290 CHAPTER 6 Normal Probability Distributions population is effectively infinite. When sampling without replacement from a finite population, we may need to adjust sx. Here is a common rule: When sampling without replacement and the sample size n is greater than 5% of the finite population size N(that is, n + 0.05N), adjust the standard deviation of sample means Sx by multiplying it by this finite population correction factor: AN - n N - 1 Except for Exercise 21 “Correcting for a Finite Population,” the examples and exercises in this section assume that the finite population correction factor does not apply, because we are sampling with replacement, or the population is infinite, or the sample size doesn’t exceed 5% of the population size. Statistical Literacy and Critical Thinking 1. Requirements Medical researchers once conducted experiments to determine whether Lisinopril is a drug that is effective in lowering systolic blood pressure of patients. Patients in a treatment group had their systolic blood pressure measured after being treated with Lisinopril. Under what conditions can the mean systolic blood pressure of this sample be treated as a value that is from a population having a normal distribution? 2.Small Sample Weights of M&M plain candies are normally distributed. Twelve M&M plain candies are randomly selected and weighed, and then the mean of this sample is calculated. Is it correct to conclude that the resulting sample mean cannot be considered to be a value from a normally distributed population because the sample size of 12 is too small? Explain. 3. Notation In general, what do the symbols mx and sx represent? What are the values of mx and sx for samples of size 36 randomly selected from the population of IQ scores with population mean of 100 and standard deviation of 15? 4.Incomes of Statistics Students Annual incomes of statistics students are known to have a distribution that is skewed to the right instead of being normally distributed. Assume that we collect a random sample of annual incomes of 50 statistics students. Can the distribution of incomes in that sample be approximated by a normal distribution because the sample is large? Why or why not? Using the Central Limit Theorem. In Exercises 5–8, assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of 1.2 kg and a standard deviation of 4.9 kg (based on Data Set 13 “Freshman 15” in Appendix B). 5. a. If 1 male college student is randomly selected, find the probability that he has no weight gain during his freshman year. (That is, find the probability that during his freshman year, his weight gain is less than or equal to 0 kg.) b. If 25 male college students are randomly selected, find the probability that their mean weight gain during their freshman year is less than or equal to 0 kg. c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30? 6-4 Basic Skills and Concepts
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