342 Chapter 7 The Normal Distribution Important Formulas z-score: z-score for a sample mean: z = x − �� �� z = ̄x − �� ��̄ x Convert z-score to raw score: Standard deviation of the sample proportion: ��̂p = √ p(1 − p) n x = �� + z�� Standard deviation of the sample mean: z-score for a sample proportion: ��̄ x = �� √ n z = ̂p − p ��̂p Chapter Quiz 1. Following is a probability density curve for a population. 0 2 4 6 8 10 0.4 0.3 0.2 0.1 0 Area = 0.09 Area = 0.59 a. What proportion of the population is between 2 and 4? 0.32 b. If a value is chosen at random from this population, what is the probability that it will be greater than 2? 0.41 2. Find the area under the standard normal curve a. To the left of z = 1.77 0.9616 b. To the right of z = 0.41 0.3409 c. Between z = −2.12 and z = 1.37 0.8977 3. Find the z-score that has a. An area of 0.33 to its left −0.44 b. An area of 0.79 to its right 0.81 4. Find the z-scores that bound the middle 80% of the area under the normal curve. z = −1.28 and z = 1.28 5. Find z0.15. 1.04 6. Suppose that salaries of recent graduates from a certain college are normally distributed with mean �� = $42,650 and standard deviation �� = $3800. What two salaries bound the middle 50%? $40,104 and $45,196 Tech: $40,087 and $45,213 7. A normal population has mean �� = 242 and standard deviation �� = 31. a. What proportion of the population is greater than 233? 0.6141 Tech: 0.6142 b. What is the probability that a randomly chosen value will be less than 249? 0.5910 Tech: 0.5893 8. Suppose that in a bowling league, the scores among all bowlers are normally distributed with mean �� = 182 points and standard deviation �� = 14 points. A trophy is given to each player whose score is at or above the 97th percentile. What is the minimum score needed for a bowler to receive a trophy? 209 9. State the Central Limit Theorem. 10. A population has mean �� = 193 and standard deviation �� = 42. Compute ��̄ x and ��̄ x for samples of size n = 64. ��̄ x = 193, ��̄ x = 5.25 11. The running time for videos submitted to YouTube in a given week is normally distributed with �� = 390 seconds and standard deviation �� = 148 seconds. a. If a single video is randomly selected, what is the probability that the running time of the video exceeds 6 minutes (360 seconds)? 0.5793 Tech: 0.5803 b. Suppose that a sample of 40 videos is selected. What is the probability that the mean running time of the sample exceeds 6 minutes? 0.8997 Tech: 0.9001 12. A sample of size n = 55 is drawn from a population with proportion p = 0.34. Let ̂p be the sample proportion. a. Find ��̂p and ��̂p. ��̂p = 0.34, ��̂p = 0.063875 b. Find P( ̂p > 0.21). 0.9793 Tech: 0.9791 c. Find P( ̂p < 0.40). 0.8264 Tech: 0.8262
navidi_monk_elementary_statistics_2e_ch7-9
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