Download Problem A diagnostic test has a probability 98% of giving a positive

Problem A diagnostic test has a probability 98% of giving a positive result when applied to a person
suffering from a certain disease, and a probability 9.4% of giving a false positive. It is estimated that
1.8% of the population are sufferers.
What are the sensitivity, the specificity, and the positive predictive value of the test? Give the probabilities
in percentage with 2 decimal digits.
Solution
Sensitivity = 98.00%
Specificity = 90.60%
Positive predictive value = 16.04%
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Problem A diagnostic test has a probability 95.6% of giving a positive result when applied to a person
suffering from a certain disease, and a probability 8.3% of giving a false positive. It is estimated that
1.8% of the population are sufferers.
What are the sensitivity, the specificity, and the negative predictive value of the test? Give the probabilities in percentage with 2 decimal digits.
Solution
Sensitivity = 95.60%
Specificity = 91.70%
Negative predictive value = 99.91%
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Problem Here we define the events took aspirin, took a placebo, had a myocardial infarction (MI), did
not have MI.
Test the hypothesis that taking aspirin is independent of having a MI. Give the p-value in percentage
with 2 decimal digits.
MI
noMI
aspirin
69
8936
placebo
99
8906
Solution The p-value is 2.46%
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Problem We have the following data on the severity of head injuries a result of bicycle accidents tabulated for the cases where the cyclist weared an helm, or did not.
Test the hypothesis whether the severity is independent of wearing an helm. Give the p-value in percentage with 2 decimal digits.
none
minimal
minor
major
with helm
118
51
27
20
without helm
392
191
95
23
Solution The p-value is 0.35%
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Problem We seek to determine whether or not there is an association between gender and preference
for ice cream flavour. The data come from a survey of 695 people that ask for their preference of 1 of 3
ice cream flavours.
Test the hypothesis whether preferences are independent of sex. Give the p-value in percentage with 2
decimal digits.
chocolate
vanilla
strawberry
men
112
132
44
women
137
169
101
Solution The p-value is 0.93%
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Problem Which of the following hypothesis guarantees that
P (A ∪ B ) = P (A) + P (B )
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: A and B are exclusive.
This is problem "probability_boolean.1"
Please report any error.
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Problem Which of the following hypothesis guarantees that
P (A ∩ ¬B ) = P (A)
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: A and B are exclusive.
This is problem "probability_boolean.2"
Please report any error.
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Problem Which of the following hypothesis guarantees that
P (¬A ∩ ¬B ) = 1 − P (A) − P (B )
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: A and B are exclusive.
This is problem "probability_boolean.3"
Please report any error.
1
Problem Which of the following hypothesis guarantees that
¡
¢
P (A ∩ ¬B ) = P (A) 1 − P (B )
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: A and B are independent.
This is problem "probability_boolean.4"
Please report any error.
1
Problem Which of the following hypothesis guarantees that
P (¬A ∩ ¬B ) = 1 − P (A)P (B )
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: A and B are independent.
This is problem "probability_boolean.5"
Please report any error.
1
Problem Which of the following hypothesis guarantees that
P (A ∩ B ) = 1 − P (¬A)P (¬B )
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: A and B are independent.
This is problem "probability_boolean.6"
Please report any error.
1
Problem Which of the following hypothesis guarantees that
P (A ∪ ¬B ) = P (A)
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: A and B are exhaustive.
This is problem "probability_boolean.7"
Please report any error.
1
Problem Which of the following hypothesis guarantees that
P (A à B ) = P (A) − P (B )
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: A ⊆ B .
This is problem "probability_boolean.8"
Please report any error.
1
¢
Problem Which of the following hypothesis guarantees that
¢
P (¬A ∩ B ) = 1
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: A ⊆ B .
This is problem "probability_boolean.9"
Please report any error.
1
Problem Which of the following hypothesis guarantees that
P (A ∪ ¬B ) = 1
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: A ⊆ B .
This is problem "probability_boolean.10"
Please report any error.
1
Problem Which of the following hypothesis guarantees that
P (¬A ∩ ¬B ) = 1 − P (A)
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: None of the above..
This is problem "probability_boolean.11"
Please report any error.
