Stats without Tears
Solutions for Chapter 11
Updated 1 Jan 2016
(What’s New?)
Copyright © 2013–2023 by Stan Brown, BrownMath.com
Updated 1 Jan 2016
(What’s New?)
Copyright © 2013–2023 by Stan Brown, BrownMath.com
(a) Use MATH200A part 5 and select 2pop binomial
. You have no
prior estimates, so enter 0.5 for p̂_{1} and p̂_{2}. E is 0.03,
and CLevel is 0.95. Answer: you need
at least 2135 per sample, 2135 people under 30 and 2135
people aged 30 and older. Here’s what it looks like, using
MATH200A part 5:
Caution! Even if you don’t identify the groups, at least you must say “per sample”. Plain “2135” makes it look like you need only that many people in the two groups combined, or around 1068 per group, and that is very wrong.
Caution! You must compute this as a twopopulation case. If you compute a sample size for just one group or the other, you get 1068, which is just about half of the correct value.
If you don’t have the program, you have to use the formula: [p̂_{1}(1−p̂_{1})+p̂_{2}(1−p̂_{2})]·(z_{α/2}/E)². You don’t have any prior estimates, so p̂_{1} and p̂_{2} are both equal to 0.5. Multiply out p̂_{1} × (1−p̂_{1}) × p̂_{2} × (1−p̂_{2}) to get .5.
Next, 1−α = 0.95, so α = 0.05 and α/2 = 0.025. z_{α/2} = z_{0.025} = invNorm(1−0.025). Divide that by E (.03), square, and multiply by the result of the computation with the p̂’s.
(b) Using MATH200A Program part 5 with .3, .45, .03, .95 gives 1953 per sample.
Alternative solution: Using the formula, .3(1−.3)+.45(1−.45) = .4575. Multiply by (invNorm(1−.05/2)/.03)² as before to get 1952.74157 → 1953 per sample.
Again, you must do this as twopopulation binomial. If you do the under30 group and the 30+ group separately, you get sample sizes of 897 and 1057, which are way too small. If your samples are that size, the margins of error for under30 and 30+ will each be 3%, but the margin of error for the difference, which is what you care about, will be around 4.2%, and that’s greater than the desired 3%.
(a) You have numeric data in two independent samples. You’re testing the difference between the means of two populations, Case 4 in Inferential Statistics: Basic Cases. (The data aren’t paired because you have no reason to associate any particular Englishman with any particular Scot.)
(1)  Population 1 = English; population 2 = Scots.
H_{0}: μ_{1} = μ_{2} (or μ_{1}−μ_{2} = 0) H_{1}: μ_{1} > μ_{2} (or μ_{1}−μ_{2} > 0) 

(2)  α = 0.05 
(RC)  The problem states that samples were random.
For English, r=.9734 and crit=.9054; for Scots, r=.9772 and
crit=.9054. Both r’s are greater than crit, so both are
nearly normally distributed. The stacked boxplot shows no outliers.
And obviously the samples of 8 are far less than 10% of the
populations of England and Scotland.

(3/4)  English numbers in L1, Scottish numbers in L2.
2SampTTest with Data; L1, L2, 1, 1, μ_{1}>μ_{2}, Pooled:No Outputs: t=1.57049305 → t = 1.58, p=.0689957991 → p = 0.0690, df=13.4634, x̅1=6.54, x̅2=4.85, s1=1.91, s2=2.34, n1=8, n2=8 
(5)  p > α. Fail to reject H_{0}. 
(6) 
At the 0.05 level of significance,
we can’t say whether English or Scots have a stronger liking for soccer.
Or, We can’t say whether English or Scots have a stronger liking for soccer (p = 0.0690). 
(b) Requirements are already covered.
2SampTInt, CLevel=.90
Results: (−.2025, 3.5775)
We’re 90% confident that, on a scale from 1=hate to 10=love, the average Englishman likes soccer between 0.2 points less and 3.6 points more than the average Scot.
(a) This is the difference of proportions in two populations, Case 5 in Inferential Statistics: Basic Cases.
(1)  Population 1 = English, population 2 = Scots.
H_{0}: p_{1} = p_{2} (or p_{1}−p_{2} = 0) H_{1}: p_{1} ≠ p_{2} (or p_{1}−p_{2} ≠ 0) 

