Chapter 11 Sampling Distributions BPS - 5th Ed. Chapter 11 1
Sampling Terminology Parameter fixed, unknown number that describes the population Statistic known value calculated from a sample a statistic is often used to estimate a parameter Variability different samples from the same population may yield different values of the sample statistic Sampling Distribution tells what values a statistic takes and how often it takes those values in repeated sampling BPS - 5th Ed. Chapter 11 2
The Law of Large Numbers Consider sampling at random from a population with true mean µ. As the number of (independent) observations sampled increases, the mean of the sample gets closer and closer to the true mean of the population. ( gets closer to µ ) x BPS - 5th Ed. Chapter 11 3
Sampling Distribution The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size (n) from the same population to describe a distribution we need to specify the shape, center, and spread we will discuss the distribution of the sample mean (x-bar) in this chapter BPS - 5th Ed. Chapter 11 4
Mean and Standard Deviation of Sample Means If numerous samples of size n are taken from a population with mean μ and standard deviation σ, then the mean of the sampling X distribution of is μ (the population mean) and the standard deviation is: σ (σ is the population s.d.) n BPS - 5th Ed. Chapter 11 5
Mean and Standard Deviation of Sample Means Since the mean of X is μ, we say that an unbiased estimator of μ X is Individual observations have standard deviation σ, but sample means X from samples of size n have standard deviation σ n. Averages are less variable than individual observations. BPS - 5th Ed. Chapter 11 6
Sampling Distribution of Sample Means If individual observations have the N(µ, σ) distribution, then the sample mean of n independent observations has the N(µ, σ/ ) distribution. If measurements in the population follow a Normal distribution, then so does the sample mean. X n BPS - 5th Ed. Chapter 11 7
Central Limit Theorem If a random sample of size n is selected from ANY population with mean μ and standard deviation σ, then when n is large the sampling distribution of the sample mean X is approximately Normal: X is approximately N(µ, σ/ n ) No matter what distribution the population values follow, the sample mean will follow a Normal distribution if the sample size is large. BPS - 5th Ed. Chapter 11 8
Central Limit Theorem: Sample Size How large must n be for the CLT to hold? depends on how far the population distribution is from Normal the further from Normal, the larger the sample size needed a sample size of 30 is typically large enough for any population distribution encountered in practice recall: if the population is Normal, any sample size will work (n 1) BPS - 5th Ed. Chapter 11 9
Detecting gypsy moths. The gypsy moth is a serious threat to oak and aspen trees. A state agriculture department places traps throughout the state to detect the moths. When traps are checked periodically, the mean number of moths trapped is only 0.5, but some traps have several moths. The distribution of moth counts is discrete and stongly skewed, with standard deviation 0.7.
Detecting gypsy moths. The gypsy moth is a serious threat to oak and aspen trees. A state agriculture department places traps throughout the state to detect the moths. When traps are checked periodically, the mean number of moths trapped is only 0.5, but some traps have several moths. The distribution of moth counts is discrete and stongly skewed, with standard deviation 0.7. a. What are the mean and standard deviation of the average number of moths x of 50 traps?
Detecting gypsy moths. The gypsy moth is a serious threat to oak and aspen trees. A state agriculture department places traps throughout the state to detect the moths. When traps are checked periodically, the mean number of moths trapped is only 0.5, but some traps have several moths. The distribution of moth counts is discrete and stongly skewed, with standard deviation 0.7. a. What are the mean and standard deviation of the average number of moths x of 50 traps? b. Use the central limit theorem to find the probability that the average number of moths in 50 traps is greater than 0.6.
Insurance. An insurance company knows that in the entire population of millions of homeowners, the mean annual loss from fire is µ = $250 and the standard deviation of the loss is σ = $1, 000. The distribution of losses is strongly right-skewed: most policies have $0 loss, but a few have large losses. If the company sells 10,000 policies, can it safely base its rates on the assumption that its average loss will be no greater than $275?