#17 22S:8 (confidence intervals) (Russo)
(1) Suppose we observe a single value X from the following population of 25:
m-2 m-1 m m+1 m+2
m-3 m-2 m-1 m m+1 m+2 m+3
m-5 m-4 m-3 m-2 m-1 m m+1 m+2 m+3 m+4 m+5
What is m? Answer: you don't know & you will NEVER know ?
There is an 84% chance that we choose a value X that is within _____ of m.
Equivalently, there is an 84% chance that m is within _____ of the chosen value X.
Equivalently, the interval [ X - , X + ] has an 84% chance of containing m. This interval is called an 84% confidence interval for the unknown parameter m.
Find a 92% CI for m Find a 68% CI for m
(2) Suppose we observe a single value X from the population:
r-4 r-2 r-1 r r+1 r+2 r+4
r-5 r-4 r-3 r-2 r-1 r r+1 r+2 r+3 r+4 r+5
Find a 90% CI for r Find a 100% CI for r
(3) Suppose we observe a single value X from the following normal population:
σ = 12 ?
Find a 90% confidence interval for μ Find a 95% confidence interval for μ
#18 22S:8 (CI's for means & proportions) (Russo)
Behavior of a sample mean ?
? Behavior of a sample proportion ______________________________________
confidence intervals: + +
(1) 124 cars are clocked passing a checkpoint on I-380. It is found that their average
speed is 72.28 mph, with SD = 4.2 mph. Find a 90% confidence interval for the avr.
speed of ALL cars passiing the checkpoint.
Find a 95% confidence interval.
(2) Suppose 234 of 500 randomly chosen New Yorkers favor Hillary C. for Senator.
Find a 95% CI for the proportion of ALL NY-ers who support Hillary.
Suppose 464 of 1000 NY-ers favor Hillary (note: sample proportion is the same).
Find a 98% CI based on the poll of 1000 voters
#19 22S:8 (Hypothesis testing) (Russo)
(1) A coin is tossed 10 times. We wish to test
Ho: the coin is fair (p = .5) vs. H1: the coin is biased in favor of heads (p > .5)
TEST 1: reject Ho if the coin lands heads 10 times.
significance level = P(reject Ho | Ho is true) =
power vs. alternative p = P(reject Ho | p is true) =
alternative p = .6 .7 .8 .9 .95 1.0
power of test =
Suppose all 10 tosses are heads. What is the p-value ?
TEST 2: reject Ho if the coin lands heads > 8 times.
A B C
(2) One of 3 spinners is spun once
Ho: spinner A was spun.
H1: B or C was spun
TEST: reject Ho if
significance level = power vs. B = power vs. C =
(3) Breaking strengths of Acme steel rods are normally distributed with unknown
mean = μ and known σ = 28 lbs. A rod is chosen randomly. We want to test
Ho: μ = 2000 vs. H1: μ > 2000 at significance level = .05
TEST: reject Ho if
Find: power vs. = 2025 power vs. = 2050
Suppose the rod has br. strength = 2042. Then p-value =
(1) A long time ago, Soutwestern Mondesi Province was occupied by two ancient tribes,
the Anhalami and the Bolgomoni. It is known that male heights among the Anhalami
were normally distributed with mean = 62 inches and SD = 1.2 inches, while male
heights among the Bolgomoni were normally distributed with mean = 66 inches and SD
= 1.8 inches. Recently, two University of Iowa archeologists discovered the skeletal
remains of a male, but were unsure as to his tribal membership. Let X denote the height
of this individual.
Devise a test with significance level = .05 of
Null Hypothesis: the male was from the Anhalami tribe
vs. Alt. Hypothesis: the male was from the Bolgomoni tribe
Test: Reject the null hyp. if
Find the power of this test against the alternative.
Suppose the male measured 65.4 inches. Find the p-value.
(2) Suppose a group of 4 male skeletons is found, and all are assumed to be from one of
the two tribes mentioned. Let denote the average height of the six. Devise an = .01 test.
Our test: Reject the null hyp. if
Find the power of this test against the alternative.
Suppose the 4 males measured 61.8, 64.0, 64.5, 67.7 inches. Find the p-value.
(3) Some archeologists are ok with the normality & mean value assumptions of (1)&(2),
but are uncomfortable with the SD assumptions. Find a test, and compute the p-value.
#20 (Hyp testing & CI problems) (Russo)
parameter Test statistic Conf. Int.
??Xzsµ (large sample) X?s/nn
??Xtsµ (small sample) use t, df = n-1 X?s/nn
22ss(X-X)-012µ - µ(large sample) (X-X)?z?12 1212nn2212ss12?nn12
ˆˆˆˆˆˆ(p?p)?0p(1?p)p(1?p)121122ˆˆp – p(p?p)?z?12 12nnp(1?p)p(1?p)12?nn12
p = pooled sample proportion
µ - µ(small sample) use t, df = n + n – 2 12 12
22(X-X)-0(n-1)s?(n?1)s11121122 (X-X)?t(?) 1222n?n?2nn(n-1)s?(n?1)s1212111122(?)n?n?2nn1212
µ (large sample) p µ (small sample)
- µ(large sample) p – pµ - µ(small sample) µ12 12 12
Find the p-value of the appropriate test . Also find 90% & 95% CI's for the parameter.
