# VCU STAT 210 - Lecture31 (84 pages)

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## Lecture31

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- Pages:
- 84
- School:
- Virginia Commonwealth University
- Course:
- Stat 210 - Basic Practice of Statistics

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STAT 210 Lecture 31 Sampling Distributions of the Sample Mean X November 8 2017 Practice Problems Sailboat Pages 220 through 230 Relevant problems VIII 1 through VIII 6 Recommended problems VIII 5 and VIII 6 Hummingbird Pages 250 through 260 Relevant problems IX 1 through IX 6 Recommended problems IX 5 and IX 6 Additional Reading and Examples Sailboat Read pages 213 through 218 Pay particular attention to page 213 Hummingbird Read pages 243 through 248 Pay particular attention to page 243 Top Hat 2 Motivating Example Students often ask questions about the average score on the STAT 210 final exam and other related questions We will return to this problem and describe the sampling distribution of the mean grade on the STAT 210 final exam for a simple random sample of 49 students Statistical Inference Statistical inference involves using statistics computed from sample data to make statements about unknown population parameters In this chapter the parameter of interest is the population mean m We are assuming that data for the entire population is not known and hence m is unknown and hence statistical inferences in the form of confidence intervals and statistical tests will need to be done Statistical Inference Statistical inference involves using statistics computed from sample data to make statements about unknown population parameters When making statistical inferences the first step is to identify the population of interest and the specific parameter of interest Consider the following three examples Example 1 Of interest is to estimate the mean age of all students at this university Example 1 Of interest is to estimate the mean age of all students at this university In this situation the population is all students at this university and the parameter of interest is m the mean age of all students at this university Example 2 It is conjectured that the mean cost of all women s swimwear purchased in 2009 was 60 and of interest is to test this conjecture versus the alternative that the mean cost of all women s swimwear purchased in 2009 was actually greater than 60 Example 2 It is conjectured that the mean cost of all women s swimwear purchased in 2009 was 60 and of interest is to test this conjecture versus the alternative that the mean cost of all women s swimwear purchased in 2009 was actually greater than 60 In this situation the population consists of all women s swimwear purchased in 2009 and the parameter of interest is m the mean cost of all women s swimwear purchased in 2009 Example 3 General Motors advertises that the 2010 Chevy Equinox averages 32 miles per gallon when driven on the highway Of interest is to test this claim versus the alternative that the mean miles per gallon when driven on the highway for all 2010 Chevy Equinox vehicles is different from 32 miles per gallon Example 3 General Motors advertises that the 2010 Chevy Equinox averages 32 miles per gallon when driven on the highway Of interest is to test this claim versus the alternative that the mean miles per gallon when driven on the highway for all 2010 Chevy Equinox vehicles is different from 32 miles per gallon In this situation the population consists of all 2010 Chevy Equinox vehicles driven on the highway and the parameter of interest is m the mean miles per gallon of all 2010 Chevy Equinox vehicles when driven on the highway Sample Data Once the population and parameter of interest are determined a sample is selected from the population and data collected on the characteristic of interest for the individuals in the sample From the sample data we compute the sample mean X and the sample mean X becomes the basis for the inferences that will be made about the unknown population mean m The sample mean X is called the point estimate of the population mean m Chapter Comparisons Last chapter PROPORTIONS Population proportion p Sample proportion p is the point estimate This chapter MEANS Population mean m Sample mean X is the point estimate Top Hat Sampling Distribution A sampling distribution of a statistic is the distribution of values taken by the statistic in a large number of simple random samples of the same size n taken from the same population Sampling Distributions To theoretically describe a sampling distribution we must describe the 1 shape 2 center 3 spread and 4 unusual features of the distribution Sampling Distribution of the Sample Mean X Suppose we know the population mean m and the population standard deviation s For example X person s age m 28 9 s 4 8 Assumptions 1 The data being used to make inferences must be a simple random sample from the population Assumptions 1 The data being used to make inferences must be a simple random sample from the population 2 The population distribution must be known to be normal or the sample size must be large enough for the central limit theorem to apply Shape of 0 1 Random Variable A 0 1 random variable has the following shape which obviously is not a normal distribution p 1 p 0 1 Central Limit Theorem The central limit theorem is a mathematical property that states that regardless of the shape of the original population if the sample size is large enough then the shape of the sampling distribution will be approximately normal CLT for Proportions Since the population is not normal we must use the central limit theorem If the sample size is large enough then the distribution of the sample proportion p is approximately normal The sample will be large enough if BOTH the expected number of successes np and the expected number of failures n 1 p are greater than or equal to 10 Central Limit Theorem for Means If the population is normal the sample size does not matter If the sample size is less than 15 then the sample size is not large enough unless the population is normal If the sample size is at least 15 then the central limit theorem will apply unless the distribution is heavily skewed due to outliers at least 15 means 15 or more If the distribution is heavily skewed due to outliers then the central limit theorem only applies if the sample size is at least 40 at least 40 means 40 or more Sampling Distribution of X If the population mean m and the population standard deviation s are both known and if both assumptions are satisfied then the sampling distribution of the sample mean X can be described as follows 1 Center the mean of X is mX m Sampling Distribution of X If the population mean m and the population standard deviation s are both known and if both

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