This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: W Dr; an on wail/ti ”Pastime EXEI‘CiSBS (53 A chain of department stores is interested in estimating the proportion of accounts re~
ceivable that are delinquent. The chain consists of four stores. So that the cost of sarnw
pling is reduced, stratiﬁed random sampiing is used, with each store as a stratum.
Because no information on population proportions is availabie before sampling, propor
tionai aiiocation is used. From the accompanying tabie, estimate p, the proportion of
delinquent accounts for the chain, and piace a bound on the error of estimation. Stratum Stratum Stratum Stratum
I II III IV
Number of N1 2 65 N2 = 42 N3 = 93 N4 : 25
accounts receivable
Sample size n1 2 14 17.2 m 9 723 m 21 R4 = 5
Sample number or" 4 2 8 i deiinquent accounts @ A corporation desires to estimate the total number of workerhours lost for a given month
because of accidents among alt employees. Because laborers, technicians, and adminism
trators have different accident rates, the researcher decides to use stratiﬁed random sam
pling, with each group forming a separate stratum. Data from previous years suggest the variances shown in the accompanying table for the number of worker~hours lost per employee in the three groups, and current data give the stratum sizes. Determine the Neyrnan allocation for a sample of n = 30 employees. 1 I] III
2 ) (laborers) (technicians) (administrators)
<5" )3 W 2 _ 2 ._ 2 _
o“ —_ 36 :72 — 25 (73 m
SZPA 5% W, N12132 N2 m 92 N3 w.» 27 uncut .. . . . . _
For Exerczse 5.2, estimate the total number of workerhours lost during the given month
and place a bound on the error of estimation. Use the data in the accompanying table, ob tained from sampiing 18 laborers, 10 technicians, and 2 administrators. Make a plot of
the data to check for unusual features. I II III
(laborers) (technicians) (administrators)
8 4 1
24 0 8
G 8
G 3
16 1
32 5
24
12
l 2
S omgwmmeeqmoo‘ .A report from the Census Bureau in October 1994 provided data on new one«family
houses for a sample of 28 metropolitan statistical areas (MSAS) and consolidated metro politan statistical areas (CMSAs) from around the country. {CMSAs tend to be larger
than MSAS and can be subdivided into other metropolitan areas for purposes of census
data summaries.) Data on total housing units sotd, median sales price, and median floor
area per house are reported, as shown in the table. The median sales price can be thought
of as a typieai price for that area. Similarly, the median ﬂoor area can be thought of as a
typical ﬂoor area for houses in that area. There were 250 MSAs and 18 CMSAs in the
United States for the year in which these data were reported. a. Plot the sales prices in parallel box plots, one for MSAs and one for CMSAS, and
comment on any unusual features you see. Do you see any reason to make adjustments
to the data before proceeding to estimate the mean typical selling price for the country? I). Treating these data as a stratified random sample, with the MSAS and CMSAs being
the two strata, estimate the mean typical sales price per house for all metropolitan
areas of the United States. Calculate a bound for the error of estimation. c. Ptot the total number of units sold in parallel box plots, one for each stratum. Do you
see any unusual features here? (1. Estimate the totai number of houses sold in alt metropolitan areas of the United States
in 1993 and calculate a bound for the error of estimation. e. Suppose you are to estimate the population mean or total for each of the three vari
ables in the data set. For which of the three outcome variablesutotal units sold, price,
or square footage—will stratiﬁcation produce the least gain in precision over simple
random sampling? Explain. ' , f. Estimate the difference in average typical selling price betwoen the two strata. Can we‘
say that houses in the CMSAs are, on the average, higher priced than those in the MSAs? Median
1993 Sales Finished
total sold price ﬂoor area
