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Unformatted text preview: Distribution of the sample mean and the central limit theorem Means of different random variables The mean, X, of 2 rolls of a die takes on various values  it is a random variable. The mean waiting time between arrivals of customers to a restaurant during a certain morning, X, may take various values in different days – it is a random variable The mean number of boys per family, X, for certain 6 families is a random variable. In this lecture we will examine how the mean of a sample, X, behaves – its probability distribution, its mean and its variance. Example The mean of all possible rolls of a die is μ=3.5 , and the standard deviation is σ=1.7 . The shape of the probability distribution is: X 1 2 3 4 5 6 P(x) 1/6 1/6 1/6 1/6 1/6 1/6 1 2 3 4 5 6 1/6 Roll a die twice: The mean outcome, , of two rolls of a die is a random variable. Sometimes this mean will be less than 3.5, sometimes higher. The sampling distribution of the mean will be centered around 3.5. It can be any number between ___ to ___ X Example Suppose a teacher asked each student in a class of 50 students to roll a die twice and to compute the average of the two rolls. Today, instead of actually rolling a die twice we will use Minitab to generate possible results of the outcomes of the roll of two dice for 50 “hypothetical students”. 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 15 10 5 mean of two rolls Frequency Histogram of the mean of 2 rolls of a die by 50 simulated students: Mean=3.37 SD=1.1285 Suppose the 50 students had to roll the die 10 times: The mean outcome, X, of ten rolls of a die is a random variable. Sometimes this mean will be less than 3.5, sometimes higher. The sampling distribution of the mean will be centered around 3.5. We expect the range of results to be (compared to the two rolls) (i) more concentrated around the mean (ii) less concentrated around the mean We, again, use Minitab to help us simulate the results of 10 rolls of a die for 50 hypothetical students: Histogram of the mean of 10 rolls of a die by 50 simulated students: 5 4 3 2 10 5 mean of 10 rolls Frequency Mean=3.394 SD= 0.62087 Now, suppose the 50 students had to roll the die 100 times: The mean outcome, X, of 100 rolls of a die is a random variable. Sometimes this mean will be less than 3.5, sometimes higher. The sampling distribution of the mean will be centered around 3.5. we expect the range of results to be (compared to the ten rolls) (i) more concentrated around the mean (ii) less concentrated around the mean Histogram of the mean of 100 rolls of a die by 50 simulated students: 4.1 4.0 3.9 3.8 3.7 3.6 3.5 3.4 3.3 3.2 3.1 15 10 5 mean of 100 rolls Frequency Mean=3.5028 SD=0.20656 As the sample size, n, increases, we expect the distribution of the sample mean to be more concentrated around the mean....
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This note was uploaded on 11/03/2009 for the course IST IST taught by Professor N/a during the Spring '09 term at Anadolu University.
 Spring '09
 N/A

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