Chapter 7, Section 7.3, Question E36 According to the U.S. Census Bureau, 22.3 p

Question

Chapter 7, Section 7.3, Question E36 According to the U.S. Census Bureau, 22.3 percent of the Spanish-surnamed population in the United States have one of these surnames: Garcia, Martinez, Rodriguez, Lopez, Hernandez, Gonzalez, Perez, Sanchez, Rivera, Ramirez, Torres, Gonzales. Suppose you take a random sample of 500 Spanish-surnamed people in the United States. [Source: David L. Word and R. Colby Perkins r., Building a Spanish Surname List for the 1990's-A New Approach to an Old Problern, Technical Working Paper no. 13, March 1996.] a. Make an accurate sketch of the sampling distribution of the proportion of people in your sample who have one of these surnames 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.20 0.22 0.24 0.26 0.18 0.20 0.22 0.24 0.26 0.18 0.20 0.22 0.24 0.26 Fig. 2 Fig. 3 Fig. 4 Choose the right figure. Figure number What is the probability of getting 20% of fewer with one of these surnames in your sample? (round to three decimal places) c. What is the probability of getting 105 or more people with one of these surnames? (round to three decimal places) d. What proportions of people with one of these surnames would be rare events (not reasonably likely)? Round to three decimal places Smaller than or larger than

Explanation / Answer

(a) Here sample proportion p^ = 0.223

Standard error of proportion sep   = sqrt [0.223 * 0.777/500] = 0.0186

so figure 3 seems appropriate.

(b) Pr(p < 0.20 ; 0.223 ; 0.0186)

Z= (0.20 - 0.223)/0.0186 = -1.2355

Pr(p < 0.20 ; 0.223 ; 0.0186) = Pr(Z < -1.2355) = 0.1083

(c) Expected number of people with one of surnames = 500 * 0.223 = 111.5

Standard deviation of number of people with one of surnames = sqrt [500 * 0.223 * 0.777] = 9.308

If x is the number of people with surname

Pr(x >=105 ; 111.5 ; 9.308) = 1 - Pr(x < 104.5 ; 111.5 ; 9.308) [employing continuity correction]

Z = (104.5 - 111.5)/9.308 = -0.752

Pr(x >=105 ; 111.5 ; 9.308) = 1 - Pr(x < 104.5 ; 111.5 ; 9.308) = 1 - Pr(Z < -0.752) = 1 - 0.2260 = 0.7740

(d) Here unsual proportion are which , those are outside the limit p^ +- 2 sep

Lower limit = 0.223 - 2 * 0.0186 = 0.186

Upper than = 0.223 + 2 * 0.0186 = 0.260

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