   Chapter 8.5, Problem 35E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# In the following problems, use a normal approximation to the binomial.Testing A standardized exam for admission to a special program has been passed by 53% of the students taking it over the last five years.(a) Suppose that 1120 of 2000 students taking the test this year pass. What is the probability that at least 1120 will pass if the pass rate is 53%?(b) Does this suggest something is different? If so, what could have happened?

(a)

To determine

To calculate: The probability that at least 1120 students out of 2000 will pass the exam if the rate of passing is 53%.

Explanation

Given Information:

53% of the student taking the program over the last five year passed in the test.

And, the rate of passing is 53%.

Refer to the Appendix C of the book.

Formula used:

For a binomial distribution,

Mean is μ=np and standard deviation is σ=npq.

And z score is, z=xμσ.

Also, q=1p.

Where n the number of trials, p is the probability of success, and q is the probability of failure.

Calculation:

Consider the rate of passing is 53%.

Thus, the probability of number of students passing the exam is,

p=53%=53100=0.53

Now, total number of student is 2000. Thus, n=2000.

Therefore, the mean is,

μ=np=2000(0.53)=1060

And, the probability of failure is,

q=1p=10.53=0.47

Thus, the standard deviation is,

σ=npq=2000(0.53)(0.47)=498.2=22.32

The binomial probability that at least 1120 students out of 2000 will pass the exam by using the normal approximation is Pr(k1119

(b)

To determine

Whether the result of the probability that at least 1120 students out of 2000 will pass the exam suggest something different or not and if it is find what could have happened.

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