(2)  α = 0.05 
(RC) 

(3/4)  2PropZTest x1=105, n1=150, x2=160, n2=200, p1≠p2
results: z=−2.159047761 → z = −2.16, p=.030846351 → p = 0.0308, p̂_{1} = 0.70, p̂_{2} = 0.80, p̂ = 0.7571428751 
(5)  p < α. Reject H_{0} and accept H_{1}. 
(6) 
The English and Scots are not equally likely to be soccer fans, at the 0.05 level of significance;
in fact the English are less likely to be soccer fans.
Or, The English and Scots are not equally likely to be soccer fans, (p = .0308); in fact the English are less likely to be soccer fans. 
(b) Requirements already checked.
2PropZInt with CLevel = .95 → (−.1919, −.0081)
That’s the estimate for p_{1}−p_{2}, English minus Scots. Since that’s negative, English like soccer less than Scots do. With 95% confidence, Scots are more likely than English to be soccer fans, by 0.8 to 19.2 percentage points.
(c) [(−.0081) − (−.1919)] / 2 = 0.0919, a little over 9 percentage points.
(d) MATH200A part 5, 2pop binomial, p̂_{1}=.7, p̂_{2}=.8, E=.04, CLevel .95 gives 889 per sample
By formula, z_{α/2} =
z_{0.025} = invNorm(1−0.025) = 1.96.
n_{1} = n_{2} =
[.7(1−.7)+.8(1−.8)]×(1/96/.04)² =
888.37 → 889 per
sample
(a) This is beforeandafter paired data, Case 3 in Inferential Statistics: Basic Cases. You’re testing the mean difference.
(1)  d = After−Before
H_{0}: μ_{d} = 0, running makes no difference in HDL H_{1}: μ_{d} > 0, running increases HDL Remark: If this was a research study, they would probably test for a difference in HDL, not just an increase. Maybe this study was done by a fitness center or a runningshoe company. They would want to find an increase, and HDL decreasing or staying the same would be equally uninteresting to them. 

(2)  α = 0.05 
(RC) 
Before in L1, After in L2, L3=L2−L1

(3/4) 
TTest 0, L3, 1, μ>0
results: t=3.059874484 → t = 3.06, p=.0188315555 → p = 0.0188, d̅=4.6, s=3.36, n=5 
(5)  p < α. Reject H_{0} and accept H_{1}. 
(6) 
At the 0.05 level of significance, running 4 miles daily for six months raises HDL level.
Or, Running 4 miles daily for six months raises HDL level (p = 0.0188). 
(b) TInterval with CLevel .9 gives (1.3951, 7.8049).
Interpretation: You are 90% confident that running an average of four miles a day for six months will raise HDL by 1.4 to 7.8 points for the average woman.
Caution! Don’t write something like “I’m 90% confident that HDL will be 1.4 to 7.8”. The confidence interval is not about the HLD level, it’s about the change in HDL level.
Remark: Notice the correspondence between hypothesis test and confidence interval. The onetailed HT at α = 0.05 is equivalent to a twotailed HT at α = 0.10, and the complement of that is a CI at 1−α = 0.90 or a 90% confidence level. Since the HT did find a statistically significant effect, you know that the CI will not include 0. If the HT had failed to find a significant effect, then the CI would have included 0. See Confidence Interval and Hypothesis Test.
(a) Each participant either had a heart attack or didn’t, and the doctors were all independent in that respect. This is binomial data. You’re testing the difference in proportions between two populations, Case 5 in Inferential Statistics: Basic Cases.
(1) 
Population 1: Aspirin takers; population 2: nonaspirin takers.
H_{0}: p_{1} = p_{2}, taking aspirin makes no difference H_{1}: p_{1} ≠ p_{2}, taking aspirin makes a difference 

(2)  α = 0.001 
(RC) 