Metmpolitan area (thousands) (doliars) {square feet)
Atlanta, GA MSA 27.8 $118,200 2120
CharlotteGastoniaRock Hill, NC~SC MSA 7.8 115,100 1945
Chicago—Garwaenosha, lL—IN—WI CMSA 18.9 159,500 2020
Colorado Springs. CO MSA 3.0 138,600 2210
Dallas—Fort Worth,TX CMSA 19.2 _ 123,000 2325
DenverB oulderwGreeiey, CO CMSA 11.9 174,600 2225
Houston—Galveston—Brazoria,TX CMSA 10.2 114,200 2680
Jacksonville, FL MSA 5.0 95,000 1995
Kansas City, MOwKS MSA 6.5 99,300 1720
Las Vegas, NVAZ MSA 18.5 121,700 1770
Los AngelesRiversideOrange County, CA CMSA 23,4 139,800 1820
Miami—Fort Lauderdale, FL CMSA 11.2 131,500 2185
MinneapolisSt. Paul, MNWl MSA 10.6 155,600 2030
New Orleans, LA MSA , 1.6 99,200 2045
New York—Northern NJwLong Isiand, NY—NJ—CT—PA CMSA 17.9 191,400 2140
NorfolkVirginia BeachNeWport News, VA—NC MSA 6.0 120,200 2245
Orlando, FL MSA 10.5 107,500 1725
Phoenix—Mesa, AZ MSA 21.9 113,900 2180
Sacramento—Yule, CA CMSA 6.5 144,000 1540
St. Louis, MO—lL MSA 6.9' 144,500 1995
Salt Lake CityOgden, UT MSA 6.5 100,300 1545
San Antonio,’1‘X'MSA 3.0 117,900 2375
San Diego, CA MSA 3.8 225,000 2375
SeattleTacoma—Bremertou, WA CMSA 10.8 159,700 1885
TampaSt. PetersburgClearwatcr, FL MSA 8.4 113,700 2240
Tucson, AZ MSA 3.8 106,600 1810
Washington—Baitirnore, DC—MD‘VA—WV CMSA 31.1 184,400 2305 West Palm Beach—Boea Raton, FL MSA 5.6 158,400 250 @ Acorporation wishes to obtain information on the effectiveness of a business machine. A number of division heads will he interviewed by telephone and asked to rate the equip
ment on a numerical scaie. The divisions are located in North America, Europe, and Asia.
Hence, stratiﬁed sampling is used. The costs are larger for interviewing division heads
loéated outside North America. The accompanying table gives the costs per interview,
approximate variances of the ratings, and N that have been established. rl‘he corporation
wants to estimate the average rating with V075,) : 0.]. Choose the sample size n that
achieves this bound, and ﬁnd the appropriate allocation. Stratum I Stratum II Stratum III
(N orth America) (Europe) (Asia)
c; 2 $9 c; 2 $25 C3 = $36
a”? = 2.25 0% :2 3.24 a; m 3.24
N;=:112 N2=6S 513239 A school desires to estimate the average score that may be obtained on a reading com—
prehension exam for students in the sixth grade. The school’s students are grouped into
three tracks, with the fast learners in track I and the slow learners in track HI. The school
decides to stratify on tracks because this method should reduce the variability of test
scores. The sixth grade contains 55 students in track I, 80 in track II, and 65 in track III.
A stratiﬁed random sample of 50 students is proportionally allocated and yields simple
random samples of m z 14, n; m 20, and :23 = 16 from tracks I, II, and III. The test is
administered to the sample of students, With the results as shown in the table. a. Estimate the average score for the sixth grade, and place a bound on the error of esti
mation. Track I Track 11 Track III so 85 42
92 32 32
68 48 36
as 75 31
72 53 65
87 73 29
85 65 43
91 7s 19
9o 49 53
31 69 14
52 72 61
79 81 31
6t 53 42
33 59 30 as 39 52 32 71 ' 61 59 42 WW 1). Construct parallel box plots for these data and comment on the patterns you see. Do
you think there could be a problem in placing students in tracks? c. Estimate the difference in average scores between track I and track II students. Are Suppose the average test score for the class in track I students signiﬁcantly better, on the average, than traclc 13 students? (3 Exercise 5.6 is to be estimated again at the
pling are equal in all strata, Iout the variances dif—
fer. Find the optimum (Neyman) allocation of a sample of size 50, using the data in Ex
ercise 5.6 to approximate the variances. a score, With a bound of four points on the error of estimation. Use proportional allocation.
5 .9 Repeat Exercise 5.8 using Neyman allocatio It. Compare the results with the answer to
Exercise 5.8. ‘ ...
View
Full Document
 Spring '11
 Staff

Click to edit the document details