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Chapter 7: Problem 46

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 is the distribution of a statistic (like a proportion) obtained from multiple random samples of the same size from a population.

Step by step solution

01

Understanding the Population and Sample

Consider the entire population of Canadians as a large group we are interested in studying. Since it's impractical to ask every single person, we take a smaller group called a 'sample.' For this example, the sample consists of 1000 Canadians, chosen randomly, to find out what proportion think the prime minister is doing a good job.
02

Conducting Multiple Samples

Imagine repeatedly selecting different random groups of 1000 Canadians. Each group is a different sample, and for each sample, we calculate the proportion of people who think the prime minister is doing a good job. These proportions might vary slightly between different samples due to the randomness of who gets selected.
03

Defining a Sampling Distribution

A 'sampling distribution' is the distribution of all these proportions from multiple samples. It shows us how these calculated proportions spread out when you take many samples. Even if the samples differ, patterns or trends in these proportions can give us information about the entire population.
04

Visualizing the Sampling Distribution

Imagine plotting the proportion from each sample on a graph. The result is a distribution that is often shaped like a bell curve. This shape helps us understand the variability and center (mean) of these sample proportions, giving insight into the true proportion of the entire population.
05

Using the Sampling Distribution

By analyzing this sampling distribution, statisticians can estimate the true proportion of all Canadians who think the prime minister is doing a good job, taking into account the variability among samples. It helps gauge the accuracy of our sample estimates.

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Key Concepts

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

Population and Sample
In statistics, the concepts of 'population' and 'sample' are essential. Understanding these ideas is crucial to comprehend how data is analyzed and utilized.
  • The **population** refers to the entire group we're interested in studying. In the context of our example, the population consists of all Canadians. The population is often too large to study in its entirety. Therefore, we cannot ask every person in the country about their opinion on the prime minister's performance.
  • A **sample**, on the other hand, is a smaller group selected from the population. The sample aims to represent the larger population accurately. In our example, we select a sample of 1000 Canadians randomly. This sample provides a manageable group we can study closely. By analyzing this smaller group, we try to gain insights that can be applied to the whole population.
By understanding the concepts of population and sample, we are building the foundation for more detailed statistical analysis.
Random Sampling
Random sampling is a pivotal method in statistics that ensures biases are minimized. Using random samples means every individual in a population has an equal chance of being chosen.
  • In our example, we conduct random sampling by selecting 1000 Canadians without any preference. This method is crucial because it helps to ensure that the sample's views reflect those of the entire population.
  • Making selections randomly reduces the chances of selecting a non-representative sample. It prevents skewing the results due to over or under-representing certain groups.
Random sampling is instrumental in ensuring the validity of statistical estimates derived from samples.
Statistical Estimation
Statistical estimation refers to the process of making inferences about a population's characteristics based on a sample. This is often necessary when collecting data from an entire population is difficult or impossible.
  • For example, if our sample of 1000 Canadians shows that 60% think the prime minister is doing a good job, we use statistical estimation to infer that the proportion for the entire Canadian population is in a similar vicinity.
  • This estimation allows us to make educated guesses about the larger population based on a smaller, manageable group.
  • Estimates are not definite conclusions but are crucial in providing insights into the population's opinions and behaviors.
Statistical estimation is integral to research and decision-making processes, facilitating understanding and predictions about populations.
Variability in Statistics
Variability is a key concept in statistics that emphasizes differences in data. It is an essential factor to consider when interpreting data from different samples.
  • In our example, by taking multiple samples, each sample might give slightly different results on the proportion of Canadians who think the prime minister is doing a good job. Variability acknowledges these differences due to chance and the natural diversity present.
  • Variability helps understand how spread out the sample data points are, indicating the reliability of an estimate. High variability might suggest that more sampling is needed or that the estimate isn't stable.
  • When statisticians analyze sampling distributions, they look for patterns that variability within samples shows. These patterns help assess the accuracy and consistency of our population predictions.
Recognizing variability in statistics allows for better conclusions and interpretations in studies, adding depth to findings and suggestions.

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Most popular questions from this chapter

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.

You'd like to estimate the proportion of the 14,201 undergraduate students at Syracuse University who are full-time students. You poll a random sample of 350 students, of whom 330 are full-time. Unknown to you, the proportion of all undergraduate students who are full-time students is \(0.951 .\) Let \(X\) denote a random variable for which \(x=1\) denotes full-time student and for which \(x=0\) denotes part-time student. (For recent enrollment numbers, go to www.syr.edu/about/ facts.html.) a. Describe the population distribution. Sketch a graph representing the population distribution. b. Describe the data distribution. Sketch a graph representing the data distribution. c. Find the mean and standard deviation of the sampling distribution of the sample proportion for a sample of size \(350 .\) Explain what this sampling distribution represents. Sketch a graph representing this sampling distribution. d. Use the Sampling Distribution app accessible from the book's website to check your answers from parts a through \(\mathrm{c}\). Set the population proportion equal to \(p=0.951\) and \(n=350 .\) Compare the population, data, and sampling distribution graph from the app with your graphs from parts a through \(\mathrm{c}\).

At a university, \(60 \%\) of the 7,400 students are female. The student newspaper reports results of a survey of a random sample of 50 students about various topics involving alcohol abuse, such as participation in binge drinking. They report that their sample contained 26 females. a. Explain how you can set up a binary random variable \(X\) to represent gender. b. Identify the population distribution of gender at this university. Sketch a graph. c. Identify the data distribution of gender for this sample. Sketch a graph. d. Identify the sampling distribution of the sample proportion of females in the sample. State its mean and standard deviation for a random sample of size \(50 .\) Sketch a graph. e. Use the Sampling Distribution app accessible from the book's website to check your answers from parts b through d. Set the population proportion equal to \(p=0.60\) and \(n=50 .\) Compare the population, data, and sampling distribution graph from the app with your graphs from parts b through d.

Each student should bring 10 coins to class. For each coin, observe its age, the difference between the current year and the year on the coin. a. Using all the students' observations, the class should construct a histogram of the sample ages. What is its shape? b. Now each student should find the mean for that student's 10 coins, and the class should plot the means of all the students. What type of distribution is this, and how does it compare to the one in part a? What concepts does this exercise illustrate?

According to the website http://www.digitalbookworld.com, the average price of a bestselling ebook increased to \(\$ 8.05\) in the week of February 18,2015 from \(\$ 6.89\) in the previous week. Assume the standard deviation of the price of a bestselling ebook is \(\$ 1\) and suppose you have a sample of 20 bestselling ebooks with a sample mean of \(\$ 7.80\) and a standard deviation of \(\$ 0.95\) a. Identify the random variable \(X\) in this study. Indicate whether it is quantitative or categorical. b. Describe the center and variability of the population distribution. What would you predict as the shape of the population distribution? Explain. c. Describe the center and variability of the data distribution. What would you predict as the shape of the data distribution? Explain. d. Describe the center and variability of the sampling distribution of the sample mean for 20 bestselling ebooks. What would you predict as the shape of the sampling distribution? Explain.
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