Chapter 8 Sampling Methods and the Central Limit Theorem

Transcription

1 Chapter 8 Sampling Methods and the Central Limit Theorem. a. 33 Louisiana, 555 S. Main, 35 Monroe, 65 W. Central b. Answers will vary c. 63 Dixie Hwy, 835 S. McCord Rd., 464 Woodville Rd. d. Answers will vary (LO ). a. Childrens Hospital Medical Center, St. Francis-St. George Hospital, Bethesda North, Good Samaritan Hospital, Mercy Hospital-Hamilton b. Answers will vary c. Jewish Hospital-Kenwood, Mercy Hospital-Anderson, Good Samaritan Hospital, St. Elizabeth Medical Center-North unit, Emerson Behavioral Service, Shriners Burns Institute d. Answers will vary (LO ) 3. a. Bob Schmidt Chevrolet, Great Lakes Ford Nissan, Grogan Towne Chrysler, Southside Lincoln Mercury, Rouen Chrysler Jeep Eagle b. Answers will vary c. Yark Automotive, Thayer Chevrolet Toyota, Franklin Park Lincoln Mercury, Matthews Ford Oregon, Inc., Valiton Chrysler (LO ) 4. a. Denker, Brett; Wood, Tom; Keisser, Keith; Priest, Harvey b. Answers will vary c. Farley, Ron; Hinckley, Dave; Priest, Harvey; and Wood, Tom (LO ) 5. a. Sample Values Sum Mean, 4, , , , , b. ( )/6 3.5 =( )/4 = 3.5 c. More dispersion with population compared to the sample means. The sample means vary from to 5 whereas the population varies from to 6. (LO 4) 8-

2 6. a. Sample Values Sum Mean, 4, , ,8 5 5, , , , ,8 6 4,8 6 b. =( )/5 = 4 ( )/ 4 c. They are equal. The dispersion for the population is greater than that for the sample means. The population varies from to 8, whereas the sample means only vary from to 6. (LO 4) 7. a. Sample Values Sum Mean,, ,, ,, ,5, ,4, ,4, ,5, ,5, ,4, ,4, b. ( ) / 4.6 =( )/5 = 4.6 c. The dispersion of the population is greater than that of the sample means. the sample means vary from.66 to 6.33 where as the population varies from to. (LO 4) 8-

3 8. a. Sample Values Sum Mean,,.33,,3 3. 3,,6 6. 4,, ,3, ,, ,3, ,3, ,,6 7.33,, b. ( ) / =( )/5 = c. The dispersion of the population is greater than the sample means. The sample means vary from.33 to 3.33, the population varies from to 6. (LO 4) 9. a. found by 6 C 3 b. Sample Cases Sum Mean Ruud,Wu,Sass 3,6,3 4. Ruud,Sass,Flores 3,3, Ruud,Flores,Wilhelms 3,3, 6. Ruud,Wilhelms,Schueller 3,, 4.33 Wu,Sass,Flores 6,3,3 4. Wu,Flores,Wilhelms 6,3, 9 3. Wu,Wilhelms,Schueller 6,, 7.33 Sass,Flores,Wilhelms 3,3, 6. Sass,Wilhelms,Schueller 3,, 4.33 Flores,Wilhelms,Schueller 3,, 4.33 Wu, Sass, Wilhelms 6,3, 9 3. Ruud,Wu,Flores 3,6,3 4. Ruud,Wu,Wilhelms 3,6, 9 3. Ruud,Wu,Schueller 3,6, 3.33 Ruud,Sass,Wilhelms 3,3, 6. Ruud,Sass,Schueller 3,3, 7.33 Ruud,Flores,Schueller 3,3, 7.33 Wu,Sass,Schueller 6,3, 3.33 Wu,Flores,Schueller 6,3, 3.33 Sass,Flores,Schueller 3,3,

4 c. d =( )/6 =.67 They are equal Population Values Probability.6.4. u # of Cases Dis tributio n o f Sample Means Sample Means # of Cases Sample Mean Number of Means Probability More of a dispersion in population compared to sample means. The sample means vary from.33 to 4.. The population varies from to 6. (LO 4) 8-4

