Section 7.3b Sample Means The Central Limit Theorem

Transcription

1 We have seen in the previous investigation that even though the distribution of the population of the ages of the 1000 pennies was skewed to the right, the distribution of the sample means appeared to be approximately normal when the sample size was large. We also know that if the original population distribution is normal, which is often the case, then the sampling distribution of the sample means is also approximately normal. The question that arises is when can we assume that the sampling distribution of sample means is approximately normal if the population isn t? We do have a rule of thumb for sampling distribution of proportions ( 10 n 1 10) but that doesn t apply to the sampling distribution of sample means. In this n and investigation we will explore the criteria for determining when the sampling distribution for the sample means is approximately normal. 1. Consider the distribution shown below. This is called a beta distribution with a mean 0.5 and There are 1000 subjects in this distribution and we will assume that these 1000 subjects represent the entire population. a. Describe the shape of this distribution. b. Open the Fathom file betadistribution.ftm and collect random samples of size 1 n. Let Fathom calculate the mean of these values and plot them on a scatterplot. Continue to collect sample means from samples of size 1 until the shape of the distribution of these sample means becomes clear. What is the apparent shape of this distribution? Page 1 of 14

2 c. Continue to collect sample means until you have 500 sample means. Record the mean and the standard deviation for this distribution below and compare this mean and standard deviation with the population mean and standard deviation. d. Now repeat parts b and c for this population using n 2, n 5, n 25, n 50, and n 100. What appears to be true about the shape, center, and spread of the sampling distribution as n increases? Notice that even with this extremely non normal population distribution we find that the sampling distribution of the sample means becomes approximately normal as the sample sizes increase. This is a fact about the sampling distribution of sample means which leads us an extremely important theorem in statistics called the Central Limit Theorem. The Central Limit Theorem(CLT) Draw an SRS of size n from any population with a mean and a standard deviation. The Central Limit Theorem says that when the sample size n is large the sampling distribution of the sample means x is approximately normal. Page 2 of 14

3 The Central Limit Theorem (CLT) allows you to use a normal probability calculation to answer questions about the sample mean from many observations even when the population is not normal. But what is meant by large sample sizes? Smaller sample sizes may not always yield a normal sampling distribution. The Central Limit Theorem may be safely applied if the sample size n exceeds 30. If the population distribution is reasonably close to a normal distribution then sample sizes of 15 to 20 are often large enough for the sampling distribution of x to have an approximately normal distribution. Of course you still need to ensure randomization and independence of the observations in your sample but the Central Limit Theorem virtually guarantees a normal sampling distribution for the means given a large enough sample size. You can consider the Central Limit Theorem as the Fundamental Theorem of Statistics it s really that important! Be careful not to confuse the distribution of the data in your sample with the sampling distribution. For example, don t mistakenly think the CLT says that the data in your sample will be normally distributed as long as your sample size is large enough. In fact, as the sample size gets larger, we expect the distribution of the data in the sample to look more and more like the data in the population distribution from which the sample was drawn- skewed, bimodal, uniform, normal, etc. You should have seen that from the previous activity with the beta distribution. The Central Limit Theorem doesn t tell you anything about the shape of the data in your sample. It is about the shape of the sampling distribution of x. Make sure you understand the difference. Assumptions and Conditions for using a normal distribution to approximate the sampling distribution of x Randomization Condition: The data must be from an SRS of the population Independence Assumption: The sample values must be independent. This is a tough one to check so if the 10% condition is met and the data is from a SRS independence is usually satisfied. Large Enough Sample Size: This is not an obvious one. General rules of thumb are 30 or more but there is no magic number. We will address this issue of sample size later. Page 3 of 14

4 Lets apply these ideas to a problem situation. Section 7.3b Sample Means The Central Limit Theorem 2. Male blue throat birds have a complex song. Let x denote the duration of a randomly selected song in seconds. Suppose that the population mean value of the song is known to be 13.8 and has a standard deviation of 11.8 seconds. The population distribution of x is also known to be non- normal. Suppose you select an SRS of 25 blue throat birds and measure their song durations. a. The first condition is that we have a random sample from the population. Do we meet this condition? Why is this condition important? b. The second condition is independence. Do we reasonably meet this condition? c. The third condition is normality. Can you safely assume that the sampling distribution of x is approximately normal? Why or why not. What if the population was approximately normal? Page 4 of 14

