Worksheet for recitation 12 STT 200, Lecture 5, Sec 23 and 24, 2013-4-2
 To construct confidence interval for 𝑝 with confidence level C :
 Determine the critical value 𝑧
∗
, either using Normal table, or R/calculator. To use R/calculator, note the total area below 𝑧 ∗
is C +
1−𝐶
2
=
𝐶+1
2
, so we can find 𝑧 ∗
using qnorm((C+1)/2) in R, or invnorm((C+1)/2) in DISTR in a Ti-83 Plus calculator. For example in a calculator:
 for 95% confidence, 𝑧
∗
= invnorm((0.95 + 1)/2) = 1.96
 for 90% confidence, 𝑧
∗
= invnorm((0.90 + 1)/2) = 1.645
 Find 𝑆𝐸(𝑝̂) = √ 𝑝̂(1−𝑝̂) 𝑛
 The margin of error is 𝑀𝐸 = 𝑧 ∗ 𝑆𝐸(𝑝̂) = 𝑧 ∗ √ 𝑝̂(1−𝑝̂) 𝑛
 The confidence interval is 𝑝̂ ± 𝑀𝐸 = (𝑝̂ − 𝑀𝐸, 𝑝̂ + 𝑀𝐸)
 To determine sample size, 𝑛 =
(𝑧 ∗ ) 2 𝑝̂(1−𝑝̂)
𝑀𝐸 2
 Relationship between margin of error (half of width of confidence interval) and proportion, sample size and confidence level:
0.0
0.2
0.4
p = 0.5
0.6
0.8
1.0
40 60 80 100 120 140 sample size n fixed confidence level = 95%, p = 0.5
fixed confidence level = 95%, n = 100
 To validate the confidence interval, we need to check several conditions:
0.80
0.85
0.90
0.95
confidence level C fixed n = 100, p = 0.5
 Independence condition: the n = 37 responses in the sample are chosen independently.
 Randomness condition: the n = 37 responses in the sample are chosen randomly. We used the table of random digits from 1 to 400.
 10% condition: the sample size n = 37 is less than 10% of the population size 400.
 Success/failure condition: the n = 37 responses in the sample contains at least 10 yeses and 10 nos. http://stt.msu.edu/users/zhangz19/stt200sp13.html Page 1

Worksheet for recitation 12 STT 200, Lecture 5, Sec 23 and 24, 2013-4-2
1.
11% of a random sample of 1003 adults approved of attempts to clone a human. a.
Find the margin of error if we want 95% confidence. b.
Explain what that margin of error means. c.
If we only need to be 90% confident, will the margin of error be larger or smaller? Explain. d.
Find that margin of error. e.
In general, if all other aspects of the situation remain the same, would smaller samples produce smaller or larger margin of error?
2.
In a random survey of 226 college students, 20 reported being “only” children. Estimate the proportion of students nationwide. a.
Check conditions for constructing a confidence interval. b.
Construct 95% confidence interval. c.
Interpret your interval. d.
Explain what “95% confidence” means in this context.
3.
74% of 1644 randomly selected college freshmen returned to college the next year. Estimate the national freshman-to-sophomore retention rate. a.
Verify that the conditions are met. b.
Construct a 98% confidence interval. c.
Interpret your interval. d.
Explain what “98% confidence” means in this context.
4.
In a University, it’s believed that 25% of adults between over 30 love Statistics. We wish to see if this percentage is the same among the 18 to 25 age group. a.
How many of this younger age group must we survey in order to estimate the proportion of those who love Statistics to within 5% with 90% confidence? b.
If we want to cut the margin of error in half, how many of this younger age group must we survey? Do you have any concerns about this sample? Explain. http://stt.msu.edu/users/zhangz19/stt200sp13.html Page 2