344 Chapter 7 The Normal Distribution a. If the claim is true, what is the probability of obtaining a sample proportion that is less than or equal to 0.44? 0.0119 Tech: 0.0118 b. If the claim is true, would it be unusual to obtain a sample proportion less than or equal to 0.44? Yes c. If the claim is true, would it be unusual for less than half of the voters in the sample to support the politician? No 12. Side effects: A new medical procedure produces side effects in 25% of the patients who receive it. In a clinical trial, 60 people undergo the procedure. What is the probability that 20 or fewer experience side effects? 0.9495 13. Defective rods: A grinding machine used to manufacture steel rods produces rods, 5% of which are defective. When a customer orders 1000 rods, a package of 1060 rods is shipped, with a guarantee that at least 1000 of the rods are good. What is the probability that a package of 1060 rods contains 1000 or more that are good? 0.8554 Tech: 0.8547 14. Is it normal? Is it reasonable to treat the following sample as though it comes from an approximately normal population? Explain. No 2.6 4.2 1.5 2.0 0.6 0.7 6.6 2.2 9.7 1.8 4.2 4.4 0.6 15. Is it normal? Is it reasonable to treat the following sample as though it comes from an approximately normal population? Explain. Yes 8.8 11.2 11.6 6.3 9.3 1.5 14.6 7.5 5.2 9.0 4.3 9.9 7.8 13.1 Write About It 1. Explain why P(a < X < b) is equal to P(a ≤ X ≤ b) when X is a continuous random variable. 2. Describe the information you must know to compute the area under the normal curve over a given interval. 3. Describe the information you must know to find the value corresponding to a given proportion of the area under a normal curve. 4. Suppose that in a large class, the instructor announces that the average grade on an exam is 75. Which is more likely to be closer to 75: i. The exam grade of a randomly selected student in the class? ii. The mean exam grade of a sample of 10 students? Explain. 5. Consider the formula for the standard deviation of the sampling distribution of ̂p given by ��̂p = √ p(1 − p) n . What happens to the standard deviation as n gets larger and larger? Explain what this means in terms of the spread of the sampling distribution. 6. Explain how to decide when it is appropriate to use the normal approximation to the binomial distribution. 7. Describe the effect, if any, that the size of a sample has in assessing the normality of a population. Case Study: Testing The Strength Of Cans In the chapter opener, we discussed a method used to determine whether shipments of aluminum cans are strong enough to withstand the pressure of containing a carbonated beverage. Several cans are sampled from a shipment and tested to determine the pressure they can withstand. Based on this small sample, quality inspectors must estimate the proportion of cans that will fail at or below a certain threshold, which we will take to be 90 pounds per square inch. The quality control inspectors want the proportion of defective cans to be no more than 0.001, or 1 in 1000. They test 10 cans, with the following results. Can 1 2 3 4 5 6 7 8 9 10 Pressure at failure 95 96 98 99 99 100 101 101 103 104 None of the cans in the sample were defective; in other words, none of them failed at a pressure of 90 or less. The quality control inspectors want to use these data to estimate the proportion of defective cans in the shipment. If the estimate is to be no more than 0.001, or 1 in 1000, they will accept the shipment; otherwise, they will return it for a refund. The following exercises will lead you through the process used by the quality control inspectors. Assume the failure pressures are normally distributed. 1. Compute the sample mean ̄x and the sample standard deviation s. ̄ x = 99.6, s = 2.8363 2. Estimate the population mean �� with ̄x and the population standard deviation �� with s. In other words, assume that the data are a sample from a normal population with mean �� = ̄x and standard deviation �� = s. Under this assumption, what proportion of cans will fail at a pressure of 90 or less? 0.0004 3. The shipment will be accepted if we estimate that the proportion of cans that fail at a pressure of 90 or less is less than 0.001. Will this shipment be accepted? Yes
navidi_monk_elementary_statistics_2e_ch7-9
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