SPA 3e_ Teachers Edition _ Ch 6

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426 C H A P T E R 6 • Sampling Distributions • The sampling distribution of p^ is less variable for larger samples. This is indicated by the !n in the denominator of the standard deviation formula. • The formula for the standard deviation of the distribution of p^ requires that the observations are independent. In practice, we are safe assuming independence when sampling without replacement as long as the sample size is less than 10% of the population size. Alternate Example Expensive ride? Mean and SD of the sampling distribution of p^ PROBLEM: Suppose that 26% of all high school students at a large school spend about half or more of their earnings on a car. A random sample of 150 students from this school is surveyed. Let p^ 5 the proportion of students in the sample who spend about half or more of their earnings on a car. (a) Calculate the mean and standard deviation of the sampling distribution of p^ . (b) Interpret the standard deviation from part (a). SOLUTION: (a) m p^ = 0.26 and 0.26(1− 0.26) s p^ = = 0.036 Å 150 (b) In SRSs of size n 5 150, the sample proportion of students who spend about half or more of their earnings on a car will typically vary by about 0.036 from the true proportion of p 5 0.26. a e XAMPLe What proportion of students have a smartphone? Mean and SD of the sampling distribution of p^ PROBLEM: Suppose that 43% of students at a large high school own a smartphone. As part of a schoolwide technology study, the principal surveys an SRS of n 5 100 students. Let p^ 5 the proportion of students in the sample who own a smartphone. (a) Calculate the mean and the standard deviation of the sampling distribution of p^ . (b) Interpret the standard deviation from part (a). SOLUTION: 0.43(1 − 0.43) (a) m p^ = 0.43 and s p^ = = 0.050 Å 100 (b) In SRSs of size n 5 100, the sample proportion of students who own a smartphone will typically vary by about 0.050 from the true proportion of p 5 0.43. d ThinK ABoUT iT Is the sampling distribution of p^ (the sample proportion of successes) related to the sampling distribution of X (the sample count of successes)? Yes! number of successes in sample p^ = = X sample size n For example, here are dotplots showing the simulated sampling distribution of X 5 the number of pennies from the 2000s and p^ 5 the proportion of pennies from the 2000s for samples of size 20 in Mr. Ramirez’s class. The distributions are exactly the same, other than the scale on the axis. d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d 6 8 10 12 14 16 18 20 Number of pennies from 2000s d d d d d d d d d d d d d d d d In this context, n 5 100 and p 5 0.43. The mean is m p^ 5 p p(1 − p) and the standard deviation is s p^ 5 . Å n FOR PRACTICE TRY EXERCISE 1. d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Proportion of pennies from 2000s Starnes_3e_CH06_398-449_Final.indd 426 Common Error Students may refer to the sampling distribution of the sample proportion of pennies as a binomial distribution. Tell them this is not technically correct because a binomial random variable is a count, not a proportion. However, these two distributions are mathematical transformations of one another. 18/08/16 5:01 PMStarnes_3e_CH0 426 C H A P T E R 6 • Sampling Distributions Starnes_3e_ATE_CH06_398-449_v3.indd 426 11/01/17 3:56 PM
L E S S O N 6.4 • The Sampling Distribution of a Sample Proportion 427 18/08/16 5:01 PMStarnes_3e_CH06_398-449_Final.indd 427 Also, the formulas for the mean and standard deviation of the sampling distribution of p^ are derived from the formulas we learned in the previous lesson: Shape s p^ 5 s X n m p^ 5 m X n = np n = p "np(1 − p) np(1 − p) p(1 − p) = = = n Å n 2 Å n Both the sample size and the proportion of successes in the population affect the shape of the sampling distribution of the sample proportion p^ . The following activity helps you explore the effect of these two factors. AcT iviT y Sampling from the candy machine Imagine a very large candy machine filled with orange, brown, and yellow candies. When you insert money, the machine dispenses a sample of candies. In this activity, you will use an applet to investigate the sample-to-sample variability in the proportion of orange candies dispensed by the machine. 1. Launch the Reese’s Pieces® applet at www.rossmanchance.com/applets. Click the button for “Proportion of orange” to have the applet calculate and record the value of p^ 5 the sample proportion of orange candies. 