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Chapter 7.1 Sampling Distributions Central Limit Theorem Name: _____________________________________________ Date: _______________ Period: _______ Sampling Distribution of Sample Means Activity Part 1: Description of Population: ________________________________________________ Population mean μ: _________ Population standard deviation σ: ____________ Sample size n: __________ Dotplot of years Shape of Distribution: _________________ Collecting the Data # of trials: 10 mean: ______________ st dev: ______________ # of trials: 25 mean: _______________ st dev: _______________ GROUP INFO for sampling distribution mean: ______________ st dev: ______________ mean: _______________ st dev: _______________ Dotplot of each – distribution of each CLASS INFO for sampling distribution mean: ______________ st dev: ______________ mean: _______________ st dev: _______________ Dotplot of each – distribution of each Analyzing the data 1. How does the shape of the distribution change as the number of means in the sampling distribution increases? 2. How does the mean of the sampling distribution change as the number of means increases? How does x compare to μ? 3. How does sx change as the number of means increases? How does sx 𝜎 compare to 𝑛 ? √ 4. What have you learned about the normality of the distribution for the sampling distribution of means? ***One of the MOST important and useful concepts in statistics is the Central Limit Theorem. It forms the foundation for estimating population parameters and hypothesis testing. Central Limit Theorem For any given population with mean μ and standard deviation σ, a sampling distribution of the sample means, with sample sizes of AT LEAST 30, will have the following characteristics. 1. The sampling distribution will approximate a normal distribution REGARDLESS of the shape of the original distribution. The larger the sample size the better the approximation. 2. The mean of the sampling distribution μx equals the mean of the population μ. 3. The standard deviation of the sampling distribution σx equals the standard deviation of the population σ divided by the square root of the sample size. Examples: 1. If the mean of a given sampling distribution is 85, what is the mean of the population with a sampling size of 100? 2. If the standard deviation of a given population is 9 and a sampling distribution is created from the population with a sampling size of 100, what is the standard deviation of the sampling distribution? 3. Some health reports claim that the average cold lasts 7 days If 120 samples of size 100 are taken from across the US, what would you expect the average of the sampling distribution to be? 4. The Federal Reserve Bank of New York conducted a study and claims that the inflation rates of American households had a population standard deviation of 0.2 percentage points in 1996. If a sampling distribution is created using samples of size 78, what would be the standard deviation of the sampling distribution? Chapter 7 Section 2 Central Limit Theorem with Population Means 1. Draw a picture that describes the information in the question using normal distribution. 2. Convert the values in the problems to standard scores using the sampling distribution mean and standard deviation. 3. Use the normal curve table and the z values to find the are under the curve. Examples 1. The body temperatures of adults are normally distributed with a mean of 98.6 and a standard deviation of 0.73. What is the probability of a sample of 36 adults having an average normal body temperature less than 98.3? 2. The body temperatures of adults are normally distributed with a mean of 98.6 and a standard deviation of 0.73. What is the probability of a sample of 40 adults having an average normal body temperature greater than 99? 3. In 2006, prices of women’s athletic shoes were normally distributed with a mean of $75.15 and a standard deviation of $17.89. What is the probability that the average price of a sample of 30 pairs of women’s athletic shoes will differ from the population mean by more than $3.00? 4. The walking gait of adult males is normally distributed with a mean of 2.4 feet and a standard deviation of 0.3 feet. A sample of 34 men’s strides is taken. a. Find the probability that an individual man’s stride is less than 2.1 feet? b. Find the probability that the mean of the sample taken is less than 2.1 feet? c. Find the probability that the mean of the sample taken is more than 2.1 feet? d. Find the probability that the sample mean differs from the population mean by more than 0.06 feet? 5. The distribution of the number of people in line at a grocery store check out has a mean of 3 and a variance of 9. A sample of 50 grocery lines is taken. a. Calculate the probability that the sample mean of the line length is more than 4. b. Calculate the probability that the sample mean of the line length is less then 2.5. c. Calculate the probability that the sample mean differs from the population mean by less than 0.5. 6. Intelligence is often cited as begin normally distributed with a mean of 100.0 and a standard deviation of 15.0. a. What is the probability of a random person on the street having an intelligence level less than 95? b. If a random sample of 50 people is taken, what is the probability that their mean intelligence level will be less than 95? c. If a random sample of 50 people is taken, what is the probability that their mean intelligence level will differ from the mean intelligent by more than 5? 