1
Problem Which of the following hypothesis guarantees that
P (A ∪ B ) = P (A)P (B )
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: None of the above..
This is problem "probability_boolean.12"
Please report any error.
1
Problem Which of the following hypothesis guarantees that
¡
¢
P (A ∩ ¬A) = P (A) 1 − P (A)
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: None of the above..
This is problem "probability_boolean.13"
Please report any error.
1
Problem Which of the following hypothesis guarantees that
P (A ∩ A) = 2P (A)
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: None of the above..
This is problem "probability_boolean.14"
Please report any error.
1
Problem Which of the following hypothesis guarantees that
P (¬A ∩ ¬B ) = P (A)
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: None of the above..
This is problem "probability_boolean.15"
Please report any error.
1
Problem Which of the following hypothesis guarantees that
P (A ∪ B ) = 1 − P (B )
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: None of the above..
This is problem "probability_boolean.16"
Please report any error.
1
Problem Which of the following hypothesis guarantees that
P (A ∩ B ) = 1 − P (A)P (B )
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: None of the above..
This is problem "probability_boolean.17"
Please report any error.
1
Problem Which of the following hypothesis guarantees that
P (¬A ∩ ¬B ) = P (A) + P (B ) − 1
(1) A and B are exclusive
(2) A and B are independent
(3) A and B are exhaustive
(4) A ⊆ B
(5) None of the above.
Solution The correct answer is: A and B are exclusive.
This is problem "probability_boolean.18"
Please report any error.
1
Problem We roll a fair die with 8 faces. The faces are labeled by distict letters. What is the probability
that the same letter (no matter which) occurs 3 times in 5 rolls ? Give the probability in percentage with
2 decimal digits.
Solution
The probability that a given letter occurs k = 3 times in n = 5 rolls is
à !
n k
q =
p (1 − p)n−k = 0.015,
where p = 1/8.
k
Notice that two letters may not occur both k times (because 2k > n).
Hence the answer is 8 · q = 12%
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Problem We roll a fair die with 6 faces. The faces are labeled by numbers as usual. What is the probability that in a sequence of 5 rolls exactly 2 numbers (no matter which) appears ? Give the probability in
percentage with 2 decimal digits.
Solution
Below f = 6 is number of faces and n = 5 is number of rolls.
The sample space is {1, . . . , f }n .
©
ªn © ªn © ªn
For 1 ≤ a < b ≤ f , define E a,b = a, b à a à b .
µ ¶n
µ ¶n
´
³
1
2
−2
= 0.0039.
Define q = P E a,b =
f
f
We are asked to calculate the probability of the event E =
[
1≤a<b≤ f
©
ª ©
ª
Observe that if a, b 6= a 0 b 0 then E a,b ∩ E a 0 ,b 0 = ∅.
à !
³
´
X
¡ ¢
f
Therefore P E =
q = 5.79%.
P E a,b =
2
1≤a<b≤ f
1
E a,b .
Problem
We roll a fair die with 7 faces. The faces are labeled by numbers as usual. What is the probability that
the number 1 appears before 2 ? Give the probability in percentage with 2 decimal digits.
Solution
Below f = 7 is number of faces and n = 6 is number of rolls.
©
ªn
The sample space is 1, . . . , f .
©
ªi −1 © ª ©
ªn−i
For i = 1, . . . , f define E i = 3, . . . , f
× 1 × 1, . . . , f
©which isªnis the event to obtain 1 at the i -th
roll and that neither 1 not 2 have come out before. Let E 0 = 3, . . . , f .
We are asked to calculate the probability of the event E =
n
[
Ei .
i =0
Observe that if i 6= j then E i ∩ E j = ∅.
n
X
¡ ¢
¡ ¢
Therefore P E =
P Ei =
i =0
µ
1
f −2
¶n
+
n µ 1 ¶i
1X
= 3.6%.
f i =1 f − 2
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Problem Within a school district, students were randomly assigned to one of two teachers T1 and T2 .
After the assignment, T1 had 28 students, and T2 had 26 students.
At the end of the year, each class took the same standardized test. The students of T1 had an average
test score of 77, with a standard deviation of 12.7; and the students of T2 had an average test score of 79,
with a standard deviation of 14.5.
