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Probability Distributions

Probability Distributions. Lesson 11-2.

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Probability Distributions

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  1. Probability Distributions Lesson 11-2

  2. “I surveyed my chemistry class and found that, on average, the students are 16.31 years old,” Sean says. “Therefore, the average age in all of the chemistry classes is 16.31 years.” “Exactly 16.31 years?” asks Yiscah. What concerns do you think Yiscah might have about Sean’s statement?

  3. Sean is making a prediction about a population based on the mean of a sample. A sample is a part of a larger collection, called a population, that is selected to represent the entire population. Values that describe a population are called parameters instead of statistics. So Sean is predicting the value of a population parameter based on a sample statistic.

  4. Sean is also considering ages. When you studied probabilities in Chapter 10, you considered random variables that had whole-number values, such as 5 heads, 3 tails, or 454 students. Those were discrete random variables. Ages have values between whole numbers. You might say that everyone in your class is 16 or 17; but actually no one is exactly 16, because a person is exactly 16 at only one instant. Ages are continuous random variables, and there are infinitely many possible ages.

  5. Pencil Lengths • In this investigation you’ll explore the difference between discrete and continuous random variables. Begin by collecting all the pencils that your group has. Step 1 Measure your pencils accurate to a tenth of a centimeter. Before you share data with other groups, predict the shape of a histogram of the class data. Step 2 Share all measurements so that the class has one set of data. On graph paper, draw a histogram with bins representing 1 cm increments in pencil length. Step 3 Divide the number of pencils in each bin by the total number of pencils. Make a new histogram, using these quotients as the values on the y-axis.

  6. Step 4 Check that the area of your second histogram is 1. Why must this be true? Step 5 Imagine that you collect more and more pencils and draw a histogram using the method described in Step 3. Sketch what this histogram of increasingly many pencil lengths would look like. Give reasons for your answer. Step 6 Imagine doing a very complete and precise survey of all the pencils in the world. Assume that their distribution is about the same as the distribution of pencils in your sample. Also assume that you use infinitely many very narrow bins. What will this histogram look like?

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