Characteristics of Binomial Distributions


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1 Lesson2 Characteristics of Binomial Distributions In the last lesson, you constructed several binomial distributions, observed their shapes, and estimated their means and standard deviations. In Investigation 1 of this lesson, you will learn how to visualize the shape of a binomial distribution if you know n and p. In Investigation 2, you will discover simple formulas for the mean and the standard deviation of a binomial distribution. But first, consider the following situation. According to an annual nationwide survey of college freshmen, twothirds of both male and female freshmen planned to earn a graduate degree (master s or doctorate) or an advanced professional degree (such as law or medicine). (Source: Higher Education Research Institute, Annual Freshman Survey UCLA, 1997.) Suppose you have a random sample of 2 college freshmen and count the number who say they plan to get an advanced degree. The graph of the binomial distribution of the number of freshmen who plan to get an advanced degree is shown below Number of Freshmen 3 UNIT 5 BINOMIAL DISTRIBUTIONS AND STATISTICAL INFERENCE
2 Think About This Situation Examine the binomial distribution on the previous page. a b c d e What is the approximate shape of the graph? Explain what the height of the bar between 14 and 141 means. How many freshmen in a random sample of 2 would you expect to say that they plan to earn advanced degrees? How can you find the standard deviation of this probability distribution? What information does the standard deviation give you? How do you think the shape, center, and spread would change if the sample size was much larger? Much smaller? How do you think the shape, center, and spread would change if you graphed the proportion of successes rather than the number of successes? INVESTIGATION 1 The Shapes of Binomial Distributions The graph of the binomial distribution for the number of freshmen who plan to get advanced degrees looks approximately normal. In this investigation, you will see that not all binomial distributions are approximately normal in shape. However, you will be able to predict when they will be if you know the probability of a success p and the sample size n. 1. According to the 2 U.S. Census, about 2% of the population of the United States are children; that is, age 13 or younger. (Source: Age: 2, Census 2 Brief, October 21 at c2kbr112.pdf) Suppose you take a random sample of people from the United States. The following graphs show the binomial distributions for the number of children in random samples varying in size from 5 to Sample Size n = Number of Children.4.3 Sample Size n = Number of Children 5 5 Sample Size n = Number of Children LESSON 2 CHARACTERISTICS OF BINOMIAL DISTRIBUTIONS 31
3 5 Sample Size n = 5 Sample Size n = Number of Children Number of Children a. Determine the exact height of the tallest bar on the graph for a sample size of 1. b. Why are there more bars as the sample size increases? How many bars should there be for a sample size of n? Why are there only 7 bars for a sample size of 1? c. What happens to the shape of the distribution as the sample size increases? d. What happens to the mean of the number of successes as the sample size increases? e. What happens to the standard deviation of the number of successes as the sample size increases? 2. Examine the following graphs. They show binomial distributions for the number of heads when a fair coin is tossed various numbers of times. n = 1 n = 2 n = 3 n = 4 n = Number of Heads x n = 6 32 UNIT 5 BINOMIAL DISTRIBUTIONS AND STATISTICAL INFERENCE
4 n = 7 n = 8 n = 9 n = Number of Heads x a. What happens to the shape of these distributions as the sample size n increases? b. What happens to the mean of the probability distribution as n increases? c. What happens to the standard deviation as n increases? 3. Now consider what happens when the sample size is fixed and the probability of success varies. The set of graphs below show the binomial distributions for a sample size of 4 and probabilities of success varying from to.9. p = 1% p = 2% p = 3% p = 4% p = 5% Number of Successes LESSON 2 CHARACTERISTICS OF BINOMIAL DISTRIBUTIONS 33
5 p = 6% p = 7% p = 8% p = 9% Number of Successes a. Which of these distributions are a bit skewed? What happens to the shape of the distributions as the probability of success p increases? Which of these distributions has a shape that is closest to normal? b. What happens to the mean of the number of successes as the probability of success p increases? c. What happens to the standard deviation as p increases? d. What symmetries do you see in this set of graphs? e. How is this set of graphs similar to the box plot charts from fixed sample sizes you made in Course 3, Unit 2, Modeling Public Opinion? The box plot chart for samples of size 4 is shown below. Population Percent 9% Box Plots from Samples of Size 4 Sample Outcome as a Proportion Sample Outcome as a Total 34 UNIT 5 BINOMIAL DISTRIBUTIONS AND STATISTICAL INFERENCE
