a given statistic based on a random sample. It's a concept that describes how the values of the statistic vary from one sample to another. In many cases, particularly with a large sample size, the sampling distribution can be approximated by a normal distribution, thanks to the Central Limit Theorem.

In polling, we're often dealing with proportions, such as the percentage of the sample that prefers a certain candidate. The sampling distribution of the sample proportion tells us about the variability of these sample percentages if we were to repeat our poll many times. It is characterized by its mean, often equal to the population proportion, and its standard deviation, which decreases with an increased sample size. Understanding the sampling distribution is vital for making probability-based predictions about the population from the sample, such as in the exercise when predicting the spread of poll results around the observed proportion of 54% who prefer Goode.
Normal Distribution
The normal distribution, also known as the Gaussian distribution, is a symmetrical bell-shaped curve that describes how values of a variable are distributed. It is defined by its mean and standard deviation, which determine the center and the spread of the distribution, respectively.

In the realm of polling and statistics, the normal distribution is significant because many variables naturally follow this pattern, especially when the sample size is large. For instance, the Central Limit Theorem allows us to treat the sampling distribution of the percentage of voters who prefer a particular candidate as approximately normal if the number of polled individuals is sufficiently large.

This approximation with the normal distribution makes it easier for us to calculate the probabilities of observing a certain sample proportion, such as the probability of more than 52% of voters preferring Goode in our initial problem. Utilizing the properties of the normal distribution gives us a powerful tool for making inferences about the preferences and behaviors of the entire population based on our sample data.

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Most popular questions from this chapter

Y o u r ~ p i c k l e ~ c o m p a n y ~ r a t e s ~ i t s ~ p i c k l e s ~ o n ~ a ~ s c a l e ~ o f spiciness from 1 to \(10 .\) Market research shows that customer preferences for spiciness are normally distributed, with a mean of \(7.5\) and a standard deviation of 1 . Assuming that you sell 100,000 jars of pickles, how many jars with a spiciness of 9 or above do you expect to sell?

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