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Chapter 7: Problem 45

What is a sampling distribution? How would you explain to someone who has never studied statistics what a sampling distribution is? Explain by using the example of polls of 1000 Canadians for estimating the proportion who think the prime minister is doing a good job.

Short Answer

Expert verified
A sampling distribution shows how sample statistics vary, helping to estimate population parameters like the true proportion of Canadians approving of the prime minister.

Step by step solution

01

Understanding Sampling Distributions

A sampling distribution is a probability distribution of a statistic obtained from a large number of samples drawn from a specific population. For a given statistic, such as the mean or proportion, the sampling distribution represents how the statistic varies from one sample to another.
02

Using an Example

Consider a poll where we take samples of 1000 Canadians several times. Each sample gives us a proportion of people who think the prime minister is doing a good job. These proportions will vary slightly from one sample to another, even though the sample size is the same. This variability creates a distribution known as the sampling distribution.
03

Characteristics of Sampling Distribution

The sampling distribution is centered around the actual population proportion, particularly if the samples are randomly and independently drawn. It will have less variability if the sample size is larger, and this distribution of sample proportions allows statisticians to make inferences about the population proportion.
04

Why Sampling Distributions Matter

Sampling distributions help us understand the variability and reliability of sample statistics. By looking at the distribution, we can estimate how close our sample proportion from a particular survey is likely to be to the actual population proportion, providing a measure of the survey's accuracy.

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Key Concepts

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

Probability Distribution
Probability distribution is a fundamental concept in statistics that describes how the probabilities of different outcomes are distributed for a random variable. In the context of a sampling distribution, the random variable is the statistic being measured from various samples, like the proportion of people who think the prime minister is doing a good job.
  • Each possible outcome of this statistic has a probability associated with it, which adds up to 1.
  • This distribution provides a visual or mathematical representation of where the statistic is most likely to fall.
A probability distribution for a sampling distribution is crucial because it helps us understand the likelihood of various outcomes. This understanding is essential for interpreting results from random samples and making predictions about the population.
Population Proportion
The population proportion is a key term in statistics that refers to the ratio of members in a population who hold a particular trait or characteristic. For instance, if we consider the example of 1000 Canadians surveyed, the population proportion could be the fraction of all Canadians who think the prime minister is doing a good job.
  • The population proportion is denoted by the symbol \( p \).
  • It represents a constant value for a particular population, although it is usually unknown and needs to be estimated from sample data.
Knowing the population proportion is vital for making accurate inferences about the broader population from sample data. By collecting data from samples, statisticians aim to draw conclusions about this unknown proportion.
Sample Variability
Sample variability is about how the results of a statistic, such as a sample proportion, can fluctuate among different samples. In our example, when several samples of 1000 Canadians are surveyed, the percentage of people who think the prime minister is doing a good job may not be the same in each sample.
  • This variation occurs because each sample may include different individuals, leading to different outcomes.
  • The more diverse a population, the greater the potential variability in sample statistics.
Sample variability is significant because it is a natural part of collecting random samples. By understanding this concept, statisticians can assess how accurately a sample statistic represents the population, and use it to calculate measures like the standard error.
Statistical Inference
Statistical inference involves making judgments about a population based on data collected from random samples. It's about moving from known sample statistics to estimating unknown population parameters. The concept of sampling distribution is central here, as it helps us understand how sample statistics can fluctuate.
  • We use the characteristics of the sampling distribution to make educated guesses about a population.
  • This process often involves estimating the population proportion or testing hypotheses about a population feature.
In practical terms, statistical inference lets us determine, for example, how likely it is that the results of a poll reflect the actual views of all Canadians. By analyzing the sample data through the lens of sampling distributions, statisticians can make more confident assertions about what is true for the entire population.

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Most popular questions from this chapter

Multiple choice: Sampling distribution The sampling distribution of a sample mean for a random sample size of 100 describes a. How sample means tend to vary from random sample to random sample of size 100 . b. How observations tend to vary from person to person in a random sample of size 100 . c. How the data distribution looks like the population distribution when the sample size is larger than 30 . d. How the standard deviation varies among samples of size 100 .

Canada lottery In one lottery option in Canada (Source: Lottery Canada), you bet on a six-digit number between 000000 and \(999999 .\) For a \(\$ 1\) bet, you win \(\$ 100,000\) if you are correct. The mean and standard deviation of the probability distribution for the lottery winnings are \(\mu=0.10\) (that is, 10 cents) and \(\sigma=100.00\). Joe figures that if he plays enough times every day, eventually he will strike it rich, by the law of large numbers. Over the course of several years, he plays 1 million times. Let \(\bar{x}\) denote his average winnings. a. Find the mean and standard deviation of the sampling distribution of \(\bar{x}\). b. About how likely is it that Joe's average winnings exceed \(\$ 1,\) the amount he paid to play each time? Use the central limit theorem to find an approximate answer.

Number of sex partners \(\quad\) According to recent General Social Surveys, in the United States the population distribution for adults of \(X=\) number of sex partners in the past 12 months has a mean of about 1.0 and a standard deviation of about 1.0 . a. Does \(X\) have a normal distribution? Explain. b. For a random sample of 100 adults, describe the sampling distribution of \(\bar{x}\) and give its mean and standard deviation. What is the effect of \(X\) not having a normal distribution?

Blood pressure Vincenzo Baranello was diagnosed with high blood pressure. He was able to keep his blood pressure in control for several months by taking blood pressure medicine (amlodipine besylate). Baranello's blood pressure is monitored by taking three readings a day, in early morning, at mid-day, and in the evening. a. During this period, the probability distribution of his systolic blood pressure reading had a mean of 130 and a standard deviation of \(6 .\) If the successive observations behave like a random sample from this distribution, find the mean and standard deviation of the sampling distribution of the sample mean for the three observations each day. b. Suppose that the probability distribution of his blood pressure reading is normal. What is the shape of the sampling distribution? Why? c. Refer to part b. Find the probability that the sample mean exceeds \(140,\) which is considered problematically high. (Hint: Use the sampling distribution, not the probability distribution for each observation.)

Baseball hitting Suppose a baseball player has a 0.200 probability of getting a hit in each time at-bat. a. Describe the shape, mean, and standard deviation of the sampling distribution of the proportion of times the player gets a hit after 36 at- bats. b. Explain why it would not be surprising if the player has a 0.300 batting average after 36 at-bats.
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