5 . a., found by (5!)/3!! b. Cars Sample Cars Sample sold mean sold mean 8,6 7 6, 8 8,4 6 6,6 6 8, 9 4, 7 8,6 7 4,6 5 6,4 5,6 8 c. 6.8 for population, 6.8 for sample means. They are identical. d. Population Means Number of Sales Cars Sold Sample Means Number of Means Cars Sold There is more dispersion in the population than in the sample. (LO 4) 8-5

6 . a b. Sample Sum Frequency 3 4 Values The mean of the sample means is 4.84, which is close to the population mean of 4.5. The sample means range form. to 7., where as the population values range from to 9. From the above graph, the sample means tend to cluster between 4 and 5. (LO 5)

7 . a. Frequency Units Sold b c. Answers will vary, below is one sample Sample Sample Values Sum,3,,3, ,3,4,, ,3,4,4, ,,5,5, ,4,4,, d. Sample mean is very close to the population mean. It is not to be expected that they are exact. e. Frequency Sample Means There is less dispersion in the sample means than the population. (LO 5) 3. Answers will vary. (LO 5) 4. Answers will vary. (LO 5) 8-7

8 5. a. z = So probability is.66, found by / 9 b z So the probability is.587, found by / 9 c..647, found by (LO 7) 6. a. z = So probability is.38, found by / 4 b. z = So probability is.794, found by (.396). 5/ 4 c. z = So probability is.98, found by / 4 d..57, found by (LO 7) 7. z = So probability is or virtually certain. (LO 7) 5 / 5 8. a. 8 sx b. z = So probability is.785, found by / 4 c. z = So probability is.78, found by / 4 d..57, found by (LO 7) 9. a. Formal Man, Summit Stationers, Bootleggers, Leather Ltd., Petries b. Answers will vary c. Elder-Beerman, Frederick s of Hollywood, Summit Stationers, Lion Store, Leather Ltd., Things Remembered, County Seat, Coach House Gifts, Regis Hairstylists (LO ). a. Jeanne Fiorito, Douglas Smucker, Jeanine S. Huttner, Harry Mayhew, Mark Steinmetz, and Paul Langenkamp. b. One randomly selected group of numbers is 5, 6, 74, 64, 66, 55, 7, and. The members of the sample are Janet Arrowsmith, David DeFrance, Mark Zilkoski, and Larry Johnson. c. Francis Aona, Paul Langenkamp, Ricardo Pena, and so on. d. Answers will vary. (LO ) 8-8

9 . a. Samples Mean Deviation from Mean Square of Deviation,...,.5.5.5,3...,.5.5.5,..., ,... 3, , b. Mean of sample means is ( )/ 9 = 8/9 =. The population mean is ( + +3)/3 = 6/3=. They are the same value. c. Variance of sample means is ( )/9 = /3. Variance of the population values is ( + + )/3 = /3. The variance of the population is twice as large as that of the sample means. d. Sample means follow a triangular shape peaking at. The population is uniform between and 3. (LO 4). a.., found by. / or./ b..9876, found by (.4938). The z-value for 6 is.5, found by (6 5.5)/.. Similarly the z-value for 5 is.5. c..686, found by (.343). The z-value for 5.7 is., found by ( )/.. Similarly the z-value for 5.3 is.. d. This is very unlikely. The z-value for 6.5 is 5., found by ( )/.. (LO 7) 3. Larger samples provide narrower estimates of a population mean. So the company with sampled customers can provide more precise estimates. In addition, they are selected consumers who are familiar with laptop computers and may be better able to evaluate the new computer. (LO 4) 4. a. The standard error of the mean declines as the sample size grows because the sample size is in the denominator and as the denominator increases the proportion decreases. b. If the sample size is increased, the Central Limit theorem guarantees the distribution of the sample means becomes more normal. c. The shape of the distribution becomes narrower since the dispersion is less and estimates of the mean are more precise. (LO 4) 5. a. We selected 6, 4, 75, 7, and 48. Answers will vary. b. We selected the third observation. So the sample consists of 75, 7, 68, 8, 48. Answers will vary. c. Number the first motels from to 9. Randomly select three numbers. Then number the last five numbers to 4. Randomly select two numbers from that group. (LO ) 6. Answers will vary. (LO ) 8-9

