Workshop Statistics: Discovery with Data, Second Edition

Topic 19: Confidence Intervals I: Proportions

Activity 19-1: Penny Spinning

(a) pennies
(b) categorical
(c) Answers will vary from class to class.
(d) This proportion is a statistic, with the symbol .
(e) no
(f) yes
(g) , the proportion of the sample
(h)-(i) Answers will vary from class to class.
(j) no
 

Activity 19-2: Critical Values

(a)
(b) .99
(c) 2.33
(d, a)
(d, b) .975
(d, c) 1.96
 

Activity 19-3: Penny Spinning (cont.)

(a) Answers will vary from class to class.
(b) no
(c)-(d) Answers will vary from class to class but to find the width, subtract the smaller endpoint from the larger endpoint.
(e) Since we add and subtract z*sqrt((1-)/n) from in forming the interval, the quantity z*sqrt((1-)/n) is already the half-width.
(f) The sample must be a simple random sample from the population of interest, and the sample size must be large relative to the value of the sample proportion (checked by calculating n>10 and n(1-)>10)
 

Activity 19-4: Halloween Practices and Beliefs (cont.)

(a) statistic
(b) categorical
(c) (.661, .718)
(d) The student's axis should have the region from approximately .661 to .718 shaded.
(e) Width: .057;  Half-width: .0285
(f) Interval: (.282, .339);  Width: .057;  Half-width: .0285
(g) These half-widths are the same, but the intervals are not the same.  They are related because the sum of the lower value of one interval with the upper value of the other interval should equal 1, i.e., subtract the endpoints of the interval from 1.  Often it will equal .999, because of error due to rounding.
(h) Answers will vary from student to student.
(i) Interval: (.652, .727);  Width: .075;  Half-width: .0375
(j) The width of the 99% confidence interval is greater than the width of the 95% confidence interval by .018.  The midpoints, however, are the same: .69.
(k) (.183,  .256)
(l) Answers will vary from student to student.
(m) Interval: (.194, .245);  Midpoint: .22;  Half-width: .0255;  The midpoint is the same, but the width is narrower.
 

Activity 19-5: Reese's Pieces (cont.)

Students' answers to (a)-(c) may differ since the data are chosen randomly.  These are meant to be sample answers.
Note, answers (b)-(g) below, are the answers to (a)-(f) in the Calculator version.
(a) Click here for sample results.
(b) 186 of the 200 intervals captured .45.
(c) .32, .293, .32, .587, .587, .60, .573, .60, .623, .333, .333, .333, .32, .293
(note, these are all on the extreme small end or the extreme high end)
(d) no
(e) When a large number of samples are taken, about 95% of the intervals will contain the actual value of the population proportion.
Students' counts in (f)-(g) may differ since the data are chosen randomly.  These are meant to be sample answers.
(f) 195;  98%;  This is reasonably close to the percentage from (a). Changing the sample size does not change the coverage proportion. However, the intervals are now more narrow.
(g) 159;  80%;  This is not reasonably close to the percentage from (a). The intervals are narrower so fewer of them capture q.
 

Activity 19-6: Literary Digest Pol (cont.)

(a) (.569, .571)
(b) The sampling method was biased so we can't expect these calculations to be accurate.