CHAPTER 7. Parameters are numerical descriptive measures for populations.

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1 CHAPTER 7 Introduction Parameters are numerical descriptive measures for populations. For the normal distribution, the location and shape are described by µ and σ. For a binomial distribution consisting of n trials, the location and shape are determined by p. Often the values of parameters that specify the exact form of a distribution are unknown. You must rely on the sample to learn about these parameters. Simple Random Sampling The sampling plan or experimental design determines the amount of information you can extract, and often allows you to measure the reliability of your inference. Simple random sampling is a method of sampling that allows each possible sample of size n an equal probability of being selected. In general, if a sample of n elements is selected from a population of N elements using a sampling plan in which each of the possible samples has the same chance of selection, then the sampling is said to be random and the resulting sample is a simple random sampling. The numerical descriptive measures that calculate from the sample are called statistics. Sampling Distributions The sampling distribution of a statistic is the probability distribution for the possible values of the statistic that results when random samples of size n are repeatedly drawn from the population. Sampling distributions for statistics can be Derived mathematically using the laws of probability Approximated with simulation techniques Used statistical theorems to derive exact or approximate sampling distribution The Central Limit Theorem is one such theorem. 1

2 Central Limit Theorem If random samples of n observations are drawn from a nonnormal population with finite µ and standard deviation σ, then when n is large, the sampling distribution of the sample mean x is approximately normally distributed, with mean µ and standard deviation σ/ n. The approximation becomes more accurate as n becomes large. Why is this important? The Central Limit Theorem also implies that the sum of n measurements is approximately normal with mean nµ and standard deviation σ n. Many statistics that are used for statistical inference are sum or average of sample measurements. When n is large, these statistics will have approximately normal distributions. This will allow us to describe their behavior and evaluate the reliability of our inferences. How large is large? If the sample is normal, then the sampling distribution of x will also be normal, no matter what the sample size. When the sample population is approximately symmetric, the distribution becomes approximately normal for relatively small values of n. When the sample population is skewed,the sample size must be at least 30 before the sampling distribution of x becomes approximately normal. The Sampling Distribution of the Sample Mean x If a random sample of n measurements is selected from a population with mean µ and standard deviation σ, the sampling distribution of sample mean x will have mean µ and standard deviation σ/ n. If the population has a normal distribution, the sampling distribution of x will be exactly normally distributed, regardless of the sample size, n. If the population distribution is nonnormal, the sampling distribution of x will be approximately normally distributed for large samples (by CLT). 2

3 Standard Error the standard deviation of a statistic used as an estimator of a population parameter is also called the standard error of the estimator, SE because it refers to the precision of the estimator. Therefore, the standard deviation of x- given by σ/ n- is referred to as the standard error of the mean Finding Probabilities for the Sample Mean If the sampling distribution of x is normal or approximately normal, standardize the interval of interest in terms of z = x µ σ/ n Find the appropriate area using table. Examples: 1. The soda filling machine is supposed to fill cans of soda with 12 fluid ounces. Suppose that the fills are actually normally distributed with a mean of 12.1 oz and a standard deviation of 2 oz. What is the probability that the average fill for a 6-pack of soda is less than 12 oz? 2. Suppose a random sample of n = 25 observations is selected from a population that is normally distributed, with mean equal to 106 and standard deviation equal to 12. a. Give the mean and the standard deviation of the sampling distribution of the sample mean x. b. Find the probability that x exceeds 110. c. Find the probability that the sample mean deviates from the population mean µ = 106 by no more than Suppose than college faculty with the rank of professor at 2-years institutions earn an average of $65,608 per year with a standard deviation of $4000. In an attempt to verify this salary level, a random sample of 60 professors was selected from a personnel database for all 2-year institutions in the United States. a. Describe the sampling distribution of the sample mean x. b. Within what limits would you expect the sample average to lie, with probability 0.95? c. Calculate the probability that the sample mean x is greater than $67,000. d. If your random sample actually produced a sample mean of $67,000, would you consider this unusual? What conclusion might you draw? - 3

4 The Sampling Distribution of the Sample Proportion The Central Limit Theorem can be used to conclude that the binomial random variable x is approximately normal when n is large, with mean np and standard deviation npq. The sample proportion, ˆp = x n divided by n. is simply a rescaling of the binomial random variable x, From the Central Limit Theorem, the sampling distribution of ˆp will also be approximately normal, with a rescaled mean and standard deviation The Sampling Distribution of the Sample Proportion A random sample of size n is selected from a binomial population with parameter p. The sampling distribution of sample proportion, ˆp = x n deviation SE(ˆp) = pq. n will have mean p and standard If n is large, and p is not too close to zero or one, the sampling distribution of ˆp will be approximately normal Finding Probabilities for the Sample Proportion If the sampling distribution of ˆp is normal or approximately normal, standardize the interval of interest in terms of z = ˆp p pq n Find the appropriate area using table Examples: 1. The soda bottler in the previous example claims that only 5% of the soda cans are underfilled. A quality control technician randomly samples 200 cans of soda. What is the probability that more than 10% of the cans are underfilled? 2. The percentage of students who used the internet as their major resource for a school project in the past year was 66%. Sup pose that you take a sample of n = 100 students, and record the number of students who used the Internet as their major resource for their school project during the past year. Let ˆp be the proportion of students surveyed who used the Internet as a major resource in the past year a. What is the exact distribution of ˆp? How can you approximate the distribution of 4

5 ˆp? b. What is the probability that the sample proportion ˆp exceeds 68%? c. What is the probability that the sample proportion lies between 64% and 68%? d. Would a sample proportion of 70% contract the reported value of 66%? - Suggested Exercises: 7.19, 7.21, 7.27, 7.29, 7.31, 7.33, 7.37, 7.43, 7.45, 7.47, 7.63, 7.65, 7.73, 7.75,