6 4. Based on your work in Activities 1 through 3, draw conclusions about the effect of sample size or value of p on the shape of the graph of a binomial distribution. a. Complete each sentence using the word more or less. With a fixed sample size, the farther p is away from.5, the skewed the binomial distribution. Usually, with a fixed value of p, the larger the sample size, the skewed the binomial distribution. b. What are the exceptions to the statement in the second item of Part a? 5. To predict whether the graph of a binomial distribution will look approximately normal, compute both np and n(1 p) and check to see if both are at least 1. a. The value np can be interpreted as the expected number of successes. How can you interpret n(1 p)? b. Which of the distributions in Activity 1 can be considered approximately normal using the above guideline? Does this agree with your visual impression? c. Which of the distributions in Activity 2 can be considered approximately normal using this guideline? Does this agree with your visual impression? d. Which of the distributions in Activity 3 can be considered approximately normal using this guideline? Does this agree with your visual impression? 6. Imagine the binomial distribution with n = 35 and p =. a. Is this distribution skewed left, skewed right, or symmetric? b. Where is it centered? Estimate its standard deviation. c. Check your answers to Parts a and b using the binomial function capabilities of your calculator to make a graph of this distribution. Checkpoint Think about a binomial distribution with probability of a success p and sample size n. a b As n increases, but the probability of a success p remains the same, what happens to the shape, center, and spread of the binomial distribution for the number of successes? As p increases from 1 to.99, but n remains the same (for example, n = 5) what happens to the shape, center, and spread of the binomial distribution for the number of successes? Be prepared to discuss your descriptions of changes in the binomial distributions. LESSON 2 CHARACTERISTICS OF BINOMIAL DISTRIBUTIONS 35
7 On Your Own Think about patterns of change in binomial distributions as the sample size or probability of success varies. a. How does the binomial distribution for the number of successes in a sample of size 25 and probability of success.3 differ from a binomial distribution with sample size 5 and probability of success.3? b. How does the binomial distribution for the number of successes in a sample of size 25 and probability of success.3 compare to the binomial distribution with sample size 25 and probability of success.7? In the first part of this investigation, you examined the shape, center, and spread of binomial distributions of the number of successes in a random sample. Now examine these same characteristics of distributions of the proportion ˆp of successes. 7. How do you convert the number of successes in a binomial situation to the proportion of successes? 8. Recall from Activity 1 (page 31) that about 2% of the population of the United States are children age 13 or younger. Shown below are the graphs from Activity 1 with the scale on the xaxes changed to one that gives the proportion ˆp of children in random samples, for sample sizes varying from 5 to Sample Size n = 5 1. Proportion of Children.4.3 Sample Size n = Proportion of Children 5 5 Sample Size n = Proportion of Children 5 5 Sample Size n = Sample Size n = Proportion of Children Proportion of Children a. What happens to the mean of the sample proportions as the sample size increases? b. What happens to the standard deviation of the sample proportions as the sample size increases? Why does this make sense? 36 UNIT 5 BINOMIAL DISTRIBUTIONS AND STATISTICAL INFERENCE
8 9. On a copy of each of the graphs from Activity 2, change the scale on the x axis to one that gives the sample proportion of heads, ˆp. a. What happens to the mean of the proportion of heads as the sample size increases? b. What happens to the standard deviation of the proportion of heads as the sample size increases? Checkpoint Consider a binomial situation with probability of success p and sample size n. a b c If n increases but p remains the same, what happens to the shape, center, and spread of the distribution of the sample proportions? Compare the graphs of the distributions of ˆp and of 1 ˆp for a fixed sample size n. Summarize the differences between the binomial distribution for the number of successes and the distribution for the corresponding sample proportions. Be prepared to discuss the variations in the distributions. On Your Own Think about patterns of change in distributions for the sample proportion ˆp as the sample size, or probability of success varies. a. How does the distribution of the sample proportion ˆp from a sample of size 25 and probability of success.3 differ from the distribution with sample size 5 and probability of success.3? b. How does the distribution of the sample proportion ˆp from a sample size of 25 and probability of success.3 compare to the distribution with sample size 25 and probability of success.7? c. Compare your answers for Parts a and b with your corresponding answers to Parts a and b of the On Your Own on page 36. INVESTIGATION 2 Simple Formulas for the Mean and the Standard Deviation Recall that the formula for the mean of a probability distribution is = x p(x) Similarly, the formula for the standard deviation of a probability distribution is = ( x ) 2 p (x ) LESSON 2 CHARACTERISTICS OF BINOMIAL DISTRIBUTIONS 37
