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Name Class 8-5 Date Sampling Distributions Extension: Confidence Intervals and Margins of Error Essential question: How do you calculate a confidence interval and a margin of error for a population proportion or mean? CC.9–12.S.IC.4 1 Video Tutor EXPLORE Developing a Sampling Distribution The table provides data about the first 50 people to join a new gym. For each person, the table lists his or her member ID number, age, and sex. ID Age Sex ID Age Sex ID Age Sex 1 30 M 11 38 F 21 74 F 2 48 M 12 24 M 22 21 M 3 52 M 13 48 F 23 29 F 4 25 F 14 45 M 24 48 M 5 63 F 15 28 F 25 37 6 50 F 16 39 M 26 52 7 18 F 17 37 F 27 8 28 F 18 63 F 28 9 72 M 19 20 M 29 10 25 F 20 81 F 30 ID Age Sex ID Age Sex 31 32 M 32 28 F 41 46 M 42 34 F 33 35 M 43 44 F 34 49 M 44 68 M M 35 18 M 45 24 F F 36 56 F 46 34 F 25 F 37 48 F 47 55 F 44 M 38 38 F 48 39 M 29 F 39 52 F 49 40 F 66 M 40 33 F 50 30 F A Use your calculator to find the mean age μ and standard deviation σ for the population of the gym’s first 50 members. Round to the nearest tenth. μ= ;σ= B Use your calculator’s random number generator to choose a sample of 5 gym __ members. Find the mean age x for your sample. Round to the nearest tenth. __ x = C Report your sample mean to your teacher. As other students report their sample means, create a class histogram below. To do so, shade a square above the appropriate interval as each sample mean is reported. For sample means that lie on an interval boundary (such as 39.5), shade a square on the interval to the right (39.5 to 40.5). 8 Frequency © Houghton Mifflin Harcourt Publishing Company 6 4 2 25.5 27.5 29.5 31.5 33.5 35.5 37.5 39.5 41.5 43.5 45.5 47.5 49.5 51.5 53.5 55.5 26.5 28.5 30.5 32.5 34.5 36.5 38.5 40.5 42.5 44.5 46.5 48.5 50.5 52.5 54.5 Sample Mean D Calculate the mean of the sample means μ__x and the standard deviation of the sample means σ_x . μ__x = Chapter 8 ; σ_x = 459 Lesson 5 E Now use your calculator’s random number generator to choose a sample of 15 gym __ members. Find the mean x for your sample. Round to the nearest tenth. __ x = F Report your sample mean to your teacher and make a class histogram below. Frequency 8 6 4 2 25.5 27.5 29.5 31.5 33.5 35.5 37.5 39.5 41.5 43.5 45.5 47.5 49.5 51.5 53.5 55.5 26.5 28.5 30.5 32.5 34.5 36.5 38.5 40.5 42.5 44.5 46.5 48.5 50.5 52.5 54.5 Sample Mean G Calculate the mean of the sample means μ__x and the standard deviation of the sample means σ_x . μ__x = ; σ_x = REFLECT 1a. In the class histograms, how does the mean of the sample means compare with the population mean? 1b. What happens to the standard deviation of the sample means as the sample size increases? © Houghton Mifflin Harcourt Publishing Company 1c. What happens to the shape of the histogram as the sample size increases? The histograms that you made are sampling distributions. A sampling distribution shows how a particular statistic varies across all samples of n individuals from the same __ population. You have worked with the sampling distribution of the sample mean, x . The mean of the sampling distribution of the sample mean is denoted μ__x . The standard deviation of the sampling distribution of the sample mean is denoted σ_x and is also called the standard error of the mean. __ You may have discovered that μ__x is close to x regardless of the sample size and that σ_x decreases as the sample size n increases. These observations were based on simulations. When you consider all possible samples of n individuals, you arrive at one of the major theorems of statistics. Chapter 8 460 Lesson 5 Properties of the Sampling Distribution of the Sample Mean If a random sample of size n is selected from a population with mean μ and standard deviation σ, then (1) μ__x = μ, (2) σ_x = ____ √σ , and n (3) the sampling distribution of the sample mean is normal if the population is normal; for all other populations, the sampling distribution of the mean approaches a normal distribution as n increases. The third property stated above is known as the Central Limit Theorem. All normal distributions have the following properties, sometimes collectively called the 68-95-99.7 rule: • 68% of the data fall within 1 standard deviation of the mean. • 95% of the data fall within 2 standard deviation of the mean. • 99.7% of the data fall within 3 standard deviation of the mean. You will learn more about the specific properties of normal distributions later in this chapter. CC.9–12.S.IC.4 2 EXAMPLE Using the Sampling Distribution of the Sample Mean Boxes of Cruncho cereal have a mean mass of 323 g with a standard deviation of 20 g. You choose a random sample of 36 boxes of the cereal. What interval captures 95% of the means for random samples of 36 boxes? • Write the given information about the population and the sample. © Houghton Mifflin Harcourt Publishing Company μ= σ= n= • Find the mean of the sampling distribution of the sample mean and the standard