Making Sense of Cents
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- Frederick Stanley
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1 Name: Date: Making Sense of Cents Exploring the Central Limit Theorem Many of the variables that you have studied so far in this class have had a normal distribution. You have used a table of the normal distribution to answer questions about a randomly selected individual or a random sample taken from a normal distribution. Many distributions, however, are not normal or any other standard shape. If the shape of a distribution isn t normal, can we make any inferences about the mean of a random sample from that distribution? Objective Discover the central limit theorem by observing the shape, mean, and standard deviation of the sampling distribution of the mean for samples taken from a distribution that is not normal. Activity Take a random sample of 25 pennies and make a list of the dates on each penny. Next to each date, write the age of the penny by subtracting the date from this year. Date of Penny Age Date of Penny Age What do you think the shape of the distribution of all the ages of pennies from students in the class will look like?
2 Make a histogram using a frequency distribution with 10 classes of the ages of all the pennies in the class. Estimate the mean and standard deviation of the distribution. Confirm your estimates by actual computation using the graphing calculator. In List 1 you should put in all of the raw data values. USE THE #YEARS OLD AS YOUR RAW DATA VALUES (Do not use the sample statistics, as you and your partner s pennies are your population, so you must compute the mean and then you may use the population statistic given on the calculator.) Mean = µ = Standard Deviation= σ =
3 Take a random sample of size 5 from your pile of pennies, and compute the mean age of your sample. Sketch the distribution of the class s data by drawing the number line on the board with the results of the class s computation. Sketch the curve that approximates the distribution. Sampling Distribution of Sample Means, n = 5 What are the actual mean and standard deviation of this distribution? Again, use the population statistic and the raw data values in your list.
4 Take a random sample of size 10 from the ages of your pennies, and compute the mean age of your sample. Place your sample means on a number line on the board. Sketch the distribution of the class s data by drawing the number line on the board with the results of the class s computation. Sketch the curve that approximates the distribution. Sampling Distribution of Sample Means, n = 10 What are the actual mean and standard deviation of this distribution? Again, use the population statistic and the raw data values in your list.
5 Take a random sample of size 25 from the ages of your pennies, and compute the mean age of your sample. Place your sample means on a number line on the board. Sketch the distribution of the class s data by drawing the number line on the board with the results of the class s computation. Sketch the curve that approximates the distribution. Sampling Distribution of Sample Means, n = 25 What are the actual mean and standard deviation of this distribution? Again, use the population statistic and the raw data values in your list. What are the actual mean and standard deviation of this distribution?
6 Look at the four histograms the class has constructed. What can you say about the shape of the histogram as n increases? What can you say about the center of the histogram as n increases? What can you say about the spread of the histogram as n increases? Shape, center, and spread of the sampling distribution make up the central limit theorem. Without looking in your textbook, write a statement of what you think the central limit theorem says. The distributions you constructed for samples of size 1, 5, 10, and 25 are called sampling distributions of the sample mean. Sketch what you anticipate the curve for the sampling distribution of the sample mean to look like for samples of size 36. Write a paragraph explaining what you learned from this activity. Find an example of a use for the Central Limit Theorem in the real world.
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