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6.3 The central limit theorem

6.3 The central limit theorem. Distribution of sample means. A sampling distribution of sample means is a distribution using the means computer form all possible random samples of a specific size taken from a population

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6.3 The central limit theorem

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  1. 6.3 The central limit theorem

  2. Distribution of sample means • A sampling distribution of sample meansis a distribution using the means computer form all possible random samples of a specific size taken from a population • Ex: A researcher selects a sample of 30 males and find the mean of the age to be 30 for the first sample, 28 for the second, and 35 for the third, etc. The means with become a random variable that we can study.

  3. Sampling Error • The difference between the sample measure and the corresponding population measure due to the fact that the sample is not a perfect representation of the population.

  4. Properties of the distribution of sample means • 1. The mean of the sample means will be the same as the population mean. • 2. The SD of the sample means will be SMALLER than the SD of the population, and it will be equation to the population SD divided by the square root of the sample size.

  5. The central limit theorem • As the sample size n increases without limit, the shape of the distribution of the sample means taken with replacement form a population will approach a normal distribution. The distribution will have a mean and standard deviation σ/ .

  6. Example • Children between the ages of 2 and 5 watch an average of 25 hours of TV per week. Assume the variable is normally distributed and the SD is 3 hours. If 20 children are randomly selected, find the probability that the mean number of hours they watch television will be greater than 26.3 hours.

  7. Step 1: Find the standard deviation of the sample means: Step 2: Find the z-value using the “new” standard deviation.

  8. The average age of a vehicle registered in the US is 96 months. Assume the standard deviation is 16 months. If a random sample of 36 vehicles if selected, find the probability that the mean of their age is between 90 and 100 months.

  9. What’s the difference? • is used for INDIVIDUAL data… so only one value • is used for SAMPLE data….. Use then there is a sample of items

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