Question

Polling 1000 people in a large community to determine if there is evidence for the claim that the percentage of people in the community living in a mobile home is greater then \(10 \%\).

Short answer

If the calculated Z score is greater than the Z critical, we can conclude that there is evidence to support the claim that the percentage of people in the community living in mobile homes is greater than \(10\%\). If it is less, then there's not enough evidence to support the claim.

Step-by-step solution

Formulate the Hypothesis

First, formulate the null hypothesis and the alternative hypothesis. The null hypothesis, denoted \(H_0\), is the claim to be tested. The alternative hypothesis (or research hypothesis), denoted \(H_1\), contradicts the null hypothesis. We have \(H_0\): The proportion of community living in mobile homes is less than or equals to \(10%\) i.e., \(P <= 0.10\) and \(H_1\): The proportion of community living in mobile homes is greater than \(10%\), i.e., \(P > 0.10\).

Interpret the survey results

After the survey has been conducted and data has been collected, count how many out of the 1000 polled people live in mobile homes. Let's denote this number \(x\). Then calculate the sample proportion \(p\), which is \(\frac{x}{1000}\).

Test the Hypothesis

To test the null hypothesis, compute the test statistic – the z-score. The formula for z-score is \( Z = \frac{p - P}{\sqrt {\frac {P(1-P)}{n}}}\), in which \(P\) is the proportion under the null hypothesis, \(p\) is the sample proportion and \(n\) is the sample size. Substitute \(P = 0.10\), \(p\) from step 2, and \(n = 1000\) to get the Z score.

Make a Decision

Compare the calculated Z score with the Z critical value for the given level of confidence (usually 95% confidence level is assumed, Z critical would be approximately 1.64 for a one-tailed test). If the calculated Z is greater than Z critical, we reject the null hypothesis in favor of the alternative hypothesis. If it is less, we fail to reject the null hypothesis.