Question

1. think about some of the examples we worked through in our lecture coverage of chapter 18. these examples involved converting a sample statistic to a z - score and then using table b. you can even find a similar example in your textbook, in chapter 18 (see example 5 in chapter 18). based on what you learned from that coverage, what is the probability of surveying a random sample of n = 75 college students and finding the proportion who were initially undecided about what major to pursue to be 0.34 or smaller? as you answer this, remember the sampling distribution for samples of size n = 75 has a mean of 0.41 and a standard deviation of 0.0568. please show your work below as you attempt to answer this question.

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{\hat{p}-\mu_{\hat{p}}}{\sigma_{\hat{p}}}\), where \(\hat{p} = 0.34\), \(\mu_{\hat{p}}=0.41\), and \(\sigma_{\hat{p}} = 0.0568\).

$$ z=\frac{0.34 - 0.41}{0.0568}=\frac{- 0.07}{0.0568}\approx - 1.23 $$

Step2: Find the probability using the standard normal table

We want to find \(P(\hat{p}\leq0.34)\), which is equivalent to \(P(Z\leq - 1.23)\)
Looking up the value in the standard normal table (Table B), for \(z=-1.23\), the cumulative probability \(P(Z\leq - 1.23)=0.1093\)

Answer:

The probability is \(0.1093\)