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5.3 The Central Limit Theorem

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Roll a die 5 times and record the value of each roll. Find the mean of the values of the 5 rolls. Repeat this 250 times.

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Don’t forget: You can copy- paste this slide into other presentations, and move or resize the poll.

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x=3.504 s=.7826 n=5

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Roll a die 10 times and record the value of each roll. Find the mean of the values of the 10 rolls Repeat this 250 times.

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Don’t forget: You can copy- paste this slide into other presentations, and move or resize the poll. Poll: Toss a die 10 times and record your resu...

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x=3.48 s=.5321 n=10

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Roll a die 20 times. Find the mean of the values of the 20 rolls. Repeat this 250 times.

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Don’t forget: You can copy- paste this slide into other presentations, and move or resize the poll.

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x=3.487 s=.4155 n=20

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What do you notice about the shape of the distribution of sample means?

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Central Limit Theorem Suppose we take many random samples of size n for a variable with any distribution--- For large sample sizes: 1.The distribution of means will be approximately a normal distribution.

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1, 2, 3, 4, 5, 6 Mean: =3.5 Standard Deviation: =1.7078 How does the mean of the sample means compare to the mean of the population? Remember for 250 trials: When n=5, x=3.504 When n=10, x=3.48 When n=20, x=3.487 How does the mean of the sample means compare to the mean of the population?

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Central Limit Theorem Suppose we take many random samples of size n for a variable with any distribution--- For large sample sizes: 1.The distribution of means will be approximately a normal distribution. 2.The mean of the distribution of means approaches the population mean, .

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1, 2, 3, 4, 5, 6 Mean: =3.5 Standard Deviation: =1.7078 How does the standard deviation of the sample means compare to the standard deviation of the population? Remember for 250 trials: When n=5, s=.7826 When n=10, s=.5321 When n=20, s=.4155 How does the standard deviation of the sample means compare to the standard deviation of the population?

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Central Limit Theorem Suppose we take many random samples of size n for a variable with any distribution--- For large sample sizes: 1.The distribution of means will be approximately a normal distribution. 2.The mean of the distribution of means approaches the population mean, . 3.The standard deviation of the distribution of means approaches.

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Cost of owning a dog Suppose that the average yearly cost per household of owning dog is $186.80 with a standard deviation of $32. Assume many samples of size n are taken from a large population of dog owners and the mean cost is computed for each sample. If the sample size is n=25, find the mean and standard deviation of the sample means. If the sample size is n=100, find the mean and standard deviation of the sample means.

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Teacher’s salary The average teacher’s salary in New Jersey (ranked first among states) is $52,174. Suppose the distribution is normal with standard deviation equal to $7500. What percentage of individual teachers make less than $45,000? Assume a random sample of 64 teachers is selected, what percentage of the sample means is a salary less than $45,000?

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Height of basketball players Assume the heights of men are normally distributed with a mean of 70.0 inches and a standard deviation of 2.8 inches. What percentage of individual men have a height greater than 72 inches? The mean height of a 16 man roster on a high school team is at least 72 inches. What percentage of sample means from a sample of size 16 are greater than 72 inches? Is this basketball team unusually tall?

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