error of the mean. μ__x = μ = The sampling distribution of the sample mean is approximately normal. In a normal distribution, 95% of the data fall within 2 standard deviations of the mean. μ__x - 2σ__x = μ__x + 2σ__x = σ__x = ____ √σ = _______ ≈ n ( + 2( - 2 ) )= = So, 95% of the sample means fall between Chapter 8 g and 461 g. Lesson 5 REFLECT 2a. When you choose a sample of 36 boxes, is it possible for the sample to have a mean mass of 315 g? Is it likely? Explain. CC.9–12.S.IC.4 3 explore Developing Another Sampling Distribution Use the table of data from the first Explore. This time you will develop a sampling distribution based on a sample proportion rather than a sample mean. A Find the proportion p of gym members in the population who are female. p= B Use your calculator’s random number generator to choose a sample of 5 gym members. Find the proportion of female members p̂ for your sample. p̂ = C Report your sample proportion to your teacher. As D Calculate the mean of the sample proportions μp̂ and the standard deviation of the sample proportions σp̂ . Round to the nearest hundredth. 8 6 4 2 -0.05 0.15 0.35 0.55 0.75 0.95 0.05 0.25 0.45 0.65 0.85 1.05 ; σp̂ = Sample Proportion E Now use your calculator’s random number generator to choose a sample of 10 gym members. Find the proportion of female members p̂ for your sample. p̂ = 10 F Report your sample proportion to your teacher. As other G Calculate the mean of the sample proportions μp̂ and the standard deviation of the sample proportions σp̂ . Round to the nearest hundredth. 8 Frequency students report their sample proportions, create a class histogram at right. 6 4 2 -0.05 0.15 0.35 0.55 0.75 0.95 0.05 0.25 0.45 0.65 0.85 1.05 μp̂= Chapter 8 Sample Proportion ; σp̂ = 462 Lesson 5 © Houghton Mifflin Harcourt Publishing Company μp̂= 10 Frequency other students report their sample proportions, create a class histogram at right. REFLECT 3a. In the class histograms, how does the mean of the sample proportions compare with the population proportion? 3b. What happens to the standard deviation of the sample proportions as the sample size increases? When you work with the sampling distribution of a sample proportion, p represents the proportion of individuals in the population that have a particular characteristic (that is, the proportion of “successes”) and p̂ is the proportion of successes in a sample. The mean of the sampling distribution of the sample proportion is denoted μp̂. The standard deviation of the sampling distribution of the sample proportion is denoted σp̂ and is also called the standard error of the proportion. Properties of the Sampling Distribution of the Sample Proportion If a random sample of size n is selected from a population with proportion of successes p, then (1) μp̂ = p, √ p(1 - p) (2) σp̂ = ________ , and n © Houghton Mifflin Harcourt Publishing Company (3) if both np and n(1 - p) are at least 10, then the sampling distribution of the sample proportion is approximately normal. CC.9–12.S.IC.4 4 EXAMPLE Using the Sampling Distribution of the Sample Proportion About 40% of the students at a university live off campus. You choose a random sample of 50 students. What interval captures 95% of the proportions for random samples of 50 students? A Write the given information about the population and the sample, where a success is a student who lives off campus. p= n= B Find the mean of the sampling distribution of the sample proportion and the standard error of the proportion. μp̂ = p = Chapter 8 √ ( ) 1 p(1 p) ___________________ σp̂= ________ ≈ = n √ 463 Lesson 5 C Check that np and n(1 - p) are both at least 10. np = · = n(1 - p) = · = Since np and n(1 - p) are both greater than 10, the sampling distribution is approximately normal. D In a normal distribution, 95% of the data fall within 2 standard deviations of the mean. μp̂- 2σp̂= μp̂+ 2σp̂= ( + 2( ) )= - 2 = So, 95% of the sample proportions fall between and . REFLECT 4a. How likely is it that a random sample of 50 students includes 31 students who live off campus? Explain. Previously, you investigated sampling from a population whose parameter of interest (mean or proportion) is known. In many real-world situations, you collect sample data from a population whose parameter of interest is not known. Now you will learn how to use sample statistics to make inferences about population parameters. CC.9–12.S.IC.4 5 explore Analyzing Likely Population Proportions A © Houghton Mifflin Harcourt Publishing Company You survey a random sample of 50 students at a large high school and find that 30% of the students have attended a school football game. You cannot survey the entire population of students, but you would like to know what population proportions are reasonably likely in this situation. Suppose the proportion p of the population that has attended a school football game is 30%. Find the reasonably likely values of the sample proportion p̂ . In this