Section 7.3 Sampling Distributions and the Central Limit Theorem 315 Step 4: We will use Table A.2. We compute the z-score for 1100. z = x − ��̄ x ��̄ x = 1100 − 1135 9.7 = −3.61 The area to the left of z = −3.61 is 0.0002. The probability that the sample mean weight is less than 1100 is 0.0002. This probability is less than 0.05, so it would be unusual for the sample mean to be less than 1100. Check Your Understanding 3. A population has mean �� = 10 and standard deviation �� = 8. A sample of size 50 is drawn. a. Find the probability that ̄x is greater than 11. 0.1894 Tech: 0.1884 b. Would it be unusual for ̄x to be less than 8? Explain. Yes 4. A population has mean �� = 47.5 and standard deviation �� = 12.6. A sample of size 112 is drawn. a. Find the probability that ̄x is between 45 and 48. 0.6449 b. Would it be unusual for ̄x to be greater than 48? Explain. No Answers are on page 318. SECTION 7.3 Exercises Exercises 1– 4 are the Check Your Understanding exercises located within the section. Understanding the Concepts In Exercises 5 and 6, fill in each blank with the appropriate word or phrase. 5. The probability distribution of ̄x is called a distribution. sampling 6. The states that the sampling distribution of ̄x is approximately normal when the sample is large. Central Limit Theorem In Exercises 7 and 8, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement. 7. If ̄x is the mean of a large (n > 30) simple random sample from a population with mean �� and standard deviation ��, then ̄x is approximately normal with ��̄ x = �� √ n . True 8. As the sample size increases, the sampling distribution of ̄x becomes more and more skewed. False Practicing the Skills 9. A sample of size 75 will be drawn from a population with mean 10 and standard deviation 12. a. Find the probability that ̄x will be between 8 and 14. b. Find the 15th percentile of ̄x. 8.56 10. A sample of size 126 will be drawn from a population with mean 26 and standard deviation 3. a. Find the probability that ̄x will be between 25 and 27. 0.9998 b. Find the 55th percentile of ̄x. 26.03 11. A sample of size 68 will be drawn from a population with mean 92 and standard deviation 24. a. Find the probability that ̄x will be greater than 90. b. Find the 90th percentile of ̄x. 95.73 12. A sample of size 284 will be drawn from a population with mean 45 and standard deviation 7. a. Find the probability that ̄x will be greater than 46. 0.0080 b. Find the 75th percentile of ̄x. 45.28 13. A sample of size 91 will be drawn from a population with mean 33 and standard deviation 17. a. Find the probability that ̄x will be less than 30. b. Find the 25th percentile of ̄x. 31.81 Tech: 31.80 14. A sample of size 82 will be drawn from a population with mean 24 and standard deviation 9. a. Find the probability that ̄x will be less than 26. b. Find the 10th percentile of ̄x. 22.73 15. A sample of size 20 will be drawn from a population with mean 6 and standard deviation 3. a. Is it appropriate to use the normal distribution to find probabilities for ̄x? No b. If appropriate find the probability that ̄x will be greater than 4. Not appropriate c. If appropriate find the 30th percentile of ̄x. Not appropriate 16. A sample of size 42 will be drawn from a population with mean 52 and standard deviation 9. a. Is it appropriate to use the normal distribution to find probabilities for ̄x? Yes b. If appropriate find the probability that ̄x will be between 53 and 54. 0.1609 Tech: 0.1608 c. If appropriate find the 45th percentile of ̄x. 51.82 17. A sample of size 5 will be drawn from a normal population with mean 60 and standard deviation 12. a. Is it appropriate to use the normal distribution to find probabilities for ̄x? Yes b. If appropriate find the probability that ̄x will be between 50 and 70. 0.9372 Tech: 0.9376
navidi_monk_elementary_statistics_2e_ch7-9
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