2. Click on the “Draw Samples” button. An animated simple random sample of n 5 25 candies should be dispensed. The screen shot shows the results of one such sample. Was your sample proportion of orange candies close to the actual population proportion, p 5 0.50? Teaching Tip: Differentiate Students who have difficulty with algebra can ignore the mathematical derivations of m p^ and s p^ at the top of this page. They are not important for understanding the big ideas in this and other lessons. 3. Click “Draw Samples” 9 more times, so that you have a total of 10 sample proportions. Look at the dotplot of your p^ values. Does the distribution have a recognizable shape? 4. To take many more samples quickly, enter 990 in the “number of samples” box. Click on the “ Animate” box to turn off the animation. Then click “Draw Samples.” You have now taken a total of 1000 samples of n 5 25 candies from the machine. Describe the shape of the simulated sampling distribution of p^ shown in the dotplot. 5. How does the shape of the sampling distribution of p^ change if the proportion of orange candies in the machine is p 5 0.10 instead of p 5 0.50? Set the probability of orange candies to p 5 0.10 and draw 1000 samples of size n 5 25. What if p 5 0.90? Describe how the value of p affects the shape of the sampling distribution of p^ . 6. How does the shape of the sampling distribution of p^ change if the sample size increases? Set the probability of orange to p 5 0.90 and the number of candies to n 5 25 and draw 1000 samples. Then, repeat with sample sizes of n 5 100 and n 5 500. Describe how the value of n affects the shape of the sampling distribution of p^ . Activity Overview Time: 15–18 minutes 18/08/16 5:01 PM Materials: An Internet-connected device for each student or group of students Teaching Advice: This activity helps students understand the shape of the sampling distribution of p^ . Because the sampling distribution of p^ is closely related to the sampling distribution of the sample count X, this activity can be skipped if your students understood Lesson 6.3 well. If you don’t have enough devices, students can work in groups or you can demonstrate the applet to the entire class. Showing the applet as a demonstration also saves time, even if it doesn’t engage students as much. This applet gives a visual of the population distribution (the candy machine), the distribution of one sample (the candies in the dish), and the sampling distribution (the dotplot). Point out these three distributions to your students. Beginning at Step 4, emphasize the difference between “number of candies” (sample size) and “number of samples.” Students will often be confused about the meaning of these terms. Make sure students are using the correct values. There is nothing special about 1000 samples. This activity would work just as well if students generated 10,000 samples. What matters is to repeat the sampling process often enough to see the pattern in the sampling distribution. The shape of the distribution of p^ follows the same rules as the shape of the distribution of the corresponding binomial random variable X. This should be evident from comparing the results of this activity to the results of the activity in Lesson 6.3. Answers: 1. Students should launch the applet and click “Proportion of orange.” 2. Student results will vary. Most students should get a sample proportion close to 0.50, but some students may not. 3. With only 10 dots on the dotplot, the distribution shouldn’t have a recognizable shape. 4. The distribution should look moundshaped and roughly symmetric. 5. When p 5 0.1, the shape of the distribution is skewed to the right. When p 5 0.9, it is skewed to the left. Values of p that are lower than 0.5 result in right-skewed distributions and values higher than 0.5 result in left-skewed distributions. 6. As n increases, the sampling distribution of p^ gets more approximately normal. This result holds true for all values of p. Teaching Tip The preceding activity helps students visualize sampling. This is a good time to remind them that the word “sample” does not refer to a single individual, but to a group of many individuals. Make sure your students don’t refer to individuals as “samples.” Lesson 6.4 L E S S O N 6.4 • The Sampling Distribution of a Sample Proportion 427 Starnes_3e_ATE_CH06_398-449_v3.indd 427 11/01/17 3:56 PM
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SPA 3e_ Teachers Edition _ Ch 6

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