case, p = μp̂ = p = and n = . √ ( ) 1 p(1 - p) ________ and σp̂ = n = ____________________ ≈ √ The reasonably likely values of p̂ fall within 2 standard deviations of μp̂ . μp̂ - 2σp̂ = μp̂ + 2σp̂ = Chapter 8 ( ( - 2 +2 ) )= = 464 Lesson 5 segment at the level of 0.3 on the vertical axis to represent the interval of likely values of p̂ that you found above. C Now repeat the process for p = 0.35, 0.4, 0.45, and so on to complete the graph. You may wish to divide up the work with other students and pool your findings. D 1 On the graph, draw a horizontal line Draw a vertical line at 0.4 on the horizontal axis. This represents p̂ = 0.4. The line segments that this vertical line intersects are the population proportions for which a sample proportion of 0.4 is reasonably likely. 0.9 Proportion of Successes in Population, p B 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Proportion of Successes in Sample, p̂ REFLECT 5a. Is it possible that 30% of all students at the school have attended a football game? Is it likely? Explain. © Houghton Mifflin Harcourt Publishing Company 5b. Is it possible that 60% of all students at the school have attended a football game? Is it likely? Explain. 5c. Based on your graph, which population proportions do you think are reasonably likely? Why? A confidence interval is an approximate range of values that is likely to include an unknown population parameter. The level or degree of a confidence interval, such as 95%, gives the probability that the interval includes the true value of the parameter. Recall that when data are normally distributed, 95% of the values fall within 2 standard deviations of the mean. Using this idea in the Explore, you found a 95% confidence interval for the proportion of all students who have attended a school football game. Chapter 8 465 Lesson 5 The above argument shows that you can find the endpoints of the confidence interval by finding the endpoints of the horizontal segment centered at p̂ . You know how to do this using the formula for the standard error of the sampling distribution of the sample proportion from earlier in this lesson. Putting these ideas together gives the following result. 1 0.9 Proportion of Successes in Population, p To develop a formula for a confidence interval, notice that the vertical bold line segment in the figure, which represents the 95% confidence interval you found in the Explore, is about the same length as the horizontal bold line segment. The horizontal bold line segment has endpoints μp̂ - 2σp̂ and μp̂ + 2σp̂ where p̂ = 0.4. Since the bold line segments intersect at (0.4, 0.4), the vertical bold line segment has these same endpoints. 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Proportion of Successes in Sample, p̂ A Confidence Interval for a Population Proportion A c% confidence interval for the proportion p of successes in a population is given by p̂ (1 - p̂ ) p̂ (1 - p̂ ) p̂ - z c _________ ≤ p ≤ p̂ + zc _________ n n √ √ where p̂ is the sample proportion, n is the sample size, and zcdepends upon the desired degree of confidence. In order for this interval to describe the value of p reasonably accurately, three conditions must be met. © Houghton Mifflin Harcourt Publishing Company 1.There are only two possible outcomes associated with the parameter of interest. The population proportion for one outcome is p, and the proportion for the other outcome is 1 - p. 2. np̂ and n(1 - p̂ ) must both be at least 10. 3.The size of the population must be at least 10 times the size of the sample, and the sample must be random. Use the values in the table below for zc . (Note that for greater accuracy you should use 1.96 rather than 2 for z95% .) Desired degree of confidence 90% 95% 99% Value of z c 1.645 1.96 2.576 Chapter 8 466 Lesson 5 CC.9–12.S.IC.4 6 EXAMPLE Finding a Confidence Interval for a Proportion In a random sample of 100 four-year-old children in the United States, 76 were able to write their name. Find a 95% confidence interval for the proportion p of four-year-olds in the United States who can write their name. A Determine the sample size n, the proportion p̂ of four-year-olds in the sample who can write their name, and the value of zc for a 95% confidence interval. n = B p̂ = zc= Substitute the values of n, p̂, and zcinto the formulas for the endpoints of the confidence interval. Then simplify and round to two decimal places. √ √ √ ( ) ( ) 1 ____________________ ≈ p̂(1 - p̂) p̂ - zc ________ = n p̂(1 - p̂) p̂ + zc ________ = n So, you can state with 95% confidence that the proportion of all four-year-olds in the United States who can write their name lies between √ - + 1 ____________________ ≈ and . REFLECT © Houghton Mifflin Harcourt Publishing Company 6a. Find the 99% confidence interval for p and describe how increasing the degree of confidence affects the range of values. Why does this make sense? You can use reasoning similar to the argument in the Explore to develop