Education professionals refer to science, technology, engineering, and mathematics as the STEM disciplines. A research group reported that 27% of freshmen entering college in a recent year planned to major in a STEM discipline. A random sample of 85 freshmen is selected. Round the answer to at least four decimal places

1) It is appropriate to use the normal curve, since np = 22.95 ≥ 10 and n(1 - p) = 62.05 ≥ 10

2) The probability that less than 28% of the freshmen in the sample are planning to major in a STEM discipline = 0.5832

Step-by-step explanation:

Concluding part of the question

1) Is it appropriate to use the normal approximation to find the probability that less than 28% of the freshmen in the sample are planning to major in a STEM discipline?

2) Find the probability that less than 28% of the freshmen in the sample are planning to major in a STEM discipline?

Solution

1) The condition for a distribution (binomial) to approximate a normal distribution is that

np ≥ 10 and n(1-p) ≥ 10

where n = sample size = 85

p = population proportion = sample proportion = 0.27

np = 85 × 0.27 = 22.95 ≥ 10

n(1-p) = 85 × 0.73 = 62.05 ≥ 10

2) To obtain the required probability, we need the standard deviation of the sampling distribution

σₓ = √[p(1-p)/n] = √(0.27×0.73/85) = 0.0481541642 = 0.04815

To obtain the probability that less than 28% of the freshmen in the sample are planning to major in a STEM discipline, we need to standardize 28%

The standardized score for any value is the value minus the mean then divided by the standard deviation.

z = (x - μ)/σ = (0.28 - 0.27)/0.04815 = 0.21

To obtain the probability that less than 28% of the freshmen in the sample are planning to major in a STEM discipline

P(x < 0.28) = P(z < 0.21)

We'll use data from the normal probability table for these probabilities

P(x < 0.28) = P(z < 0.21) = 0.58317 = 0.5832 to 4 d.p

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