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Problem Set Section 5.1 Continuous Random Variables 1. State whether the variable is discrete or continuous. a. The number of cups of coffee sold in a cafeteria during lunch. b. The height of a basketball player. c. The cost of a Statistics textbook. d. The blood pressure of a group of students before a Statistics test. e. The number of goals scored in a soccer game. 2. Write the following in symbolic form using x as the random variable. These are statements for continuous random variables. a. Between four and eight b. more than four c. fewer than four d. at least four e. at most four f. exactly four 3. Suppose the class length follows a uniform distribution and is between 50 minutes and 52 minutes. Find the following probabilities. a. Find the probability density function. b. Sketch the probability density function. 4. Using the probability density function in number 3, find the following probabilities. a. Greater than 50.5 b. Between 50.5 and 52 c. Less than 52 d. Greater than 52 5. Suppose you are trying to find a short in a 4m length of wire. Supposing you have no reason to suspect one part of the wire over another. a. Find the probability density function for this uniform random variable. b. What is the probability you find the short in the last half-meter of the wire? 1 Problem Set Section 5.2 Standard Normal Distribution 1. Using your calculator NormCdf function, determine these probabilities of a normal distribution in the scenario where the mean μ = 0 and the standard deviation σ = 1. This distribution has already been normalized. Use symmetry and complements if needed. a. P(-1 < z < 1) b. P(0 < z < 1) c. P(-1 < z < 0) Notice anything consistent about the above three? d. Less than .25 e. greater than 1.96 f. less than -2.33 g. less than -.25 h. between .50 and 1.50 i. between -2 and -1 j. between -1.05 and -.52 k. Between -2.67 and 1.50 l. between -1 and 2 m. between 0 and 1.50 n. P(-1.96 < z < 0) o. P(-2 < z < 2) p. P(z > -2.575) q. P(z < -1.645) r. P(1.96 < z < 2.33) 2. Answer the following questions regarding a standard normal distribution. a. About how much area is between z = -1 and z = 1? b. About how much area is between z = -2 and z = 2? c. About how much area is between z = -3 and z = 3? d. About how much area is between z = -3.5 and z = 3.5? e. About how much area is within 1 standard deviation of the mean? f. About how much area is with 1.96 standard deviations of the mean? g. About how much area is between 1 and 3 standard deviations of the mean? 3. Find the z score for the following percentiles from a standard normal distribution. a. P90 b. P20 c. P5 4. Using your calculator invNorm function, find the z scores associated with the following probability statements. Include a sketch. a. P(0 < z < a) = .2907 Find a b. P(-b < z < b) = .8664 Find b c. P(z > c) = .0656 Find c d. P(z > d) = .9802 Find d e. P(z < e) = .3600 Find e 2 Problem Set Section 5.3 Applications of Normal Distributions 1. If IQ is normally distributed with a mean of 100 and a standard deviation of 15. a. Calculate the probability of finding a genius (IQ > 130). b. Calculate the probability of average intelligence (between 90 and 110). c. Calculate the probability of above intelligence (between 110 and 120). d. Calculate the probability of IQ’s being less than 90. e. Calculate the 60th percentile of IQ. f. Calculate the 99th percentile of IQ. g. Find the IQ score separating the top 10% from the others. h. Find the IQ score separating the bottom 20% from the others. 2. Soda Machine: The amount of soda a filling machine pours into 12 ounce cans is normally distributed with a mean of 12 ounces and standard deviation of 0.15 ounces. What is the probability that a randomly selected can contains between 11.97 and 11.99 ounces of soda? 3. Weight of Males: If the weight of males is normally distributed with μ=150 and σ=10, what is the probability that a randomly selected male will weight between 140 lbs and 155 lbs? 4. SAT Scores: SAT scores have a normal distribution with a mean of 1020 and a standard deviation of 153. a. If a college required a minimum score of 1200, what percentage of students would satisfy the requirement? b. If 2000 people applied to the college, how many would you expect to meet the requirement? c. What score would a student need to attain if the requirement was changes so that a score in the top 30% qualified for entry. 