1. Researchers are interested in the effectiveness of a new allergy drug. In particular, they
are interested in knowing whether most allergy patients would benefit from taking it. 432 of 800
randomly chosen allergy patients benefited by taking the new drug.
hyp: 50% of all allergy patients would benefit from taking this drug vs.
alt: more than 50% would benefit p ________________________________________________________________________
2. Average life of 400 ACME car batteries is 2004 days with SD = 35.5 days.
hyp: average life of ALL bateries = 2000 days vs.
alt: average life of ALL bateries > 2000 days µ (large sample) _______________________________________________________________________
3. 54 of 225 randomly chosen boron absorbing rods have cracks.
hyp: percentage of ALL rods with cracks = 20% vs.
alt: percentage of ALL rods with cracks > 20% p ________________________________________________________________________
4. Average caloric content of 6 fast food burgers at a local chain = 768, SD = 171.4
hyp: average caloric content of ALL burgers = 900 vs.
alt: average caloric content of ALL burgers < 900 µ (small sample) ________________________________________________________________________
5. 15 of 60 randomly chosen Iowans are worried about the economy. 24 of 80 randomly
chosen Californians are worried about the economy.
hyp: same percentage of worry in each state vs.
alt: higher percentage in CA p – p 12 ________________________________________________________________________
6. 50 Friday customers at Backwater Mike's: avr. tip = $2.40 SD = .60
80 Saturday customers at Backwater Mike's: avr. tip = $2.82 SD = .80
hyp: Friday & Saturday tips the same., on average vs.
hyp: average tip higher on Saturday µ - µ(large sample) 12 ________________________________________________________________________
7. Use a chi-square test (based on 100 spins) to determine if the spinner is working, so that
P(red) = 1/2 P(green) = 1/4 P(blue) = 1/4 Observe 58 red, 20 green, 22 blue ________________________________________________________________________
8. 108 of 144 randomly chosen cars in Iowa are found to be in violation of U.S. emission
standards. Let p = the proportion of cars in Iowa in violation. DOT wishes to test
Ho: p = .8 vs. H1: p < .8 at the α = .05 level. p
Find the power of the test vs. p = .7 [ power vs. p = .7 = prob that we reject the
hypothesis when p = .7]
9. In pblm (8), let µ denote the average odometer reading of observed cars. DOT also
wishes to test Ho: = 45,000 vs H1 > 45,000 Avr reading of the cars is
47,186 miles, SD = 12,024 miles µ (large sample) ________________________________________________________________________
10. The 5 readings (in meters) given by a rangefinder in repeat measurements of the distance
between an observer and a bridge are independent n(, 4) random variables. Suppose
average reading is 1802.6 m. We wish to test Ho: µ = 1800 vs. H1: µ > 1800
where µ denotes the true distance. µ (small sample) ________________________________________________________________________
11. Continuing (8), suppose 82 of 100 randomly chosen Minnesota cars are in violation of the U.S.
standard. DOT wishes to test Ho: The proportions of Iowa and Minn. are the same vs.
H1: the proportion in Minn. is higher. p – p 12 ________________________________________________________________________
12. The mean breaking strength of 120 ACME steel rods is 1026 lbs. with SD = 28 lbs. The mean
breaking strength of 200 GlaxCo steel rods is 1018 lbs. with SD = 16 lbs. Are ACME rods
better, on average, that GlaxCo? µ - µ(large sample) 12 ________________________________________________________________________
13. The number of particles of a pollutant in a sample of one cubic meter of air collected near
Schaeffer Hall was measured. Results : 126 154 163 133 118 125 158 143
The observations are assumed to be from a normal distribution whose mean is the actual number
of particles present in the air sample. We wish to test Ho: μ = 150 vs. H1: μ < 150
µ (small sample) ________________________________________________________________________
14. 10 subjects are given drug A, and 8 subjects drug B. The change in their diastolic blood pressures
are recorded and the following statistics calculated:
Drug A: avr. change = +8.2, SD = 2.1 Drug B: avr. change = +5.8, SD = 1.6
Let μ and μ denote the avr pressure change (in the population) caused by A & B, respectively AB
We wish to test H: μ - μ = 0 vs. H: μ - μ > 0 What assumptions must we make ? oABaAB
µ - µ(small sample) 12 ________________________________________________________________________
15. Anxiety levels are measured in 10 subjects before and after viewing a horror film.
before: 56 68 42 70 58 60 60 62 59 68
after: 64 75 40 78 55 62 65 61 70 75
Let d = the average change in anxiety level within the horror film viewing public.
