Case Study. The National Center for Health Statistics (NCHS) keeps records of many health-related aspects of people, including the birth weights of all babies born in the United States. The birth weight of a baby is related to its gestation period (the time between conception and birth). For a given gestation period, the birth weights can be approximated by a normal distribution. The means and standard deviations are shown below. Gestation period Under 28 weeks Mean birth weight Standard deviation 1.88 lbs 1.19 lbs 28 to 31 weeks 4.07 lbs 1.87 lbs 32 to 35 weeks 5.73 lbs 1.48 lbs 36 weeks 6.46 lbs 1.20 lbs 37 to 39 weeks 7.33 lbs 1.09 lbs 40 weeks 7.72 lbs 1.05 lbs 41 weeks 7.83 lbs 1.08 lbs 42 weeks and over 7.65 lbs 1.12 lbs 5. Birth Weights. What percent of the babies born with each gestation period have low birth weight (under 5.5 pounds)? Illustrate each with a graph: (Draw, label and shade). a. 32 to 35 weeks b. 40 weeks 6. Birth Weights. Describe the weights of the top 10% of the babies born with each gestation period. Illustrate each with a graph: (Draw, label and shade). a. 28 to 31 weeks b. 42 weeks and over 7. Birth weights. A birth weight of less than 3.3 pounds is classified by the NCHS as a "very low birth weight." What is the probability that a baby has a very low birth weight for each gestation period. Illustrate each with a graph: (Draw, label and shade). a. Under 28 weeks b. 32 to 35 weeks 8. Normal Distribution. In a large lecture-hall section of a statistics class, the points for the final exam are normally distributed with a mean of 72 and a standard deviation of 9. Grades are to be assigned according to the following rule: The top 10% receive A's The next 20% receive B's The middle 40% receive C's The next 20% receive D's The bottom 10% receive F's Find the lowest score on the final exam that would qualify a student for an A, a B, a C, and a D. Draw and label ONE normal curve to illustrate your results. 9. Selling Cars: You sell a brand of automobile tire that has a life expectancy that is normally distributed with a mean life of 30,000 miles and a standard deviation of 2500 miles. You want to give a guarantee for free replacement tires if the tires wear out too soon, but you can only afford to give free replacements on approximately 5% of the tires you sell. How should you word your free replacement guarantee? Draw, shade and label a normal curve to illustrate your results. 10. BONUS QUESTION: A vending machine dispenses coffee into an 8-ounce cup. The amount of coffee dispensed into the cup is normally distributed with a standard deviation of 0.03 ounces. You can allow the cup to overfill 1% of the time. What amount should you set as the mean amount of coffee to be dispensed? Draw, shade and label a normal curve to illustrate your results. Section 7.3 The Central Limit Theorem 1. When samples are repeatedly taken: the mean of each chosen sample, is itself a random variable, with the following properties: XIXI XXXXX XXXXX a. The shape of the distribution of sample means is b. The mean of the sample means is (Notation) IX c. The standard deviation of the sample means is population standard deviation. The standard deviation of the sample means is larger/smaller often referred to as (Notation) than the original 1. Oil Changes. Suppose that the mean time for an oil change at a "10-minute oil change joint" is 11.4 minutes with a standard deviation of 3.2 minutes. (Illustrate each with a graph!) a) If a random sample of n = 35 oil changes is selected, describe the sampling distribution of the sample mean. random variable: X= distribution: X- (E 0= b) If a random sample of n = 35 oil changes is selected, what is the probability the mean oil change time is less than 11 minutes? X =>