a formula for a confidence interval for a population mean. A Confidence Interval for a Population Mean A c% confidence interval for the mean μ in a normally distributed population is given by __ σ ≤ μ ≤ __x σ x - z c ____ + zc ____ √ n √ n __ where x is the sample mean, n is the sample size, σ is the population standard deviation, and zc depends upon the desired degree of confidence. Note that it is assumed that the population is normally distributed and that you know the population standard deviation σ. In a more advanced statistics course, you can develop a confidence interval formula that does not depend upon a normally distributed population or knowing the population standard deviation. Chapter 8 467 Lesson 5 CC.9–12.S.IC.4 7 EXAMPLE Finding a Confidence Interval for a Mean In a random sample of 20 students at a large high school, the mean score on a standardized test is 610. Given that the standard deviation of all scores at the school is 120, find a 99% confidence interval for the mean score among all students at the school. __ A Determine the sample size n, the sample mean x , the population standard deviation σ, and the value of zc for a 99% confidence interval. __ n= σ= zc= B x = __ Substitute the values of n, x , σ, and zc into the formulas for the endpoints of the confidence interval. Then simplify and round to the nearest whole number. __ σ = x - zc ____ √ n __ σ = x + zc ____ √ n - __________ ≈ √ + __________ ≈ √ So, you can state with 99% confidence that the mean score among all students at the school lies between and . REFLECT 7a. What is the 99% confidence interval when the sample size increases to 50? Describe how increasing the sample size affects the confidence interval. © Houghton Mifflin Harcourt Publishing Company 7b. What do you assume about the test scores of all students at the school in order to use the formula for the confidence interval? In the previous example, you found the 99% confidence interval 541 ≤ µ ≤ 679, which is a range of values centered at µ = 610. You can write the confidence interval as 610 ± 69, where 69 is the margin of error. The margin of error is half the length of a confidence interval. Chapter 8 468 Lesson 5 Margin of Error for a Population Proportion The margin of error E for the proportion of successes in a population with sample proportion p̂ and sample size n is given by √ p̂ (1 - p̂ ) E = z c ________ n where zc depends on the degree of the confidence interval. Margin of Error for a Population Mean The margin of error E for the mean in a normally distributed population __ with standard deviation σ, sample mean x , and sample size n is given by σ E = zc ____ √ n where zc depends on the degree of the confidence interval. From the above formulas, it is clear that the margin of error decreases as the sample size n increases. This suggests using a sample that is as large as possible; however, it is often more practical to determine a margin of error that is acceptable and then calculate the required sample size. pra c t i c e Bags of SnackTime Popcorn have a mean mass of 15 ounces with a standard deviation of 1.5 ounces. A quality control inspector selects a random sample of 40 bags of popcorn at the factory. Find each of the following. 1. What is the population mean? © Houghton Mifflin Harcourt Publishing Company 2. What is the sample mean? 3. What is the sample standard deviation? 4. The interval that captures 95% of the proportions for random samples of 40 bags of popcorn 5. In a random sample of 100 U.S. households, 37 had a pet dog. a. Do the data satisfy the three conditions for the confidence interval formula for a population proportion? Why or why not? b. Find a 90% confidence interval for the proportion p of U.S. households that have a pet dog. Chapter 8 469 Lesson 5 c. Find a 95% confidence interval for the proportion p of U.S. households that have a pet dog. 6. In a quality control study, 200 cars made by a particular company were randomly selected and 13 were found to have defects in the electrical system. a. Give a range for the percent p of all cars made by the company that have defective electrical systems, assuming you want to have a 90% degree of confidence. b. What is the margin or error? c. How does the margin of error change if you want to report the range of percents p with a 95% degree of confidence? 7. The mean annual salary for a random sample of 300 kindergarten through 12th grade teachers in a particular state is $50,500. The standard deviation among the state’s entire population of teachers is $3,700. Find a 95% confidence interval for the mean annual salary μ for all kindergarten through 12th grade teachers in the state. 