Workshop Statistics: Discovery with Data, Second Edition

Topic 20: Confidence Intervals II: Means

Activity 20-1: Christmas Shopping (cont.)

(a) quantitative
(b) In this case, $857 is a statistic, with symbol , because it is the mean of a sample size of 922 people.
(c) The parameter of interest in this study is the mean amount expected to be spent on Christmas presents in 1999 by all American adults, with symbol m.
(d) No, but is more likely to be close to $857 than far from it.
(e) 250/sqrt(922) = $8.23
(f) 857 - 2(8.23), 857 + 2(8.23) = $840.54 - $873.46
(g) s, the standard deviation of the sample.
 

Activity 20-2: Parents' Ages (cont.)

(a)

(b) Shade middle 95% of sketch.
(c) .025
(d) t*=2.032
(e) t* is greater than z*
(f) 22.31 + 2.032(5.60/sqrt(35)) = (20.39,  24.24)
Note: Answers to (g)-(i) below are the answers to (h)-(j) in the Calculator version.
(g) No, the distribution of ages at which the population of mothers had their first child probably does not follow a normal distribution (evidence: ages in sample are skewed to the right).  Normality of the population is not required for the use of this procedure to be valid.  If the population is not distributed normally, the procedure will be considered valid as long as n > 30.  In this case, n = 35.
(h) I'd be uncomfortable since the sample definitely has some skewness and an outlier. We need the larger sample size for the central limit theorem to kick in for the procedure to be valid.
(i) In the long-run, over many samples, 95% of the intervals generated would contain the mean age at first of first child for all mothers in the population.
 

Activity 20-3: Parents' Ages (cont.)

(a) Answers will vary from student to student.
(b) 22.52 + 1.96(4.89/sqrt(1199)) =  (22.24, 22.80);  this interval is much narrower than the confidence interval from Activity 20-2.
(c) Answers will vary from student to student.
(d) 22.52 + 1.645(4.89/sqrt(1199)) = (22.29, 22.75); this interval is narrower than the 95% confidence interval.
 

Activity 20-4: Sleeping Times

(a)
Sample number
Sample size
Sample mean
Sample std. dev.
3
30
6.6
.825
1
10
6.6
.825
2
10
6.6
1.597
4
30
6.6
1.597
(b) The sample means are equal.
(c) standard deviation
(d) sample size
(e) Sample 1 produces a more accurate estimate of m because it has a smaller standard deviation.
(f) Sample 3 produces a more accurate estimate of m because it has a larger sample size.
 

Activity 20-5: Sleeping Times (cont.)

Answers will vary from class to class.