Question

calculating probabilities for sampling distribution
quiz complete
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in a certain town, 62% of the residents enjoy riding roller coasters. alfred takes a sample of 75 residents from this town. what is the probability that fewer than 58% of the residents in the sample enjoy riding roller coasters?
find the z-table here.
○ 0.238
○ 0.433
○ 0.714
○ 0.762

Explanation:

Step1: Define given values

Population proportion $p=0.62$, sample proportion $\hat{p}=0.58$, sample size $n=75$

Step2: Calculate standard error

$$SE=\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{0.62(1-0.62)}{75}}=\sqrt{\frac{0.62\times0.38}{75}}=\sqrt{\frac{0.2356}{75}}\approx0.0563$$

Step3: Compute z-score

$$z=\frac{\hat{p}-p}{SE}=\frac{0.58-0.62}{0.0563}\approx-0.71$$

Step4: Find corresponding probability

Use z-table: $P(z<-0.71)=0.2389\approx0.238$

Answer:

0.238