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Central Limit Theorem Notes Spring

# Central Limit Theorem Notes Spring - 2 of 8 1.1...

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Unformatted text preview: 2 of 8 1.1 Introduction Example 1 . Generate 3 sets of 8 random random numbers from a normally distributed population with a mean of 10 and standard deviation of 2. R: rnorm (8 , mean = 10 , sd = 2) [ 1 ] 9.6562 6.9938 8.3588 8.1606 9.8177 9.5088 10.8508 [ 8 ] 6.4181 R: rnorm (8 , mean = 10 , sd = 2) [ 1 ] 7.0716 6.9356 6.6494 12.9266 12.0327 10.1264 12.6311 [ 8 ] 10.2955 R: rnorm (8 , mean = 10 , sd = 2) [ 1 ] 9.6340 8.3440 7.1764 9.1673 7.3618 10.7930 7.5256 [ 8 ] 12.5522 Comparison of histogram of 5,000 randomly generated numbers to density function. R: x . rand = rnorm (5000 , mean = 10 , sd = 2) R: curve (dnorm(x , mean = 10 , sd = 2) , 0 , 16 , lty = ”dashed ” , + ylab = ”f (x) ” , main = ”Histogram of random numbers vs ←- theoreti cal density ”) R: hist (x . rand , prob = T, add = T) 5 10 15 0.00 0.05 0.10 0.15 0.20 Histogram of random numbers vs theoretical density x f(x) POPULATION MEANS AND SAMPLE MEANS College aged women’s systolic blood pressures (in mm Hg) are normally dis- tributed with a mean of 114.8 and a standard deviation of 13.1. Question 1 . If we randomly measured 10 women in this subgroup, would the mean blood pressure be 114.8? Question 2 . If we conducted another random sample of 10 women would the mean blood pressure be the same? Anthony Tanbakuchi MAT167 The Central Limit Theorem 3 of 8 Question 3 . If we repeatedly sample all the possible combinations of 10 women what would the distribution of sample means look like? Sample means Take 5 random samples each with 5 women from the subgroup and computing the mean to estimate the population mean. ¯ x = { 104 , 116 , 119 , 115 , 110 } Question 4 . Is the sample mean the same each time? Why? Question 5 . What is the range of sample mean values? Question 6 . What is a quick estimate of their standard deviation? Question 7 . How could we improve the sample mean’s estimate of the popula- tion mean?...
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Central Limit Theorem Notes Spring - 2 of 8 1.1...

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