Test the hypothesis that T1 and T2 are equally effective teachers.
Solution
n 1 = 28
n 2 = 26
first/second sample sample size
s 1 = 12.7
s 2 = 14.5
s
s 12
s2
+ 2 = 3.7
s =
n1 n2
x̄ 1 = 77
first/second sample standard deviation
joint sample standard deviation
x̄ 2 = 79
first/second sample sample mean
x̄ 1 − x̄ 2
= −0.54
s
©
ª
ν = min n 1 , n 2 − 1 = 25
¡
¢
2 · P Tν < −|t | = 59.57 %
t =
value of the test statistic
degrees of freedom
p-value
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Problem A machine fills milk into 1000ml packages. It is suspected that the machine is not working
correctly and that the amount of milk filled differs from the setpoint 1000ml. A sample of 11 packages
filled by the machine are collected. The sample mean is equal to 997.8 ml and the sample variance is
equal to 2.92 ml.
With what significance level can we reject the hypothesis that the amount filled corresponds on average
to the setpoint. Give the p-value in percentage with 2 decimal digits.
Solution
µ = 1000
population mean (if H0 true)
n = 11 < 30
sample size
s = 2.92
sample standard deviation
x̄ = 997.8
t=
sample mean
x̄ − µ
p = −2.50
s/ n
value of the test statistic
n − 1 = 10
¡
¢
2 · P Tn−1 < −|t | = 3.15 %
degrees of freedom
p-value
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Problem Suppose the manufacturer claims that the mean lifetime of a light bulb is on average at least
10000 hours. In a sample of 11 light bulbs, it was found that they only last 9971 hours on average.
Assume the sample standard deviation is 101 hours.
With what significance level can we reject the hypothesis that the lifetime is at least 10000 hours? Give
the p-value in percentage with 2 decimal digits.
Solution
µ = 10000
population mean (if H0 true)
n = 11 < 30
sample size
s = 101
sample standard deviation
x̄ = 9971
t=
sample mean
x̄ − µ
p = −0.95
s/ n
value of the test statistic
n − 1 = 10
¡
¢
P Tn−1 < t = 18.17 %
degrees of freedom
p-value
1
Problem A machine fills water into 1000ml bottles. It is suspected that the machine is not working
correctly and that the amount of water filled is less than the setpoint 1000ml. A sample of 13 bottles
filled by the machine are collected. The sample mean is equal to 998.1 and the sample variance is equal
to 3.82.
With what significance level can we reject the hypothesis that the amount filled is at least the setpoint.
Give the p-value in percentage with 2 decimal digits.
Solution
µ = 1000
population mean
n = 13 < 30
sample size
s = 3.82
sample standard deviation
x̄ = 998.1
t=
sample mean
x̄ − µ
p = −1.79
s/ n
value of the test statistic
n − 1 = 12
¡
¢
P Tn−1 < t = 4.91 %
degrees of freedom
p-value
1
Problem Assume the height of a given population is normally distributed with standard deviation 8.74 cm.
We want to estimate the population mean with margin of error ± 1.3 cm. Find the minimal sample size
needed for 90 % confidence level.
Solution
α = 1 − (confidence level)/100 = 0.1
significance level
σ = 8.74
population standard deviation
is the value such that P (−z α/2 < Z ) = α/2 = 0.05
z α/2 = 1.64
n =
z α/2 σ2
²2
minimal sample size
The sample size needs to be at least 123.
1
Problem A survey is conducted to analyze the daily expenses of summer tourists. A sample of 133
tourists spend on average 153.8 €. The sample variance is 71.53 €.
Determine a 99 % confidence interval for the average daily expenses of a tourist. Give lower and upper
bound. Round the result to two decimal digits.
Solution
α = 1 − (confidence level)/100 = 0.01
significance level
n = 133 > 30
sample size
σ ≈ s = 71.53
population standard deviation
x̄ = 153.8
sample mean
z α/2 = 2.58
is the value such that P (−z α/2 < Z ) = α/2 = 0.005
σ
ε = z α/2 p
n
margin of error
x̄ − ε = 137.82
lower bound
x̄ + ε = 169.78
upper bound
1