322 Chapter 7 The Normal Distribution 0.4 0.51 z = –1.91 Figure 7.28 Step 2: ��̂p = p = 0.51. ��̂p = √ p(1 − p) n = √ 0.51(1 − 0.51) 75 = 0.057723 Step 3: Figure 7.28 presents the normal curve with the area shaded in. Step 4: We will use Table A.2. We compute the z-score for 0.40. z = ̂p − ��̂p ��̂p = 0.40 − 0.51 0.057723 = −1.91 The area to the left of z = −1.91 is 0.0281. It would be unusual for the sample proportion to be less than 0.40. Check Your Understanding 3. The General Social Survey reported that 56% of American adults saw a doctor for an illness during the past year. A sample of 65 adults is drawn. a. What is the probability that more than 60% of them saw a doctor? 0.2578 Tech: 0.2580 b. Would it be unusual if more than 70% of them saw a doctor? Yes 4. For a certain type of computer chip, the proportion of chips that are defective is 0.10. A computer manufacturer receives a shipment of 200 chips. a. What is the probability that the proportion of defective chips in the shipment is between 0.08 and 0.15? 0.8173 Tech: 0.8179 b. Would it be unusual for the proportion of defective chips to be less than 0.075? No Answers are on page 324. SECTION 7.4 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. If n is the sample size and x is the number in the sample who have a certain characteristic, then x∕n is called the sample . proportion 6. The probability distribution of ̂p is called a distribution. sampling 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. The distribution of ̂p is approximately normal if np ≥ 10 and n(1 − p) ≥ 10. True 8. If n is the sample size, p is the population proportion, and ̂p is the sample proportion, then ��̂p = np. False Practicing the Skills In Exercises 9–14, n is the sample size, p is the population proportion, and ̂p is the sample proportion. If appropriate, use the Central Limit Theorem to find the indicated probability. 9. n = 147, p = 0.13; P( ̂p < 0.11) 0.2358 Tech: 0.2354 10. n = 65, p = 0.86; P( ̂p < 0.80) 0.0823 Tech: 0.0816 11. n = 270, p = 0.57; P( ̂p > 0.61) 0.0918 Tech: 0.0922 12. n = 103, p = 0.24; P(0.20 < ̂p < 0.23) 0.2341 Tech: 0.2352 13. n = 145, p = 0.05; P(0.03 < ̂p < 0.08) Not appropriate 14. n = 234, p = 0.75; P(0.77 < ̂p < 0.81) 0.2219 Tech: 0.2229 Working with the Concepts 15. Coffee: The National Coffee Association reported that 63% of U.S. adults drink coffee daily. A random sample of 250 U.S. adults is selected. a. Find the mean ��̂p. 0.63 b. Find the standard deviation ��̂p. 0.0305
navidi_monk_elementary_statistics_2e_ch7-9
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