5. GPA: The GPA of college students is normally distributed with μ=2.70 and σ=0.25. a. What proportion of students have a GPA between 2.40 and 2.50? b. Calculate the 97.5th percentile. c. 10% of students have a score above _________? [90th percentile] 6. Heights of Girls: The height of a 5 year old girl is normally distributed with mean of 106 cm and a standard deviation of 5 cm. a. What is the probability that a randomly selected 5 year old girl has a height greater than 110 cm? b. What is the probability for a height greater than 120 cm? c. Calculate the 10th percentile for height. d. Calculate the 3rd percentile for height. 3 7. Salaries: Suppose that the average salary of college graduates is normally distributed with μ=$40,000 and σ=$10,000. a. What proportion of college graduates will earn less than $24,800? b. What proportion of college graduates will earn more than $53,500? c. What proportion of college graduates will earn between $45,000 and $57,000? d. Calculate the 80th percentile. e. Calculate the 27th percentile. 8. Serum Albumin: Serum albumin in healthy 20 year olds is normally distributed with a mean of 4.4 and a standard deviation of 0.2. a. What is the probability that a healthy 20 year old patient will have a serum albumin of 3.8 or less? b. A "reference range" is defined as the 2.5th to the 97.5th percentiles. Calculate a reference range for serum albumin in healthy 20 year olds. 9. Warranty Period: The life of a calculator has a normal distribution with a mean of 54 months and a standard deviation of 8 months. a. Find the probability that a randomly selected calculator will be replaced in less than 40 months. b. What should the warranty period be to replace a malfunctioning calculator if the company does not want more than 1% of all calculators sold to be replaced? 10. Length of Pregnancy: An expert witness in a paternity suit testifies that the length (in days) of pregnancy (that is, the time from impregnation to the delivery of the child) is approximately normally distributed with parameters μ = 270 and σ = 10. The defendant in the suit is able to prove that he was out of the country during a period that began 290 days before the birth of the child and ended 240 days before the birth. If the defendant was, in fact, the father of the child, what is the probability that the mother could have had the very long or very short pregnancy indicated by the testimony? 4 Problem Set Section 5.4 Determining Normality 1. Using technology, construct a Normal Probability Plot for the white blood cell count in the BLOOD dataset. Do the data appear to come from a normal population? Explain. 5 Problem Set Section 5.5 Central Limit Theorem 1. If the weight of males is normally distributed with μ=150 and σ=35, a. If one man is randomly selected, what is the probability that his weight is less than 140 lbs? b. If 100 men are randomly selected is the probability that their mean weight is less than 140 lbs? 2. The amount of soda a filling machine pours into 12 ounce cans is normally distributed with a mean of 12 ounces and standard deviation of 0.15 ounces. a. If one can is randomly selected, what is the probability that it contains between 11.97 and 11.99 ounces of soda? b. If 81 cans are randomly selected, what is the probability that their mean contains between 11.97 and 11.99 ounces of soda? c. If 16 cans are randomly selected, what is the probability that their mean contains between 11.97 and 11.99 ounces of soda? d. Why can the central limit theorem still be used in part c) even though the sample size does not exceed 30? 3. Breaking Strength of Plastic Bags: Plastic bags used for packaging produce are manufactured so that the breaking strength of the bag is normally distributed with a mean of 5 pounds per square inch and a standard deviation of 1.5 pounds per square inch. A sample of 100 bags is selected. a. What is the probability that the average breaking strength is between 5 and 5.1 pounds per square inch? b. What is the probability that the average breaking strength is between 4.8 and 4.9 pounds per square inch? c. What is the probability that the average breaking strength is less than 4.8 pounds per square inch? d. Between what two values symmetrically distributed around the mean will 95% of the average breaking strengths be? 6 4. Gas Mileage of Hybrid Cars: Introduced in the 2000 model year, the Honda Insight was the first hybrid automobile sold in the United States. The mean gas mileage for the model year 2005 Insight with an automatic transmission was 56 miles per gallon on the highway with a standard deviation of 3.2 miles per gallon. Suppose fifty 2005 Honda Insights with automatic transmissions are randomly selected. a. Why is the sampling distribution of mean approximately normal? b. What is the mean and standard deviation of the sampling distribution of the mean assuming µ=56 and σ=3.2? c. What proportion of 2005 Honda Insights with automatic transmission get over 57 miles per gallon on the highway? d. What proportion of 2005 Honda Insights with automatic transmission gets less than 55 miles per gallon on the highway? e. What proportion of 2005 Honda Insights with automatic transmission will get between 55 and 57 miles per gallon on the highway? 