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Chapter 7: Problem 39

Annual per capita (average per person) chewing gum consumption in the United States is 200 pieces (http://www.iplcricketlive.com/). Suppose that the standard deviation of per capita consumption is 145 pieces per year. Let \(\bar{x}\) be the average annual chewing gum consumption of 84 randomly selected Americans. Find the mean and the standard deviation of the sampling distribution of \(\bar{x}\). What is the shape of the sampling distribution of \(\bar{x}\) ? Do you need to know the shape of the population distribution to make this conclusion? Explain why or why not.

Short Answer

Expert verified
The mean of the sampling distribution of \(\bar{x}\) is 200 pieces. The standard deviation of the sampling distribution of \(\bar{x}\) is approximately 15.78 pieces. The shape of the sampling distribution of \(\bar{x}\) is approximately normal, and this conclusion can be made without knowing the shape of the population distribution, thanks to the Central Limit Theorem.

Step by step solution

01

Understand the Provided Data

From the given information, we know that the average (mean) annual chewing gum consumption in the United States per person is 200 pieces. The standard deviation of annual consumption is 145 pieces. The student is asked to consider 84 randomly chosen Americans and determine certain properties about their chewing gum habits.
02

Calculate the Mean of the Sampling Distribution

The mean of the sampling distribution (also called the expected value of the sample mean) is equal to the population mean. Thus, the mean of the sampling distribution of \(\bar{x}\) is the same as the average annual per capita chewing gum consumption, which is 200 pieces.
03

Calculate the Standard Deviation of the Sampling Distribution

The standard deviation of the sampling distribution (also called the standard error) is the standard deviation of the population divided by the square root of the number of individuals in the sample. In this case, it's 145 divided by the square root of 84, which approximately equates to 15.78.
04

Determine the Shape of the Sampling Distribution

The Central Limit Theorem states that, for large enough sample sizes, the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution. Since the sample size in this case is 84, which is large enough (more than 30), we can say that the shape of the sampling distribution of \(\bar{x}\) is approximately normal.
05

Conclude on Whether the Population Distribution Shape is Needed

We do not need to know the shape of the population distribution to make this conclusion because, as per the Central Limit Theorem, the sampling distribution will be approximately normal provided the sample size is sufficiently large, which in this case it is.

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Key Concepts

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

Sampling Distribution
The concept of a sampling distribution is crucial in understanding how sample data can provide estimates about a population. Imagine we take multiple samples, each of the same size, from a population.
Each of these samples will have a mean, and if you plot these means, you get what is called the sampling distribution of the sample mean.
  • This distribution describes how the sample mean varies from sample to sample, meaning it's a distribution of a statistic.
  • It allows us to make inferences about the population mean even though we are using data from a limited sample size.
For instance, in the problem, by using the sampling distribution of the sample mean of 84 Americans, we can determine how precise our estimate of the average annual chewing gum consumption is likely to be.
Sample Mean
The sample mean, denoted as \( \bar{x} \), is a central value that represents the average of a sample.
It plays a key role when dealing with the sampling distribution, as it is the main statistic being observed and analyzed.
  • In the context of our chewing gum problem, \( \bar{x} \) represents the average gum consumption by the sample of 84 Americans.
  • The Central Limit Theorem tells us that the sample mean is an unbiased estimator of the population mean.
  • In our scenario, the sample mean is used as an estimate for the overall population's average gum consumption, which is 200 pieces.
Understanding how the sample mean behaves helps us make meaningful predictions about the population from which the sample was drawn.
Standard Error
The standard error is a measure of the dispersion or spread of the sampling distribution of a statistic, commonly the sample mean.
It's calculated using the formula:
\[ SE = \frac{\sigma}{\sqrt{n}} \]where \( \sigma \) is the standard deviation of the population, and \( n \) is the sample size.
  • In our example, this calculation gives the standard error as approximately 15.78 pieces.
  • The standard error informs us of the variability of the sample mean estimates around the population mean.
  • Smaller standard errors suggest that the sample mean is a more precise estimate of the population mean.
By understanding the standard error, we can quantify the reliability of our sample mean as an indicator of the population mean.
Normal Distribution
A normal distribution is a continuous probability distribution that is symmetric around the mean, commonly known as the bell curve because of its shape.
One of the reasons the normal distribution is important is due to the Central Limit Theorem.
  • This theorem states that the sampling distribution of the sample mean will approximate a normal distribution as the sample size becomes large, typically over 30.
  • In our exercise, the sample size of 84 is more than sufficient, hence the sampling distribution of \( \bar{x} \) is approximately normal.
  • This property allows us to make statistical inferences about the population mean with known reliability.
The normal distribution supports making probabilistic statements and calculating confidence intervals based on the sample data, even if the original population distribution is not normal.

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Most popular questions from this chapter

Refer to Exercise \(7.100 .\) Seventy percent of adults favor some kind of government control on the prices of medicines. Assume that this percentage is true for the current population of all adults. a, Find the probability that the proportion of adults in a random sample of 400 who favor some kind of government control on the prices of medicincs is i. less than \(.65\) ii. between \(.73\) and \(.76\) b. What is the probability that the proportion of adults in a random sample of 400 who favor some kind of government control is within 06 of the population proportion? c. What is the probability that the sample proportion is greater than the population proportion by \(.05\) or more? Assume that sample includes 400 adults.

Gluten sensitivity, which is also known as wheat intolerance, affects approximately \(15 \%\) of people in the United States (Source: http://www.foodintol.com/wheat.asp). Let \(\hat{p}\) be the proportion in a random sample of 800 individuals who have gluten sensitivity. Find the probability that the value of \(\hat{p}\) is a. within \(.02\) of the population proportion b. not within \(.02\) of the population proportion c. greater than the population proportion by \(.025\) or more d. less than the population proportion by 03 or more

For a population, \(N=18,000\) and \(p=.25\). Find the \(z\) value for each of the following for \(n=70\). a. \(\hat{p}=.26\) b. \(\hat{P}=.32\) c. \(\hat{p}=.17\) d. \(\hat{p}=.20\)

As mentioned in Exercise \(7.22\), the average cost of going to a minor league baseball game for a family of four was \(\$ 55\) in 2009 . Suppose that the standard deviation of such costs is \(\$ 13.25 .\) Find the probability that the average cost of going to a minor league baseball game for 33 randomly selected such families is a. more than \(\$ 60\) b. less than \(\$ 52\) c. \(\$ 54\) to \(\$ 57.99\)

In a population of 1000 subjects, 640 possess a certain characteristic. A sample of 40 subjects sclected from this population has 24 subjects who possess the same characteristic. What are the values of the population and sample proportions?
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