An administrator takes a random sample of 21 students from one section STAT 250.

Question

An administrator takes a random sample of 21 students from one section STAT 250. The random variable is defined as the number of students out of 21 that earn an A in STAT 250. Assume that it is known that probability of earning an A in STAT 250 is 12% and assume that the students’ grades are independent. Answer the following questions.

a) Check if this situation fits the binomial setting. Write four complete sentences addressing each requirement in one sentence each.

b) Assuming it does, build the probability distribution in table form in StatCrunch. There are two ways to do this. You may use Data Compute Expression and choose the function dbinom. This method relies on you entering the values of the random variable in the first column of your data table. The other way to do this is to use the binomial calculator and calculate the probability of each of the values of the random variable from X = 0 to X = 9. Since the probabilities will be very close to 0 after about X = 9, you should cut off the table after X = 9. You may present this table horizontally or vertically.

c) Calculate the probability that exactly three students earned an A using the StatCrunch binomial calculator. Copy this image from StatCrunch into your document. Then, once you obtain your answer, write one sentence to explain what the probability means in context of the question.

d) Calculate the probability that at least one student earned an A. Again, provide a StatCrunch binomial calculator graph to display your answer. Then, once you obtain your answer, write one sentence to explain what the probability means in context of the question.

e) Calculate the probability that between 6 and 9 students (inclusive) earned an A. Show your work using the probability distribution you built in part (b) to answer this question. Then, verify it with a StatCrunch binomial calculator graph and include this image in your document as well. Finally, once you obtain your answer, write one sentence to explain what the probability means in context of the question.

f) Calculate the mean and standard deviation of this probability distribution. Show your work using the binomial mean and standard deviation formulas and provide your answers in your document. (No need to use StatCrunch for this part).

Explanation / Answer

Here

Answer a)

Number of sample n=21,

probability of earning an A in STAT 250 is 12% ie P is the probability of success =0.12 and

q= 1-p =0.88

Answer c)

Calculate the probability exactly three student earned ie x=3

P(x=3) = nCx* p^x * q^(n-x) here x=3  

            = 21C3* (0.12)^3 * (0.88)^(21-3)

            =0.2301

23% of the student that earned an A in STAT 250.

Answer d)

      P(x> =1 ) = nCx* p^x * q^(n-x)   

            = 21C1* (0.12)^1 * (0.88)^(21-1)+...+ 21C21* (0.12)^21 * (0.88)^(21-21)

            =0.9317

93% off the student earning an A in STAT 250.

Answer e)

calculate the prob between 6 and 9

P(x=6,7,8,9)=21C6* (0.12)^6 * (0.88)^(21-6)+21C7* (0.12)^7 * (0.88)^(21-7)+ 21C8* (0.12)^8 * (0.88)^(21-8)+ 21C9* (0.12)^9 * (0.88)^(21-9)

=0.0238+0.0069+0.0016+0.003

=0.0326

3% of student earning an A in STAT 250 between 6 to 9.

Answer f) mean of binomial distribution is =np = 21*0.12=2.52

varinace =npq= 21*0.12*0.88 =2.21

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