5. Rates of Return in the S&P 500: The S&P 500 is a collection of the largest 500 publically traded companies. The rates of return of the S&P 500 since 1950 are normally distributed with a mean of 0.007233 (0.7233%) and standard deviation of 0.04135 (4.135%). a. What is the probability that a randomly selected stock will have a positive rate of return? That is, what is the P(x>0)? b. Treating the next 12 months as a random sample, what is the probability that the mean rate of return will be positive? c. Treating the next 24 months as a random sample, what is the probability that the mean rate of return will be positive? d. Treating the next 36 months as a random sample, what is the probability that the mean rate of return will be positive? e. Describe the likelihood of a positive rate of return as the investment time increases. 7 6. Time Spent in the Fast Food Drive-Through: The quality-control manager of a Long John Silver’s restaurant wishes to analyze the length of time a car spends at the drivethrough window waiting for an order. It is determined that the mean time spent at the window in 59.3 seconds with a standard deviation of 12.1 seconds. The distribution of time at the window is skewed right. a. To obtain probabilities regarding a sample mean using the central limit theorem, what size sample is required? b. A random sample of 40 cars results in a sample mean time spent at the window of 56.8 seconds. What is the probability of obtaining a sample mean of 56.8 or less? c. Based on the results from b), has the length of time spent at the drive through window been significantly reduced? Why or why not? 7. Home Prices: A study of 800 homeowners in a certain zip code showed that the average value of the home is $250,000 and the standard deviation is $20,000. Fifty homes are currently for sale in this zip code. a. Why is the sampling distribution of the mean approximately normal? b. What is the mean and standard deviation of the sampling distribution of the mean if µ=$250,000 and σ=$20,000? c. What is the probability that the mean value of the homes for sale is less than $247,000? Is this unusual? 8. Oil Change: The time required to get an oil change at a 10-minute oil-change facility has a mean of 11.4 minutes and a standard deviation of 3.2 minutes. The shape of the distribution of the time is unknown. a. Based on a random sample of 50 cars, what is the probability that the sample mean time is greater than 10 minutes? b. Should the oil-change facility be concerned? Why or why not? 8 Problem Set Section 5.6 Normal Approximation to the Binomial 1. Decide whether you can use the normal distribution to approximate the binomial distributions. If so, find the mean and standard deviation. Check nq and np a. n=12 and p=0.6 b. n=20 and p=0.3 c. n=25 and p=0.2 d. A recent survey found that 70% of all adults over 50 wear glasses. You randomly select 30 adults over 50 and ask if he or she wears glasses. e. In a recent survey, 80% of the community favored building a police substation in their neighborhood. You randomly select 15 citizens and ask each if he or she things the community needs a police substation. 2. Using the Normal Approximation: The failure rate in a statistics class is 20%. In a class of 30 students, a. find the probability that fewer than 3 will fail. b. find the probability that between 3 and 10 will fail. c. find the probability that exactly 5 will fail. 9 Selected Answers: Section 5.1 1. a) Discrete 2. a) 4 ≤ X ≤8 3. a) y = ½ 4. a) 0.75 d) Continuous b) X>4 c) X<4 d) X≥4 b) 0.75 c) 1 d) 0 Section 5.2 1a) 0.683 1g) 0.4013 1m) 0.433 2a) 0.683 3a) z=1.28 4a) 0.809 1b) 0.341 1h) 0.242 1n) 0.475 2b) 0.954 3c) z=-1.65 4b) 1.50, -1.50 Section 5.3 1a) 0.0228 2) 0.0527 8a) 0.00135 9a) 0.04 1b) 0.4950 1d) 0.2525 4a) 11.97% 4b) about 239 8b) between 4.008 and 4.792 9b) 35.39 e) X≤4 1c) 0.341 1i) 0.136 1o) 0.954 2c) 0.997 1d) 0.599 1J) 0.155 1p) 0.995 2d) 1 1e) 0.025 1k) 0.929 1q) 0.05 2e) 0.683 4c) 1.51 4d) -2.06 4e) -0.358 1e) 104 4c) about 1100 1g) 119 f) X=4 1f) 0.009903 1l) 0.819 1r) 0.0151 2f) 0.95 Section 5.4 Section 5.5 1a) 0.388 6b) 0.0957 1b) 0.00214 6c) no 3a) 0.248 3b) 0.161 7b) µx=$250,000, x=2,828.4 Section 5.6 1a) no 1b) yes 1c) yes 1d) yes 1e) no 2) use np for mean and npq for variance 10 3c) 0.0912 7c) 0.144, no 3d) 4.7 and 5.3 7d) yes