8. You survey a random sample of 90 students at a university whose students’ grade- point averages (GPAs) have a standard deviation of 0.4. The surveyed students have a mean GPA of 3.1. © Houghton Mifflin Harcourt Publishing Company a. How likely is it that the mean GPA among all students at the university is 3.25? Explain. b. What are you assuming about the GPAs of all students at the university? Why? 9. The margin of error E for the proportion of successes in a population may be estimated by ____ √1 where n is the sample size. Explain where this estimate comes n from. (Hint: Assume a 95% confidence interval.) Chapter 8 470 Lesson 5 8-5 Name ________________________________________ Class __________________ Name Class Date __________________ Date Chapter Practice Additional Practice Sampling Distributions 2 For Exercises 1–3, use the following information: The mean height of the population of male high school students in a school district is 67.2 inches with a standard deviation of 2.8 inches. 1. Random samples are drawn from the population. What is the mean of the sampling distribution of the sample mean? 2. What is the standard error of the mean for a random sample of 25 students? for a random sample of 64 students? ____________________________ _____________________________ 3. What interval captures 95% of the sample means for a sample size of 64 students? ________________________________________________________________________ For Exercises 4 and 5, use the following information: From a random sample of 30 customers who bought a soft-serve frozen yogurt cone, 26% chose chocolate-vanilla swirl. 4. Find npˆ and n ( 1 − pˆ ) where n is the sample size and p̂ is the sample proportion. ________________________________________________________________________ © Houghton Mifflin Harcourt Publishing Company 5. Is it reasonable to use this sample to construct a confidence interval for the population proportion of customers who would choose chocolate-vanilla swirl? Explain. ________________________________________________________________________ 6. In a random sample of 44 students from a school of 1290 students, 60% chose red and blue for the new school colors over red and white. Find a 90% confidence interval for the proportion p of students at the school who prefer red and blue. Can you conclude with 90% confidence that the majority of students prefer red and blue? ________________________________________________________________________ 7. In a random sample of 550 students in a state, the mean math score on a standardized test was 22.6. Find a 95% confidence interval for the population mean score μ for the state given that the standard deviation is 4.8. ________________________________________________________________________ Find the margin of error E for the statistic at the given confidence level. 8. population proportion, if sample proportion p̂ = 0.11 and sample size n = 124; 99% ________ 9. population mean, if standard deviation σ = 6.8 and sample size n = 1000; 95% ________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. Chapter 8 471 49 Lesson 5 Holt McDougal Algebra 1 Name ________________________________________ Date __________________ Class __________________ Problem Solving Problem Solving Chapter Sampling Distributions 2 Out of a random sample of n = 40 students from a high school with more than 2000 students, p̂ = 58% say that their morning travel time to school is less than 30 minutes. 1. Can you use the data to construct a reasonably accurate confidence interval for the proportion of students with a morning travel time of less than 30 minutes? Explain. _________________________________________________________________________________________ _________________________________________________________________________________________ 2. Can you conclude that the morning travel time for the majority of the students is less than 30 minutes at the 90%, 95%, or 99% confidence level? Explain. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 3. If you increase the sample size to n = 125 students and obtain the same sample proportion of p̂ = 0.58, does this change your answer to Exercise 2? Explain. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ For Exercises 4 and 5, a random sample is drawn from a population with a normally distributed statistic with mean µ and standard deviation σ = 20. 4. What is the margin of error E for estimating μ from the sample mean at the 95% confidence level for a sample size of n = 100? ____________ 5. How can you use the margin of error formula to find the minimum sample size so that E ≤ 2.0 at the 95% confidence level? What is this sample size? ________________________________________________________________________ 6. What is the relationship between the margin of error E for a population proportion when the sample proportion is p̂ = 0.2 and when the sample proportion is p̂ = 0.5 for a given confidence level and sample size? _________________________________________________________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. Chapter 8 472 50 Lesson 5 Holt McDougal Algebra 1 © Houghton Mifflin Harcourt Publishing Company ________________________________________________________________________