We wish to test Ho: d = 0 vs. H1: d > 0
What distributional assumption must we make ? µ (small sample)
16. Is "heads" more likely when a coin is spun on a table than when it is flipped in the air ?
100 spins on the table, 62 heads 100 flips in the air, 54 heads p – p 12 ________________________________________________________________________
17. Are ACME steel rods stronger (on average) than Ming rods ?
100 ACME rods: avr. breaking strength = 1006 lbs. SD = 28 lbs.
80 Ming rods: avr. breaking strength = 1002 lbs. SD = 18 lbs.
µ - µ(large sample) 12
18. Is drug A the same as drug B ? or is A better ? 400 patients take drug A, 224 improve
250 take drug B, 118 improve. p – p 12 ________________________________________________________________________
19. Is gas the same price in Iowa (on average) as it is in Minnesota ? Or is it more expensive
in Minnesota ? A random sample of 40 gas prices in Iowa & 50 gas prices in Minnesota
resulted in the observations: Iowa avr. = 1.04, SD = .02 Minn. avr. = 1.06, SD = .025
µ - µ(large sample) 12
20. The scores on an aptitude test are assumed to be normally distributed. Eight prospective
employees are randomly chosen from a large group of applicants. The sample mean =
76 and the sample SD = 4.2 . H: μ = 80 vs. H: μ < 80. µ (small sample) oa
21. Two types of tire treads are compared. 6 cars are each equipped with a "type A" & a
"type B" tire (randomly placed in the front). The differences in wear between the tires is
noted for each car. We wish to test Ho: A same as B vs. H1: A better than B.
CAR 1 2 3 4 5 6
A 48 55 49 51 56 50
B 45 53 50 48 52 46 µ (small sample)
22. A large company believes that more than 20% of its employees arrive late for work. On a
Monday morning in November, 256 randomly chosen employees are observed. Of these,
60 are late. p
23. A cereal company claims that 40% of its boxes contain a prize. A consumer group
doubts this claim. 625 randomly chosen boxes are opened. 228 contain a prize. p ________________________________________________________________________
24. The consumer group in (23) measured the weights of the sampled boxes of cereal. They
found avr. weight = 15.88 ounces, SD = .36 . Ho: µ = 16 oz vs. H1: µ < 16 oz
µ (large sample)
25. A quick check of the records shows that the 256 employees of problem (22) averaged
12.4 days absent in 1996, with SD = 2.8 . Ho: µ = 12 vs. H1: µ > 12.
µ (large sample)
26. Five 1999 BMW's are crashed into a brick wall at 10 mph. Average damage = $2,162
SD = $268. Ho: = $2000 vs. H1: > $2000. µ (small sample)
27. 52 of 100 randomly chosen college graduates support O'Brien for Congress.
92 0f 150 randomly chosen non-college graduates support O'Brien for Congress.
Does a higher percentage of non-college grads support O'Brien ? p – p 12 ________________________________________________________________________
28. 50 Crinkly french fries: Avr calories = 12.4 SD = 2.6
80 Home Style french fries: Avr. calories = 13.2 SD = 1.8
µ - µ(large sample) 12
Does the average Home Style FF have more calories than the average Crinkly ?
29. 8 patients take a drug. Their before & After blood pressures are measured.
before: 68 62 78 56 90 88 71 70
after 72 60 86 62 88 90 78 78
What can you conclude ? Does the drug increase BP ? µ (small sample) ________________________________________________________________________
30. A simple random sample of 4 "one-pound" cans of ACME coffee were weighed. The
measurements (in ounces) were 15.8 15.7 15.3 16.1
Does this data indicate that the true average weight of all "one-pound" cans of ACME
coffee is less than 16 ounces? µ (small sample)
31. Eight deluxe burgers purchased at Sky Burger were measured for their caloric content.
The following results were obtained: average caloric content = 1032, SD = 171.4 Is
this significant evidence that the average caloric content of Sky Burger deluxe burgers
exceeds 950? (You may assume that caloric content of Sky Burger deluxe burgers is
normally distributed.) µ (small sample)
#21 22S:8 (The Poisson random variable) (Russo)
Consider a random variable X with the following pdf:
??ke?P(X?k)?fork?0,1,2,... (recall k! = k(k-1)(k-2)…(3)(2)(1), 0! = 1) k!
is a positive constant (a parameter). It can be shown that E(X) = Var(X) = . ??
The Poisson variable is used to model the occurrences of events over intervals of time.
Example 1 Suppose that the number of customers entering a clothing store over a one hour period is Poisson distributed with mean = 12. Find the probability that exactly 10
customers enter the store between 1PM & 2PM.
Find the probability that no customers enter the store between 3PM & 4PM.
Example 2 The number of earthquakes occurring in Umbria during a 10 year period is Poisson distributed with mean = 4. Find the probability that exactly 2 quakes occur in
Umbria during the next three years.
3The number E over the next three years is Poisson() =Poisson(1.2) distributed, so 4?10
Example 3 In (1), find the probability that exactly 12 customers enter the store between 1PM and 2:30PM.
The number W entering the store over a 1.5 hour period has a Poisson(18) distribution, so
Example 4 In (2), find the probability that at least one quake occurs over a two year