5 3. A hotdog manufacturer asserts that one of its brands of hotdogs has an average fat content of 18 grams per hotdog with a standard deviation 1 gram. Consumers of this brand would probably not be disturbed if the mean fat content is less than 18 grams but would certainly be unhappy if the mean fat content exceeded 18 grams. Suppose we let x denote the fat content of a randomly selected hotdog and that we select a SRS of 36 hotdogs. Suppose that we find from our sample of 36 hotdogs that x a. If the manufacture s claim is correct that the mean fat content is 18 grams then what is probability that from our sample we would get a mean fat content of 18.4 grams or greater. In other words, under these conditions P x 18.4? Check the conditions for the sampling distribution of what is x before doing any calculations. Show your work. b. Based on your results from part a, do you think the manufacturers claim is correct? Explain your reasoning. Page 5 of 14

6 The figure below summarizes the facts about the sampling distribution of x. It illustrates the big idea of a sampling distribution. Keep taking random samples of size n from a population with a mean of and standard deviation. Find the mean x for each sample and collect all these sample means and display their distribution. That s the sampling distribution of x. As we have seen, this distribution is approximately normal when the population is normal, and when the population is not normal as long as our sample sizes equal or exceed 30 (another rule of thumb) then by the Central Limit Theorem we can safely assume an approximately normal sampling distribution. Page 6 of 14

7 CHECK YOUR UNDERSTANDING Building Better Batteries. Section 7.3b Sample Means The Central Limit Theorem Everyone wants to have the latest technological gadget. That s why ipods, digital cameras, smartphones, Game Boys, and the Wii have sold millions of units. These devices require lots of power and can drain batteries quickly. Battery manufacturers are constantly searching for ways to build longer-lasting batteries. A particular manufacturer produces AA batteries that are designed to last an average of 17 hours with a standard deviation of 0.8 hours. Quality control inspectors select a random sample of 30 batteries during each hour of production, and they drain them under conditions that mimic normal use. Below is one such sample. The average lifetime (in hours) of the batteries from this sample is 16.7 hours. Do these data suggest that the production process is working properly? Are plant managers safe sending out all the batteries produced in this hour for sale? Use what you have learned in this chapter to answer these questions. Page 7 of 14

8 We have seen in the previous investigation that even though the distribution of the population of the ages of the 1000 pennies was skewed to the right, the distribution of the sample means appeared to be approximately normal when the sample size was large. We also know that if the original population distribution is normal, which is often the case, then the sampling distribution of the sample means is also approximately normal. The question that arises is when can we assume that the sampling distribution of sample means is approximately normal if the population isn t? We do have a rule of thumb for sampling distribution of proportions ( 10 n 1 10) but that doesn t apply to the sampling distribution of sample means. In this n and investigation we will explore the criteria for determining when the sampling distribution for the sample means is approximately normal. 1. Consider the distribution shown below. This is called a beta distribution with a mean 0.5 and There are 1000 subjects in this distribution and we will assume that these 1000 subjects represent the entire population. a. Describe the shape of this distribution. b. Open the Fathom file betadistribution.ftm and collect random samples of size 1 n. Let Fathom calculate the mean of these values and plot them on a scatterplot. Continue to collect sample means from samples of size 1 until the shape of the distribution of these sample means becomes clear. What is the apparent shape of this distribution? Page 8 of 14

9 c. Continue to collect sample means until you have 500 sample means. Record the mean and the standard deviation for this distribution below and compare this mean and standard deviation with the population mean and standard deviation. d. Now repeat parts b and c for this population using n 2, n 5, n 25, n 50, and n 100. What appears to be true about the shape, center, and spread of the sampling distribution as n increases? Notice that even with this extremely non normal population distribution we find that the sampling distribution of the sample means becomes approximately normal as the sample sizes increase. This is a fact about the sampling distribution of sample means which leads us an extremely important theorem in statistics called the Central Limit Theorem. The Central Limit Theorem(CLT) Draw an SRS of size n from any population with a mean and a standard deviation. The Central Limit Theorem says that when the sample size n is large the sampling distribution of the sample means x is approximately normal. Page 9 of 14

