Download Binomial Part I

Binomial Part I
In calculating probabilities use Excel, a TI calculator or some other device.
Excel: P(X = k) = Binomdist(k,n,p,false)
TI-83: P(X = k) = binomialpdf(n,p,k)
Excel: P(X  k) = Binomdist(k,n,p,true)
TI83: P(X ≤ k) = binomialcdf(n,p,k)
Excel: : P(X  k) = 1 - Binomdist(k - 1,n,p,true) TI83: P(X  k) = 1 - binomialcdf(n,p,k - 1)
1. An breast cancer test is 95% effective at detecting the cancer in people who have the cancer.
a. Suppose that 50 people who have the cancer take the test. How many are you expecting to test positive?
b. Let the random variable X count how many people out of the 50 that have cancer and take the test, result in a
positive test (cancer detected). Go through the criteria posed in your book to determine that the random
variable X is binomial. What is the mean, , and standard deviation, , of X?
c. What is the probability that 48 people out of the 50 test positive? Use function notation to indicate what you
are doing.
d. What is the probability that 48 or more out of the 50 test positive? Use function notation to indicate what
you are doing.
X
. If we only sample 4 people with breast cancer, what are the possible values of
n
p̂ (what is the sample space of p̂ )? What is the standard deviation of p̂ , the sample proportion? What is the
e. The variable p̂ =
mean of p̂ ?
2. A fair die is tossed 120 times, and X counts the number of times a “one” appears.
a. Verify that the random variable X is binomial. Go through the four requirements.
b. How many times do you expect a “one” to appear if it is tossed 120 times?
c. What is the probability that a “one” appears 20 times?
d. What is the probability that a “one” appears 15 times or less when you toss it 120 times?
e. What is the probability that a “one” appears 25% of the time or less?
3. The length of a human pregnancy is approximately normally distributed with a mean of 266 days and a
standard deviation of 16 days. What is the probability that a sample average of 40 pregnancies last less than
262 days?
4. The length of a human pregnancy is approximately normally distributed with a mean of 266 days and a
standard deviation of 16 days. What is the probability that in a sample of 40 pregnancies, less than 10, last more
than 266 days?
5. A ten-question multiple choice quiz has five choices per question. The quiz is written in a language you do
not understand, (and please don’t say it is written in Statistics) so you decide to guess at every question.
a. What is the probability that you guess right on any question?
b. Let the random variable Y count the number of times you answer a question correctly. Is the random variable
Y binomial? Go through the four requirements.
c. What is the expected number of questions that you expect to answer correctly?
d. What is the probability that you answer half of the question correctly?
e. What is the probability that you answer over half of the questions correctly?
f. What is the probability that you answer 70% of the questions correctly (a passing grade)?
g. What is the mean of the counts for ten-question quiz if you guess at every answer (5 choices per question)?
What is the mean of the sample proportion for ten-question quiz if you guess at every answer (5 choices per
question)?
h. What is the standard deviation of the counts for ten-question quiz if you guess at every answer (5 choices per
question)? What is the standard deviation for the sample proportion for ten-question quiz if you guess at every
answer (5 choices per question)?
Answers
1a. 47.5 tests should come back positive.
1b. μ X = 47.5, σ X = 1.541
1c. P(X = 48) = binomdist(48, 50, 0.95, false) = binomialpdf(50, 0.95, 48) = 0.2611
1d. P(X  48) = 1 – binomdist(47, 50, 0.95, true) = 0.5405
1e. {0, 0.25, 0.5, 0.75, 1}, σ p̂ 
2b. 20
0.95(0.05)
 0.1541 ,
4
μ p̂ = 0.95
2c. P(X = 20) = binomdist(20, 120, 1/6, false) = binompdf(120, 1/6, 20) = 0.097301
2d. P( X ≤ 15) = binomdist(15, 120, 1/6, true) = 0.1334
2e. P(X ≤ 30) = 0.9929



262  266 
3. P( Y < 262) = P  Z 
 = P(Z< -1.58) = 0.0571 Where Y measures the length of a pregnancy.
16




40


4. The probability that one pregnancy lasts more than 266 days is 0.5, since the distribution is normal with  =
266. Let X count the number of pregnancies out of a sample of 40 that last more than 266 days.
P(X ≤ 9) = binomdist(9, 40, 0.5, true) = binomialcdf(40, 0.5, 9) = 0.00034.
5a. 0.2
5c. 2
5d. P(X = 5) = binomdist(5, 10, 0.2, false) = 0.0264
5e. P(X ≥ 6) = 1 – binomdist(5, 10, 0.2, true) = 0.0064.
5g. 2, 0.2. 5h. 1.265, 0.1265.
5f. P(X = 7) = 0.0008