91Ó°ÊÓ

Suppose we are testing \(H_{0}: p=0.5\) versus \(H_{0}: p \neq 0.5 .\) Suppose that \(p\) is the true value of the population proportion. (a) Using \(\alpha=0.05,\) find the power of the test for \(n=100\), 150, and 300 assuming that \(p=0.6\). Comment on the effect of sample size on the power of the test. (b) Using \(\alpha=0.01\), find the power of the test for \(n=100\), 150 , and 300 assuming that \(p=0.6\). Compare your answers to those from part (a) and comment on the effect of \(\alpha\) on the power of the test for different sample sizes. (c) Using \(\alpha=0.05\), find the power of the test for \(n=100\), assuming \(p=0.08\). Compare your answer to part (a) and comment on the effect of the true value of \(p\) on the power of the test for the same sample size and \(\alpha\) level. (d) Using \(\alpha=0.01\), what sample size is required if \(p=0.6\) and we want \(\beta=0.05 ?\) What sample is required if \(p=0.8\) and we want \(\beta=0.05\) ? Compare the two sample sizes and comment on the effect of the true value of \(p\) on sample size required when \(\beta\) is held approximately constant.

Step by step solution

01

Understanding Power and Parameters

The power of a test is the probability of correctly rejecting the null hypothesis when it is false. This depends on the significance level (\(\alpha\)), sample size (\(n\)), and true population parameter (\(p\)).

02

Calculating Power with α = 0.05 (Part a)

For \(n=100\), 150, and 300 with \(p=0.6\):1. Calculate the Z score for each sample size using the formula: \[Z = \frac{{p - 0.5}}{{\sqrt{0.5(1-0.5)/n}}}\]2. Identify the critical value for \(\alpha = 0.05\) (\(Z_{0.025} = 1.96\)).3. Compute the power using standard normal distribution tables: \[\text{Power} = P(Z > Z_{0.025} - Z) + P(Z < -Z_{0.025} - Z)\]

03

Comment on Sample Size Effect

As the sample size increases, the power of the test increases, meaning there is a greater probability of correctly rejecting the null hypothesis when false. This is due to the variance reduction in the sampling distribution of \(p\).

04

Calculating Power with α = 0.01 (Part b)

Repeat Step 2 using \(\alpha = 0.01\). The critical value changes to \(Z_{0.005} = 2.576\). Calculate the power for \(n=100\), 150, and 300 using this new critical value.

05

Comment on α Effect

Using \(\alpha = 0.01\) results in lower power compared to \(\alpha = 0.05\) for the same sample size, reflecting the trade-off between type I error risk and type II error rate.

06

Power Calculation with Different p (Part c)

For \(n=100\) and \(p=0.08\), with \(\alpha=0.05\):1. Calculate the Z score using the formula for \(p=0.08\). 2. Find the power using steps similar to Step 2.

07

Comment on True Value of p Effect

A smaller true value of \(p\) (like 0.08) compared to 0.6 reduces the power because the deviation from the null hypothesis becomes less pronounced.

08

Sample Size Calculation for β = 0.05 (Part d)

Using \(\alpha=0.01\): For \(p=0.6\) and \(p=0.8\), 1. Find the sample size \(n\) using the formula: \[n = \left(\frac{Z_{\alpha/2} + Z_{\beta}}{p_1 - 0.5}\right)^2 \cdot (0.5)(0.5)\] for each \(p\), where \(Z_{\beta} = 1.645\).2. Calculate the required \(n\).

09

Comment on p Effect on Sample Size

Higher true values of \(p\) decrease the required sample size for the same power level, as demonstrated between \(p=0.6\) and \(p=0.8\). This occurs because the effect size is larger when \(p\) deviates more from 0.5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Size

Sample size refers to the number of observations or data points included in a statistical test. It plays a vital role in determining the reliability and validity of the results. In hypothesis testing, the sample size can significantly influence the power of a test, which is the likelihood of correctly rejecting a false null hypothesis.

When the sample size increases, the standard error decreases, leading to a more precise estimate of the population parameter. This reduction in variability makes it easier to detect an effect or difference if one exists, thereby increasing the power of the test. Think of it as taking a closer look at a painting; the more pixels you have, the clearer the image becomes.

For instance, in the provided exercise, as the sample size increases from 100 to 300, the power of the test increases. This means there's a greater chance of detecting that the true population proportion is different from 0.5 if indeed it is the case.

Significance Level

The significance level, often denoted by \(\alpha\), represents the probability of committing a type I error, which is rejecting the null hypothesis when it is actually true. Common choices for \(\alpha\) are 0.05 and 0.01, indicating a 5% or 1% risk, respectively, of concluding a false positive.

Selecting an \(\alpha\) level involves a trade-off. A lower significance level (e.g., 0.01) reduces the risk of a type I error but increases the risk of a type II error, which is failing to reject a false null hypothesis. This, in turn, affects the power of the test. A lower \(\alpha\) generally results in lower power, as illustrated in part b of the exercise, where reducing \(\alpha\) from 0.05 to 0.01 led to decreased power for the same sample sizes.

Ultimately, the choice of \(\alpha\) level should reflect the balance between the willingness to risk a false positive and the importance of avoiding false negatives in specific research contexts.

Hypothesis Testing

Hypothesis testing is a method used to make statistical inferences about a population parameter. It involves setting up two competing hypotheses: the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)). The goal is to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative.

In the context of the exercise, the null hypothesis is that the population proportion \(p = 0.5\), while the alternative hypothesis is \(p eq 0.5\). Hypothesis testing uses the sample data to assess the likelihood of the observed data under \(H_0\). If this likelihood is low (below the significance level \(\alpha\)), we reject \(H_0\).

An essential part of this process is calculating the power of a test, which is the probability of correctly rejecting \(H_0\) when it is false. In the exercise, power calculations show how adjustments in sample size, significance level, and true population proportion can affect the test's outcomes.

Population Proportion

Population proportion is a parameter that reflects the fraction of the population that shares a certain characteristic. For example, in a population of voters, the proportion who prefer a particular candidate is a population proportion.

In hypothesis testing, particularly with binomial data, you'll often test hypotheses about population proportions. The exercise presented revolves around determining the power of hypothesis tests for different assumed true population proportions and sample sizes. Given a hypothesis like \(H_0: p = 0.5\), deviations in the true proportion \(p\) can significantly affect the power of a test.

For instance, when \(p = 0.08\), the power might be lower compared to when \(p = 0.6\), since \(p = 0.08\) is closer to the null hypothesis value, making it harder to detect a difference. When selecting a sample size for a desired power, the true population proportion estimate is crucial, especially when balancing the resources available for data collection with the need for accurate statistical inference.