10 7. a. 5 found by 6 C b. Sample Value Sum Mean 79, , , , , , , , , , , , , , , c =478/6 = They are equal d. No, the student is not graded on all available information. He/she is as likely to get a lower grade based on the sample as a higher grade. (LO 4) 8-

11 8. a., found by 5 C b. Sample Value Sum Mean,3 5.5, , , , , , , ,5 5. 3, c. 36/ 3.6 =8/5 = 3.6 They are equal. (LO 4) 9. a., found by 5 C b. Shutdowns Mean Shutdowns Mean 4, ,3 3. 4, ,.5 4, ,3 4. 4, 3. 5, 3.5 3,5 4. 3,.5 c. ( ) / 3.4 =( )/5 = 3.4 The two means are equal. d. The population values are uniform in shape. The distribution of the sample means tends toward normality. (LO 4) 3. a. 5, found by 6 C b. # Sold Mean # Sold Mean 54,5 5 5,5 5 54,5 53 5, ,48 5 5,5 5 54,5 5 5,5 5 54, ,5 49 5,5 5 48,5 5 5, ,5 5 5,5 5 c. Sample Means Frequency Probability d. =5 5 e. Tending toward normal f. Sample means. Somewhat normal (LO 4) 8-

12 3. a. The distribution will be normal. b x 5 c z.9 So probability is.84, found by d z.45 So probability is.6736, found by e..49, found by (LO 7) 3. a. The distribution will be normal. b. 8 x 6 c z.5 So probability is.6, found by d z 3.5 So probability is e..9938, found by (LO 7) z. So probability is.986, found by (LO 7) a. 4, s x 5 b. The distribution will be normal c.,, z.35 So probability is.363, found by , 5 d.,, z.77 So probability is.966, found by , 5 e..5984, found by (LO 7) z.93 So probability is.838, found by (LO 7) a. 3 z. and z So probability is.859, found by b. Since the sample size is small, you assume the population is normally distributed. (LO 7) 37. Between 5954 and 646, found by (LO 7) 4 8-

13 38. a z. So probability is.7, found by b z.4 So probability is.937, found by c. 5 Between.33 and 4.67, found by (LO 7) z.78 So probability is.375, found by (LO 7) a. 36 b. Sample st roll nd roll Mean

14 c Frequency Frequency d. Both means are 3.5. The standard deviation of individual rolls is.78, while the standard deviation of sample means is.8. (LO 4) 4. a. Virginia, Kansas, Georgia, South Carolina, Utah, Nebraska, Connecticut, and Alaska; found by selecting the first eight, non-repeating numbers between and 49, inclusive. b. (Arizona),8(Florida),4(Iowa),(Massachusetts),6(Nebraska),3(North Carolina),38(Rhode Island), and 44(Vermont). (LO ) 8-4

15 4. a.. 9 x b. z. 94 and z So probability is.9476, found by c. z. 77 and z So probability is.5588, found by d. z 3. 7 So probability is virtually. (LO 7) a. z So probability is virtually b. z. 4.8 Probability is.9864, found by c..36, found by (LO 7) 44. a x 4 b z. 77 and z Probability is.5588, found by c... z. 9 and z Probability is.56, found by d z. 9 Probability is.74, found by (LO 7).5 8-5

16 45. a x b. z. 4 and z Probability is.9676, found by c. z. 7 and z Probability is.754, found by d. Probability is.6, found by (LO 7) 46. Answers will vary. (LO 7) 47. Mean attendance is.436 million. Likelihood of a sample mean this large or larger is , found by.5.995, where z.84. (LO 7) Likelihood of a sample mean this small or smaller is.77, found by.5.83, where z. 78. (LO 7)