I. Determine whether each of t
I. Determine whether each of the following random variables hasa binomial distribution. If it does, identify the values of theparameters n and p. If not, explain.
a) X is the number of times five is rolled in 15 rolls of afair, six-sided die.
b) X is the number of multiple-choice questions a student getsright on a 30-question test, when each question has four choicesand the student is completely guessing on each question.
c) X is the same as (b), but the first 15 questions have fourchoices and the last 15 have three answer choices.
d) X is the number of women in a random sample of 12 students,from a class comprising 25 women and 20 men.
e) X is the total weight of 16 randomly selected oranges
f) X is the number of oranges, out of a random sample of 16,that weight more than 170 grams
Answer:
Binomial Distribution
If ‘X’ is the randomvariable representing the number of successes, the probability ofgetting ‘r’ successes and‘n-r’failures, in ‘n’ trails,‘p’probability of success ‘q’=(1-p) is givenby the probability function
a.X is the number of times five is rolled in 15 rolls of a fair,six-sided die.
Success : Five is rolled
X :Number of successes: Number of times five is rolled:
n : Number of trails : 15 rolls of a fair die
p: Probability of success : Probability of rolling a five=1/6
Hence ‘X’ has a binomial distribution with n=15 and p=1/6
b. X is the number of multiple-choice questions a student getsright on a 30-question test, when each question has four choicesand the student is completely guessing on each question.
Success : questions a student gets right
X :Number of successes: number of multiple-choice questions astudent gets right
n : Number of trails : 30-question test
p: Probability of success : Probability of getting a questionright = 1/4 (As number of choices is 4)
Hence ‘X’ has a binomial distribution with n= 30 and p=1/4
c) X is the same as (b), but the first 15 questions have fourchoices and the last 15 have three answer choices
p : Probability of success : Probability of getting a questionright for the first 15 questions= 1/4 (As number of choices is 4)and : Probability of getting a question right for the last 15questions= 1/3 (As number of choices is 3)
As ‘p’: probability is not same across the trails , ‘X’ does notfollow Binomial distribution.
d)
X is the number of women in a random sample of 12 students, froma class comprising 25 women and 20 men
Success : Having a women in the random sample
X :Number of successes: number of women in a randomsample
n : Number of trails : sample size of the random sample = 12
p: Probability of success : Probability of selecting a women =Number of women in the class / Total number of (women+Men) =25/25+20=25/45 = 5/9
Hence ‘X’ has a binomial distribution with n=12 and p=5/9
e)
X is the total weight of 16 randomly selected oranges
X : Number of successes ;
As X : Total weight can not be defined as a success;
‘X’ does not follow Binomial distribution.
f) X is the number of oranges, out of a random sampleof 16, that weight more than 170 grams
Success : Orange that weight more than 170 grams
X : Number of success : Number of oranges whose weight more than170 grams
n : number of trails = Number of oranges in the random sample =16
p: Probability that an orange weighs more than 170 grams.(Notknown)
Therefore ‘X’ has a binomial distribution with n= 12 and p :Probability that an orange weighs more than 170 grams: notknown
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