Question

The city council in a large city has become concerned about the trend toward exclusion of renters with children in apartments within the city. The housing coordinator has decided to select a random sample of 125 apartments and determine for each whether children are permitted. Let \(\pi\) be the true proportion of apartments that prohibit children. If \(\pi\) exceeds . 75 , the city council will consider appropriate legislation. a. If 102 of the 125 sampled apartments exclude renters with children, would a level \(.05\) test lead you to the conclusion that more than \(75 \%\) of all apartments exclude children? b. What is the power of the test when \(\pi=.8\) and \(\alpha=.05\) ?

Short answer

a. The null hypothesis would not be rejected since the p-value > alpha (0.053 > 0.05). Hence, it cannot be concluded that over 75% of the apartments exclude children. b. The power of the test is approximately 0.87 when \(\pi = 0.8\) and \(\alpha = 0.05\).

Step-by-step solution

State the Null and Alternate Hypotheses

Let \(\pi_0\) be the hypothesized apartment exclusion proportion. We are testing the null hypothesis \(H_0: \pi \leq \pi_0\) against the alternate hypothesis \(H_a: \pi > \pi_0\) where \(\pi_0 = 0.75\).

Conduct the Hypothesis Test

For part a, we use the given data of 102 apartments out of 125 excluding children. The sample proportion \(\hat{\pi}\) here is 102/125 which equals 0.816. The standard error of the proportion is \(\sqrt{\pi_0 \cdot (1-\pi_0)/n}\) which equals \(\sqrt{(0.75 \cdot (1-0.75))/125} = 0.041\). The Z score is then \((\hat{\pi} - \pi_0) / \text{se}\) which equals \((0.816 - 0.75) / 0.041 = 1.61\). Using a z-table or statistical software to find the p-value for a two-tailed test at Z=1.61 yields a p-value of approximately 0.053. Given an alpha level of 0.05, as the p-value > \(\alpha\), we do not reject the null hypothesis.

Power of the Test

For part b, to find the power of the test when \(\pi = 0.8\) and \(\alpha = 0.05\), we need to find the probability that we correctly reject the null hypothesis given that the true proportion is 0.8. This equates to finding the probability of a Z score less than the critical value at this new proportion. The critical value for an \(\alpha = 0.05\) one-sided test is approximately 1.645. Then, let's calculate the Z score at \(\pi = 0.8\) which is \(Z = (0.8 - 0.75)/ \sqrt{(0.75*(1-0.75))/125} = 1.21\). Lastly, the power of the test is the area to the right of this Z-score under the standard normal curve, or using a Z-table, the power of the test is approximately 0.87.

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is crucial for hypothesis testing in statistics. The null hypothesis ( H₀) represents a default position that there is no effect or no difference. It's a statement of 'no change' or 'no effect' which we assume to be true until shown otherwise. In contrast, the alternative hypothesis ( H_a) conveys that there is an effect or a difference; it is what we aim to support with our evidence.
In our exercise, the null hypothesis suggests that the proportion of apartments that prohibit children, denoted as π₀, is less than or equal to 0.75. The alternative hypothesis posits that this proportion exceeds 0.75, suggesting a potentially concerning trend that the council may need to legislate against.
The hypothesis we choose affects how we interpret data and the conclusions we draw. A clear definition of these hypotheses prior to analyzing data is essential for an unbiased and structured approach to hypothesis testing.

Sample Proportion

The sample proportion is a fundamental concept in statistics, representing the ratio of a particular outcome within a sample. In hypothesis testing, it’s used to estimate the population proportion and can often be the basis for further statistical analysis.
In our scenario, the sample proportion, denoted as π̂, in the exercise is 0.816 which indicates that 81.6% of the apartments sampled prohibit children. This proportion ultimately serves as a stepping stone for calculating further statistics such as the test statistic and p-value.
Sample proportions, like any estimators, come with their own variability and it’s important to assess their reliability, often through calculating the standard error, before drawing conclusions about the population.

Statistical Power

Statistical power is the probability that a test correctly rejects a false null hypothesis. High power means a lower risk of a Type II error (failing to reject the null when it is false), which is crucial for effective hypothesis testing.
In our worked example, we’re examining the power of a hypothesis test regarding city housing policies. The test’s power is influenced by factors such as sample size, significance level, effect size, and the true population proportion. When the power is high, it reassures us that the test is sensitive enough to detect an effect when one exists. The power of the test in our exercise scenario was calculated to be approximately 0.87, indicating a strong likelihood of detecting an actual proportion when π=0.8. This information is useful for researchers to determine if their studies are capable of providing meaningful results and can influence decisions on study design and sample size.

Z Score

The Z score provides insight into how many standard deviations an element is from the mean. In the context of hypothesis testing, a Z score helps determine how unusual or unlikely our sample result is, assuming the null hypothesis is true.
The computation in our example involves taking the observed sample proportion and subtracting the hypothesized proportion ( π₀), then dividing by the standard error. The resulting Z score of 1.61 suggests that our sample proportion is 1.61 standard deviations above the hypothesized proportion. The Z score is used for comparing to critical values or finding the corresponding p-value, which ultimately aids in decision-making concerning the hypotheses.

P-value

The p-value quantifies the probability of obtaining a result at least as extreme as the one observed, assuming that the null hypothesis is true. It’s a critical number in hypothesis testing — a low p-value suggests that our observed data is improbable under the null hypothesis.
In the exercise example, the p-value was approximately 0.053 for a Z score of 1.61. This p-value is slightly greater than the alpha level of 0.05, which typically represents the threshold for statistical significance. Since our p-value does not fall below the alpha level, we did not have enough evidence to reject the null hypothesis that less than 75% of apartments exclude children. It’s a nuanced outcome that points to the importance of understanding and correctly interpreting the p-value in research findings.

Standard Error

Standard error measures the precision of a sample estimate and reflects the sample's variability. It tells us how far we might expect the sample proportion to be from the true population proportion.
For instance, in our housing policy exercise, the standard error of the sample proportion was calculated to be 0.041. This value is crucial as it is used to compute the test statistic (Z score) and assess the p-value. A smaller standard error often denotes a more precise estimate, leading to more trustworthy conclusions drawn from the sample about the population. This concept underscores why the calculation of standard error is a fundamental step in the process of statistical analysis.