10 The Central Limit Theorem (CLT) allows you to use a normal probability calculation to answer questions about the sample mean from many observations even when the population is not normal. But what is meant by large sample sizes? Smaller sample sizes may not always yield a normal sampling distribution. The Central Limit Theorem may be safely applied if the sample size n exceeds 30. If the population distribution is reasonably close to a normal distribution then sample sizes of 15 to 20 are often large enough for the sampling distribution of x to have an approximately normal distribution. Of course you still need to ensure randomization and independence of the observations in your sample but the Central Limit Theorem virtually guarantees a normal sampling distribution for the means given a large enough sample size. You can consider the Central Limit Theorem as the Fundamental Theorem of Statistics it s really that important! Be careful not to confuse the distribution of the data in your sample with the sampling distribution. For example, don t mistakenly think the CLT says that the data in your sample will be normally distributed as long as your sample size is large enough. In fact, as the sample size gets larger, we expect the distribution of the data in the sample to look more and more like the data in the population distribution from which the sample was drawn- skewed, bimodal, uniform, normal, etc. You should have seen that from the previous activity with the beta distribution. The Central Limit Theorem doesn t tell you anything about the shape of the data in your sample. It is about the shape of the sampling distribution of x. Make sure you understand the difference. Assumptions and Conditions for using a normal distribution to approximate the sampling distribution of x Randomization Condition: The data must be from an SRS of the population Independence Assumption: The sample values must be independent. This is a tough one to check so if the 10% condition is met and the data is from a SRS independence is usually satisfied. Large Enough Sample Size: This is not an obvious one. General rules of thumb are 30 or more but there is no magic number. We will address this issue of sample size later. Page 10 of 14

11 Lets apply these ideas to a problem situation. Section 7.3b Sample Means The Central Limit Theorem 2. Male blue throat birds have a complex song. Let x denote the duration of a randomly selected song in seconds. Suppose that the population mean value of the song is known to be 13.8 and has a standard deviation of 11.8 seconds. The population distribution of x is also known to be non- normal. Suppose you select an SRS of 25 blue throat birds and measure their song durations. a. The first condition is that we have a random sample from the population. Do we meet this condition? Why is this condition important? b. The second condition is independence. Do we reasonably meet this condition? c. The third condition is normality. Can you safely assume that the sampling distribution of x is approximately normal? Why or why not. What if the population was approximately normal? Page 11 of 14

12 3. A hotdog manufacturer asserts that one of its brands of hotdogs has an average fat content of 18 grams per hotdog with a standard deviation 1 gram. Consumers of this brand would probably not be disturbed if the mean fat content is less than 18 grams but would certainly be unhappy if the mean fat content exceeded 18 grams. Suppose we let x denote the fat content of a randomly selected hotdog and that we select a SRS of 36 hotdogs. Suppose that we find from our sample of 36 hotdogs that x a. If the manufacture s claim is correct that the mean fat content is 18 grams then what is probability that from our sample we would get a mean fat content of 18.4 grams or greater. In other words, under these conditions P x 18.4? Check the conditions for the sampling distribution of what is x before doing any calculations. Show your work. b. Based on your results from part a, do you think the manufacturers claim is correct? Explain your reasoning. Page 12 of 14

13 The figure below summarizes the facts about the sampling distribution of x. It illustrates the big idea of a sampling distribution. Keep taking random samples of size n from a population with a mean of and standard deviation. Find the mean x for each sample and collect all these sample means and display their distribution. That s the sampling distribution of x. As we have seen, this distribution is approximately normal when the population is normal, and when the population is not normal as long as our sample sizes equal or exceed 30 (another rule of thumb) then by the Central Limit Theorem we can safely assume an approximately normal sampling distribution. Page 13 of 14

14 CHECK YOUR UNDERSTANDING Building Better Batteries. Section 7.3b Sample Means The Central Limit Theorem Everyone wants to have the latest technological gadget. That s why ipods, digital cameras, smartphones, Game Boys, and the Wii have sold millions of units. These devices require lots of power and can drain batteries quickly. Battery manufacturers are constantly searching for ways to build longer-lasting batteries. A particular manufacturer produces AA batteries that are designed to last an average of 17 hours with a standard deviation of 0.8 hours. Quality control inspectors select a random sample of 30 batteries during each hour of production, and they drain them under conditions that mimic normal use. Below is one such sample. The average lifetime (in hours) of the batteries from this sample is 16.7 hours. Do these data suggest that the production process is working properly? Are plant managers safe sending out all the batteries produced in this hour for sale? Use what you have learned in this chapter to answer these questions. Page 14 of 14