Question

1. in a random sample of 402 world campus stat 200 students, 174 were pennsylvania residents. is the proportion of pennsylvania residents in the population of all world campus stat 200 students less than 50%? use alpha = 0.05.

Explanation:

Step1: Calculate the sample proportion

The sample proportion $\hat{p}=\frac{x}{n}$, where $x = 174$ (number of Pennsylvania residents) and $n=402$ (sample size). So, $\hat{p}=\frac{174}{402}\approx0.433$.

Step2: State the hypotheses

The null hypothesis $H_0:p = 0.5$ and the alternative hypothesis $H_1:p<0.5$.

Step3: Calculate the test - statistic

The test - statistic for a proportion is $z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}$. Substituting the values: $p = 0.5$, $\hat{p}=0.433$, $n = 402$.

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Step4: Find the critical value and make a decision

For a one - tailed test with $\alpha = 0.05$, the critical value $z_{\alpha}=-1.645$. Since $z=-2.69<-1.645$ (the test - statistic falls in the rejection region).

Answer:

We reject the null hypothesis. There is sufficient evidence at the $\alpha = 0.05$ level to conclude that the proportion of Pennsylvania residents in the population of all World Campus STAT 200 students is less than $50\%$.