9 In this investigation, you will see that these formulas can be simplified in the case of binomial distributions. 1. According to the 2 U.S. Census, about 2% of the population of the United States are children (aged 13 or younger). Suppose you take a random sample of four people living in the United States and count the number of children. a. Complete this probability distribution table for the number of children in your sample of size 4. Number of Children x p(x) b. Use the formulas at the bottom of page 37 to compute the mean and standard deviation of the probability distribution in Part a. c. In computing the mean of the above distribution, you could also simply reason that if 2% of the population are children, the mean number of children in a random sample of four people should be 2% of 4 or.8. Compare this value with the mean value you computed in Part b. 2. Part c of Activity 1 illustrates a general formula for computing the mean of a binomial distribution. The mean number of successes in n trials with probability of success p is = np There is also a much simpler formula for the standard deviation of a binomial distribution: = n p (1 p ) a. Verify that this formula for the standard deviation gives the same result as that in Activity 1 Part b. b. Now refer back to the graphs in Activity 1 of Investigation 1 of this lesson. Complete the table at the top of page 39 for those graphs. In each case, p =. 38 UNIT 5 BINOMIAL DISTRIBUTIONS AND STATISTICAL INFERENCE
10 Sample Size n Mean Standard Deviation c. If you double the sample size, what happens to the mean? To the standard deviation? d. How does the mean vary with the sample size? How does the standard deviation vary with the sample size? 3. Consider this table for binomial distributions with n = 9 and various values of p. The variance 2 is the square of the standard deviation. of a Success p Mean = np Variance a. Complete the table. b. What patterns do you see in the table? c. Plot the points (p, 2 ). What type of function would model the pattern of change shown in the graph? d. Describe at least three possible methods for finding an equation showing the relationship between the probability of success and the variance. e. Use two of these methods to find an equation. 4. About 26% of the U.S. population 25 and over have completed a bachelor s degree. (Source: Educational Attainment in the United States (Update). Current Population Reports, March 2, U.S. Census Bureau.) a. Describe the shape, mean, and standard deviation of the binomial distribution for samples of size 1, for this situation. b. What numbers of people with bachelor s degrees would be rare events? LESSON 2 CHARACTERISTICS OF BINOMIAL DISTRIBUTIONS 39
11 5. You can use the formulas for the mean and standard deviation of the number of successes to derive corresponding formulas for the proportion of successes. a. If the mean number of successes is = np, what computation would you do to get the mean proportion of successes? Write the formula for the mean proportion of successes. b. Why does the result in Part a make sense intuitively? c. If the standard deviation of the number of successes is = n p (1 p ),what computation would you do to get the standard deviation of the proportion of successes? Write the formula in the simplest form possible. d. By examining your formula from Part c, what can you determine about the standard deviation for the proportion of successes as the sample size increases? e. Why does the result in Part d make sense intuitively? Checkpoint When studying a binomial situation, it is often helpful to know the mean and standard deviation of the number of successes or the proportion of successes. a b Compare the formulas for the mean and standard deviation of the number of successes in a binomial distribution to the corresponding formulas for the proportion of successes. Suppose you fix a value of p and n and construct the probability distributions for the number of successes and the proportion of successes. Describe how the shape, mean, and standard deviation of each distribution changes if you increase n but keep p fixed. Be prepared to share your comparison and explanation with the class. On Your Own Consider a binomial distribution with n = 3 and p =.45. a. Make a probability distribution table giving the number of successes. b. Find the mean of the number of successes using the formula = np and then using the formula = x p(x). c. Find the standard deviation of the number of successes using the formula = n p (1 p ) and then by using the formula = ( x ) 2 p (x ). d. Find the mean and standard deviation of the distribution of the sample proportion ˆp. 31 UNIT 5 BINOMIAL DISTRIBUTIONS AND STATISTICAL INFERENCE
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