You're Reading a Free Preview Show
You're Reading a Free Preview You're Reading a Free Preview You're Reading a Free Preview You're Reading a Free Preview You're Reading a Free
Preview You're Reading a Free Preview You're Reading a Free Preview
You're Reading a Free Preview You're Reading a Free Preview You're Reading a Free Preview You're Reading a Free Preview You're Reading a Free Preview You're Reading a Free Preview You're Reading a Free Preview Recommended textbook solutions
 Statistics for Business and Economics10th EditionDavid R. Anderson, Greg Anderson 1,919 solutions
Applied Multivariate Statistical Analysis4th EditionDean W Wichern, Richard A. Johnson 116 solutions Understandable Statistics9th EditionCharles Henry Brase, Corrinne Pellillo Brase 3,048 solutions
Mathematical Reasoning for Elementary Teachers7th EditionCalvin T. Long, Duane W. DeTemple, Richard S. Millman 1,968 solutions ObjectivesBy the end of this lesson, you will be able to...
For a quick overview of this section, watch this short video summary: Sampling DistributionsConsider the following three news items.
All three of these are estimates based on samples In fact, they're probably not correct, due to sampling error. From Section 1.4, Sampling error is the error that results from using a sample to estimate information regarding a population. The idea is this - unless we sample every single individual in the sample, there will be some error in our results. Our goal in this section will be to characterize the distribution of the sample mean. The Distribution of the Sample MeanLet's look again at the definition of a random variable, from Section 6.1. A random variable is a numerical measure of the outcome of a probability experiment whose value is determined by chance. Think about the sample mean, To investigate these, let's look at a particular population. Consider the heights of the players from the starting line-up from the 2008 Men's Olympic Basketball gold medal game - Jason Kidd (76"), LeBron James (80"), Kobe Bryant (78"), Carmelo Anthony (78"), and Dwight Howard (83"). (Source: NBC Sports) The mean of the population is 79", with a standard deviation of 2.37" First, let's consider the different samples of size 2. There are 10 such samples (5C2 = 10), shown below, along with their corresponding sample means.
Interestingly, the mean of the sample means of size 2 is 79" as well. This is actually reasonable, though, because we know that the mean of a random variable is also its expected value, and it makes perfect sense that the value we should expect from the sample mean is the same as the population mean! The standard deviation, though, is very different. It helps to look at things visually. The image below represents all possible sample means for samples of size 1 (individuals), 2, 3, 4, and 5 (the population). Pay particular attention to the standard deviation. The interesting things to note here are that The Law of Large NumbersAs n increases, the difference between We're now ready to investigate the standard deviation of The Central Limit TheoremThe Central Limit TheoremRegardless of the distribution shape of the population, the sampling distribution of This is very interesting! So it doesn't matter if the distribution shape was left-skewed, right-skewed, uniform, binomial, anything - the distribution of the sample mean will always become normal as the sample size increases. What an amazing result! Exploring the Distribution of the Sample MeanOnce you have the player installed and the Central Limit Theorem demonstration downloaded, move the slider for the sample size to get a sense of its affect on the distribution shape. You can also move the new sample slider to get a different sample. The Distribution of the Sample MeanWe can even be more specific about the distribution of The Sampling Distribution of
If a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ, the sampling distribution of where Key fact: If the population is normally distributed, then the sample mean Now let's apply this distribution to various problems. Using the Central Limit TheoremIn order to find probabilities about a normal random variable, we need to first know its mean and standard deviation. With the results of the Central Limit Theorem, we now know the distribution of the sample mean, so let's try using that in some examples. Let's see a couple examples. Example 1 Let's consider again the distribution of IQs that we looked at in Example 1 in Section 7.1. We saw in that example that tests for an individual's intelligence quotient (IQ) are designed to be normally distributed, with a mean of 100 and a standard deviation of 15. What is the probability that a randomly selected sample of 20 individuals would have a mean IQ of more than 105? Solution: To answer this question, we need to find P( Here's what the distribution of Now that we have the standard deviation, we can find the probability. Using StatCrunch... Example 2 In Example 2 in Section 7.1, we were told that weights of 1-year-old boys are approximately normally distributed, with a mean of 22.8 lbs and a standard deviation of about 2.15. (Source: About.com) Suppose the sample mean of the 10 1-year-old boys at the Kiddie Care day care center is 22.3 lbs. Is that unusual? Solution: In order to determine if an event is unusual, we need to find its probability. If the probability of the event is less than 5%, we can classify it as an unusual event. In this case, we want to find the probability of observing a sample mean of 22.3 or less. Using the distribution of the sample mean, So we'd observe a sample mean of 22.3 lbs or less from a sample of 10 1-year-old boys about 23% of the time, which is not very unusual at all. Here's one for you to try: Example 3 Suppose that a particular professor's Statistics exam on probability traditionally has a mean score of 74, with a standard deviation of 11. The professor suspects that his current crop of students is very strong. To compare, he gives them the same exam he has in the past. The sample mean of the 28 students in his current class was a 78. Was the professor correct? Is his current class of students unusual compared to those from the past? [ reveal answer ] Like the previous example, we need to find the probability of our event in order to determine if it was unusual. Using the distribution of the sample mean, we know that Since the probability of observing a class of 28 students with an average score of 78 is less than 0.05, it does look like this particular class is unusual. (But in a good way!) What happens to the shape of sampling distribution of sample means as n increase?In other words, as the sample size increases, the variability of sampling distribution decreases. Also, as the sample size increases the shape of the sampling distribution becomes more similar to a normal distribution regardless of the shape of the population.
What happens to the sampling distribution as n gets large?As the sample size (n) increases, the standard deviation of the sampling distribution becomes smaller because the square root of the sample size is in the denominator. In other words, the sampling distribution clusters more tightly around the mean as sample size increases.
What is the shape of the sampling distribution of the sample mean?The Shape of the Sample Mean Distribution is Normal!
The sample mean distribution is a heap shaped, as in the shape of the normal distribution, and centered on the population mean. If the sample size is 30 or more, then the sample means are NORMALLY distributed even when the underlying data is NOT normally distributed!
|
What happens to the shape of a sampling distribution of sample means as n increases quizizz
zusammenhängende Posts
Urheberrechte © © 2024 toptenid.com Inc.
