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Chapter 7: Problem 23

The September \(21,2006,\) USA Today article "Average home has more TVs than people" stated that Americans watch an average of 4.58 hours of television per person per day. If the standard deviation for the number of hours of television watched per day is 2.1 and a random sample of 250 Americans is selected, the mean of this sample belongs to a sampling distribution. a. What is the shape of this sampling distribution? b. What is the mean of this sampling distribution? c. What is the standard deviation of this sampling distribution?

Short Answer

Expert verified
a. The shape of the sampling distribution is approximately normal. b. The mean of this sampling distribution is 4.58 hours. c. The standard deviation of this sampling distribution can be calculated as \(\frac{2.1}{\sqrt{250}}\) hours.

Step by step solution

01

Identify the Shape of the Sampling Distribution

According to the Central Limit Theorem, if the sample size is sufficiently large (generally n > 30), the sampling distribution of the sample means will be approximately normally distributed irrespective of the shape of the population distribution. As our sample size here is 250, which is a large sample size, the shape of the sampling distribution will be approximately normal.
02

Compute the Mean of the Sampling Distribution

The mean of a sampling distribution (also known as the expected value of the sample mean) is equal to the population mean. Hence, the mean of the sampling distribution in this case will be equal to the average number of hours Americans watch TV per day, which is 4.58 hours.
03

Calculate the Standard Deviation of the Sampling Distribution

The standard deviation of sampling distribution (also known as the standard error) can be calculated by dividing the population standard deviation by the square root of the sample size. In this case, the population standard deviation is given as 2.1 hours and the sample size is 250. Hence, the standard deviation of the sampling distribution is calculated as follows: \(\frac{2.1}{\sqrt{250}}\).

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Key Concepts

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

Sampling Distribution
When studying statistics, the concept of a sampling distribution is fundamental. Imagine you take many samples from a population. Each of these samples will have its own mean. The distribution of these sample means is what we call the sampling distribution.
The Central Limit Theorem (CLT) plays a significant role here. It states that if your sample size is large enough, typically greater than 30, the sampling distribution of the sample mean will tend to be normally distributed.
This is true even if the population from which you draw samples is not normal. In our example with a sample size of 250, the sampling distribution is approximately normal, thanks to the CLT.
Population Mean
The population mean is the true average of the entire population regarding the variable you are investigating. In our example, it is the average number of hours Americans watch TV per day, which is 4.58 hours.
This value doesn't change when you draw a sample. Instead, it serves as a constant reference point.
In the context of a sampling distribution, the mean of the sampling distribution, also known as the expected value of the sample mean, will always equal the population mean, which is 4.58 hours in this scenario.
Standard Deviation of the Sampling Distribution
The standard deviation of the sampling distribution is more commonly referred to as the standard error. It measures how much variability you can expect in your sample means. Essentially, it tells you how much the sample mean would vary from sample to sample.
To calculate it, you divide the population standard deviation by the square root of the sample size. In our case, with a population standard deviation of 2.1 hours and a sample size of 250, the formula becomes: \[\frac{2.1}{\sqrt{250}}\]
This calculation serves a crucial role in statistical inference, affecting how we make predictions about a population based on our samples.
Sample Size Significance
Sample size is critically important in statistical analysis. The larger your sample size, the more accurately your sample means will represent the population mean. This is because a larger sample size tends to lower the standard error, reducing variability.
  • A large sample size increases reliability and produces a more precise estimate of the population parameter.
  • According to the Central Limit Theorem, larger sample sizes ensure the sampling distribution becomes approximately normal.
In our example with a sample size of 250, we're well beyond the typical threshold of 30. This means we can comfortably rely on our sampling distribution being normally distributed, making any statistical predictions robust.

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Most popular questions from this chapter

a. Use a computer to draw 500 random samples, each of size \(20,\) from the normal probability distribution with mean 80 and standard deviation 15. b. Find the mean for each sample. c. Construct a frequency histogram of the 500 sample means. d. Describe the sampling distribution shown in the histogram in part \(c,\) including the mean and standard deviation.

Consider the set of even single-digit integers \(\\{0,2,4,6,8\\}\). a. Make a list of all the possible samples of size 3 that can be drawn from this set of integers. (Sample with replacement; that is, the first number is drawn, observed, and then replaced [returned to the sample set \(]\) before the next drawing.) b. Construct the sampling distribution of the sample medians for samples of size \(3 .\) c. Construct the sampling distribution of the sample means for samples of size 3.

Salaries for various positions can vary significantly, depending on whether or not the company is in the public or private sector. The U.S. Department of Labor posted the 2007 average salary for human resource managers employed by the federal government as \( 76,503 .\) Assume that annual salaries for this type of job are normally distributed and have a standard deviation of \(8850\) a. What is the probability that a randomly selected human resource manager received over \(100,000\) in \(2007 ?\) b. \(\quad\) A sample of 20 human resource managers is taken and annual salaries are reported. What is the probability that the sample mean annual salary falls between \(70,000\) and \(80,000 ?\)

Observations per sample" to "4." Using batch and \(500,\) take 1000 sam鈥� # Simulates sampling from a skewed population, where \(\mu=6.029\) and \(\sigma=10.79\). a. Change the "# Observations per sample" to "4." Using batch and \(500,\) take 1000 samples of size 4. b. Compare the mean and standard deviation for the sample means with \(\mu\) and \(\sigma .\) Compare the sample standard deviation with \(\sigma / \sqrt{n},\) which is \(10.79 / \sqrt{4}\) Does the histogram have an approximately normal shape? If not, what shape is it? c. Using the "clear" button each time, repeat the directions in parts a and b for samples of size 25 \(100,\) and \(1000 .\) Table your findings for each sample size. d. Relate your findings to the SDSM and the CLT.

According to the June 2004 Readers' Digest article "Only in America," the average amount that a 17 -year old spends on his or her high school prom is \(638\) Assume that the amounts spent are normally distributed with a standard deviation of \(175\) a. Find the probability that the mean cost to attend a high school prom for 36 randomly selected 17-year-olds is between \(550\) and \(700\). b. Find the probability that the mean cost to attend a high school prom for 36 randomly selected 17 -yearolds is greater than \(\$ 750\). c. Do you think the assumption of normality is reasonable? Explain.
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