Applicability of Normal Approximation in Statistics Assignment
The image shows a question from a statistics assignment, which reads:
"About 41% of employed adults in the United States hold multiple jobs. A random sample of 78 employed adults is chosen. Use the TI-84 Plus calculator as needed.
Part 1 of 5:
Is it appropriate to use the normal approximation to find the probability that less than 65% of the individuals in the sample hold multiple jobs? If not, find the probability. If not, explain why not."
To determine if it is appropriate to use the normal approximation for the binomial distribution, we often use the rule of thumb that the sample size (n) is large enough if both np and n(1-p) are greater than 5.
Let's check these conditions using the data provided:
n = number of trials = 78 (the sample size)
p = probability of success = 0.41 (probability that an individual holds multiple jobs)
q = 1 - p = 0.59 (probability that an individual does not hold multiple jobs)
Calculating np and nq:
np = 78 * 0.41 ≈ 31.98
nq = 78 * 0.59 ≈ 46.02
Both np and nq are greater than 5, thus satisfying the rule of thumb for the normal approximation to be used.
Hence, it is appropriate to use the normal approximation to find the probability in question. The next steps in the calculation would involve finding the mean, μ = np, and the standard deviation, σ = sqrt(npq), and then using these to find the probability that less than 65% of the individuals in the sample hold multiple jobs using a z-score.
Since the question is part of a series and requires the use of a TI-84 Plus calculator, further calculations might be needed to complete the assignment as directed. However, we have established that it is indeed appropriate to use the normal approximation for this problem.
