Respuesta :
Answer:
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
Hope this Helps!!!