7. A tea bag manufacturer needs to place 2 g of tea in each bag. If the machinery places an average of 2.6 g of tea in each bag with a standard deviation of 0.3 g, what is the probability that a randomly chosen bag will have between 2.0 and 2.8 g of tea? 8. A medical journal lists the average fetal heart rate of 140 beats per minutes with a standard deviation of 12 bpm. In a sample of 200, what is the probability that a fetal heart rate differs from the mean by more than 25 bpm? 9. The average wait time in a drive-thru chain is 193.2 seconds with a standard deviation of 29.5 seconds. What is the probability that in a random sample of 45 wait times, the mean is between 185.7 and 206.5 seconds? Chapter 7 Section 3 Population Proportions and the Central Limit Theorem A _______________________________________________ is the percentage of the population that has a certain characteristic. The _____________________________________________ is the percentage of the sample that has that certain characteristic. We denote sample proportions with decimal places. which is read “p hat” and is typically 2 In order to use the Central Limit Theorem for sample proportions, we must make sure that the following conditions have been met. If so, then the sampling distribution of sample proportions can be assumed to be normal, thus allowing us to use the normal distribution and z scores to calculate probability for population parameters. The formula for z scores for population parameters is (not in the calculator!!) Examples: 1. In a certain conservative precinct, 79% of the voters are registered Republicans. What is the probability that in a sample of 100 voters from this precinct, more than 68 of the voters would vote for the Republican Candidate? 2. In another precinct across town, the population is very different. In this precinct 81% of the voters are registered Democrats. What is the probability that, in a sample of 100 voters from this precinct, less than 80 of the voters would vote for the Democratic Candidate? 3. It is estimated that 7% of all Americans have diabetes. Suppose that a sample of 74 Americans is taken. What is the probability that the proportion of the sample that is diabetic differs from the population proportion by less than 1%? 4. It is estimated that 7% of all Americans have diabetes. Suppose that a sample of 74 Americans is taken. What is the probability that the proportion of the sample that is diabetic differs from the population proportion by more than 2%? 5. A large car dealership claims that 47% of their customers are looking to buy a sport utility vehicle (SUV). A sample of 61 customers is surveyed. What is the probability that les than 40% are looking to buy an SUV? 6. The local nursery is waiting for its spring annuals to be delivered, and 20% of the plants ordered are petunias. If the first truck contains 120 plants packed at random, what is the probability that no more than 30 of the plants are petunias? 7. At one private college, 34% of students are business majors. Suppose that 260 students are randomly selected from a list in the registrar’s office. What is the probability that the proportion of business students in the sample differs from the population proportion by less than 2%? 8. At a large grocery store, 72% of shoppers are women. In order to obtain information about spending habits, 40 shoppers are randomly chosen for a survey. What is the probability that the proportion of women in the sample differs from the mean by more than 3%? Chapter 7 Section 4 Approximating the Binomial Distribution Using the Normal Distribution If the conditions np≥ 5 and n(1 – p) ≥ 5 are met for a given binomial distribution, then a normal distribution can be used to approximate its probability distribution with a given mean and a standard deviation. μ = np σ = √𝑛𝑝( 1 − 𝑝) So when we need to calculate a probability for a large value of a binomial random variable, we do not need to use the binomial formula (calculators cannot calculate this for large values of n). Instead we use the normal distribution. We are trying to use the normal distribution, which is ________________________ to approximate the binomial distribution which is _____________________. In order to do this a CONTINUITY CORRECTION must be used to convert the whole number value of the discrete binomial random variable to an interval range of the continuous normal random variable by using x + 0.5 and x – 0.5. So…. If the value includes the number, it includes the interval. If it doesn’t include the number, then it doesn’t include the interval. Examples: 1. Use the continuity correction factor to describe the are under the normal curve that approximates the probability that at least 2 people in a statistics class of 50 cheated on the last test. Assume that the number of people who cheated is binomial distribution with a mean of 5 and a standard deviation of 2.12. 2. Use the continuity correction factor to describe the are under the normal curve that approximates the probability that less than 5 of the sitcoms playing on TV tonight are reruns. Assume that the number of reruns on TV tonight is a binomial distribution with a mean of 7 and a standard deviation of 2.16. 3. Calculate the probability of more than 55 girls being born in 100 births. Assume that the probability of a girl born in an individual birth is 50%. 4. After many hours of studying for your statistics test, you believe that you have a 90% probability of answering any given question correctly. Your test includes 50 true/false questions. What is the probability that you will miss no more than 4 questions? 5. Many toothpaste commercials advertise that 3 out of 4 dentists recommend their brand of toothpaste. What is the probability that out of a random survey of 400 dentists, 300 will have recommended Brand X toothpaste? Assume that the commercials are correct, and therefore, there is a 75% chance that any given dentist will recommend Brand X. 6. What is the probability that more than 150 out of the 230 eighth graders at a local middle school have been exposed to drugs? Assume that a previous study at this school reported that the probability of an individual eighth grade student having been exposes to drugs is 63%. 7. What is the probability that more than 100 out of 300 elections will contain voter fraud? One report suggests that there is a 32% chance of an individual elections containing voter fraud. 8. What is the probability that more than 20 out of a class of 347 high school seniors will drive under the influence of alcohol on prom night. The local chapter of MADD fears that the probability of a high school senior drinking and driving on Prom night is 38%. 9. What is the probability that at least 67 out of 100 cars stopped at a roadblock will not be given a ticket? Local authorities report that tickets usually are given to 23% of cars stopped. 10. What is the probability that no more than 130 out of 2300 tax returns filed at a local CPA’s office will be inaccurate? Previous records indicate only a 7% probability that a given tax return from this office in inaccurate?