Unit 6 Section 5.4

5.4: The Central Limit Theorem  Sampling Distribution – the probability distribution of a sample statistic that is formed when samples of size n are repeatedly taken from a population.  If the sample statistic is the sample mean, then it is a Sampling Distribution of Sample Means  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. Section 5.4

Properties of the Distribution of Sample Means  The mean of the sample means will be the same as the population mean.  The standard deviation of the sample means will be equal to the population standard deviation divided by the square root of the sample size. Section 5.4

 Example 1 : Suppose a professor gave an 8 point quiz to a small class of 4 students. The results of the quiz were 2, 4, 6, and 8. a)Determine the mean of the population. b)Determine the standard deviation of the population. Section 5.4

c)Now, if all samples of size 2 are taken with replacement, construct a a table to represent all the possibilities and their means. d)Use your table to create a frequency distribution for the sample means. e)Determine the mean for the samples f)Determine the standard deviation for the samples. g)Take the standard deviation for the population and divide it by the square root of the sample size. What do you notice? Section 5.4

The standard deviation of the sample means is also known as the standard error of the mean. The formula for the standard error of the mean is: Section 5.4

The Central Limit Theorem  If samples of size n (where n is greater than or equal to 30) are drawn from any population (with mean μ and standard deviation σ), then the sampling distribution of sample means approximates a normal distribution.  If the population is normally distributed, then the sampling distribution of sample means is normally distributed for ANY sample size n. This distribution will have a mean μ and a standard deviation Section 5.4

 Example 2 : Cell phone bills for residents of a city have a mean of $47 and a standard deviation of $9, as shown in the figure. Random samples of 100 cell phone bills are drawn from this population, and the mean of each sample is determined. Find the mean and standard deviation of the sample means. Then sketch a graph of the sampling distribution. Section 5.4

When finding the probabilities for a sample mean within a sampling distribution, you will need to find z-scores. When the population is sufficiently large, you can locate a z-score using the following formula: Section 5.4

Things to remember when using the Central Limit Theorem:  When the original variable is normally distributed, the distribution of the sample means will be normally distributed for any sample size n. (Normal Method)  When the distribution of the original value might not be normal, a sample size of 30 or more is needed to use a normal distribution to approximate the distribution of sample means.  The larger the sample, the better the approximation Section 5.4

 Example 3 : A.C. Neilsen reported that children between the ages of 2 and 5 watch an average of 25 hours of television per week. Assume the variable is normally distributed and the standard deviation is 3 hours. If 30 children between the ages of 2 and 5 are randomly selected, find the probability that the mean of the number of hours they watch television will be greater than 26.3 hours. Section 5.4

 Step 1 : Determine the standard deviation of the sample means.  Step 2 : Draw a picture to represent the situation.  Step 3: Find the z-score  Step 4 : Determine the probability Section 5.4

 Example 4 : The average age of a vehicle registered in the United States is 8 years, or 96 months. Assume the standard deviation is 16 months. If a random sample of 36 vehicles is selected, find the probability that the mean of their age is between 90 and 100 months. Section 5.4

Homework:  Pg  #’s 1 – 29 ODD Section 5.4