(3/4) 
2PropZTest: x1=139, n1=11037, x2=239, n2=11034, p1≠p2
results: z=−5.19, pvalue = 2×10^{7}, p̂_{1} = .0126, p̂_{2} = .0217, p̂ = .0171

(5)  p < α. Reject H_{0} and accept H_{1}. 
(6) 
At the 0.001 level of significance, aspirin does make a difference to the likelihood of heart attack.
In fact it reduces it.
Or, Aspirin makes a difference to the likelihood of heart attack (p < 0.0001). In fact, aspirin reduces the risk. 
Remark The study was conducted from 1982 to 1988 and was stopped early because the results were so dramatic. For a nontechnical summary, see Physicians’ Health Study (2009) [see “Sources Used” at end of book]. More details are in the original article from the New England Journal of Medicine (Steering Committee 1989 [see “Sources Used” at end of book]).
(b) 2PropZInt
with CLevel .95 gives (−.0125,
−.0056).
We’re 95% confident that 325 mg of aspirin every other day reduces the chance of heart attack by 0.56 to 1.25 percentage points.
Caution! You’re estimating the change in heartattack risk, not the risk of heart attack. Saying something like “with aspirin, the risk of heart attack is 0.56 to 1.25%” would be very wrong.
(a) You’re estimating the difference in means between two populations. This is Case 4 in Inferential Statistics: Basic Cases. Requirements:
Population 1 = Cortland County houses, population 2 = Broome
County houses.
2SampTInt, 134296, 44800, 30, 127139, 61200, 32, .95, No
results: (−20004, 34318)
June is 95% confident that the average house in Cortland County costs $20,004 less to $34,318 more than the average house in Broome County.
(b) A 95% confidence interval is the complement of a significance test for ≠ at α = 0.05. Since 0 is in the interval, you know the pvalue would be >0.05 and therefore June can’t tell, at the 0.05 significance level, whether there is any difference in average house price in the two counties or not.
If both ends of the interval were positive, that would indicate a difference in averages at the 0.05 level, and you could say Cortland’s average is higher than Broome’s. Similarly, if both ends were negative you could say Cortland’s average is lower than Broome’s. But as it is, nada.
Remark: Obviously Broome County is cheaper in the sample. But the difference is not great enough to be statistically significant. Maybe the true mean in Broome really is less than in Cortland; maybe they’re equal; maybe Broome is more expensive. You simply can’t tell from these samples.
The immediate answer is that
those are proportions in the sample, not the proportions among all voters.
This is twopopulation binomial data, Case 5 in Inferential Statistics: Basic Cases.
Requirements check:
Population 1 = Red voters, population 2 = Blue
voters.
2PropZInt 520, 1000, 480, 1000, .95
Results: (−.0038, .08379), p̂_{1}=.48, p̂_{2}=.52
With 95% confidence, the Red candidate is somewhere between 0.4 percentage points behind Blue and 8.4 ahead of Blue. The confidence interval contains 0, and so it’s impossible to say whether either one is leading.
Remark: Newspapers often report the sample proportions p̂_{1} and p̂_{2} as though they were population proportions, but now you know that they aren’t. A different poll might have similar results, or it might have samples going the other way and showing Blue ahead of Red.
(b) For a hypothesis test, we often use “at least 10 successes and 10 failures in each sample” as a shortcut requirements test, but the real requirement is at least 10 successes and 10 failures expected in each sample, using the blended proportion p̂. If the shortcut procedure fails, you must check the real requirement. In this problem, the blended proportion is
p̂ = (x_{1}+x_{2})/(n_{1}+n_{2}) = (7+18)/(28+32) =25/60, about 42%.
For sample 1, with n_{1} = 28, you would expect 28×25/60 ≈ 11.7 successes and 28−11.7 = 16.3 failures. For sample 2, with n_{2} = 32, you would expect 32×25/60 ≈ 13.3 successes and 32−13.3 = 18.7 failures. Because all four of these expected numbers are at least 10, it’s valid to compute a pvalue using 2PropZTest.
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