Question

Suppose that 62% of all college seniors have a job prior to graduation. If a random sample of 55 college seniors is taken, approximate the probabdity that fewe tham 31 have a job prior to graduation. Use the normal approximation to the binomial with a correction for contiauity. Round your answer to at least three decimal places. Do not round any intermediate steps: (if necessary, consult a isf at formsing.)

Answer 1

The probability that fewer than 31 college seniors out of 55 have a job prior to graduation, using the normal approximation to the binomial with continuity correction, is approximately 0.007.

Step-by-step explanation:

Given the percentage of college seniors with jobs before graduation as 62%, the expected number of college seniors with jobs in a sample of 55 can be found by multiplying the sample size by the probability: \(55 \times 0.62 = 34.1\).

To apply the normal approximation to the binomial, we calculate the standard deviation of the binomial distribution:
`\(√(np(1-p)) = √(55 * 0.62 * (1-0.62)) \approx 4.291\)`.

Applying the continuity correction, we adjust the boundaries for the normal approximation:
`\((31 - 0.5 - 34.1)/(4.291) = (-3.6)/(4.291) \approx -0.84\)`.

Using a standard normal distribution table or a calculator, we find the probability that a standard normal random variable is less than -0.84, which is approximately 0.007.