Question

The same sample statistic is used to test a hypothesis, using different sample sizes. In each case, use StatKey or other technology to find the p-value and indicate whether the results are significant at a \(5 \%\) level. Which sample size provides the strongest evidence for the alternative hypothesis? Testing \(H_{0}: p=0.5\) vs \(H_{a}: p>0.5\) using \(\hat{p}=0.58\) with each of the following sample sizes: (a) \(\hat{p}=29 / 50=0.58\) (b) \(\hat{p}=290 / 500=0.58\)

Short answer

Using the z-statistics method, calculate the p-value for case (a) and case (b) and then compare them. The test with the smaller p-value provides the strongest evidence for the alternative hypothesis.

Step-by-step solution

Calculation of p-value for case (a)

Using the z-statistics method, the z-score is defined as \(Z = \frac\{(\hat{p}-p_0)}{ \sqrt\{(p_0*(1-p_0)) / n} } \), where \(p_0\) is the assumed true population proportion in the null hypothesis, and \(n\) is the sample size. For case (a), we can substitute \(\hat{p} = 0.58, p_0 = 0.5\) and \(n = 50 \) in the z-score formula. After we estimated the z-score, we calculate the p-value , which is the probability under the standard normal curve to the right of the calculated z-score. If the p-value \(< 0.05\), the result is considered significant at the \(5 \%\) level.

Calculation of p-value for case (b)

The same method applied for case (b), but this time the sample size \(n=500\). All another values stay the same: \(\hat{p} = 0.58, p_0 = 0.5\). After we estimated the z-score, calculate the p-value. If the p-value \(< 0.05\), the result is considered significant at the \(5 \%\) level.

Comparing the statistical impact of two sample sizes

After having the p-values for both sample sizes, compare them. The smaller the p-value, the stronger the evidence against the null hypothesis. Hence, the sample size providing the smaller p-value would be providing the strongest evidence for the alternative hypothesis.

Key concepts

P-Value Significance

When conducting hypothesis testing, understanding the p-value can seem complex, but it's crucial in determining statistical significance. The p-value indicates the probability of obtaining a result at least as extreme as the one in your sample data, assuming that the null hypothesis is true. In simpler terms, it measures how compatible your data is with the null hypothesis.

Within hypothesis testing, if the p-value is low, it suggests that the observed data is unusual under the assumption of the null hypothesis. Traditionally, researchers use a significance level, often set at 0.05 or 5%, to decide whether or not to reject the null hypothesis. If the p-value falls below this cutoff, the results are deemed 'statistically significant,' implying there's evidence to support the alternative hypothesis. This doesn't prove the alternative hypothesis, but rather suggests it's a more plausible explanation for the data compared to the null hypothesis.

Imagine we're tossing a coin and want to check if it's fair. If we find a significant p-value after many tosses, it might indicate the coin is biased. Remember to consider other factors like sample size, which can affect the p-value, as we'll discuss later.

Alternative Hypothesis

In the realm of hypothesis testing, the alternative hypothesis (\( H_a \) or \( H_1 \)) represents what we aim to support, based on sample data. It's defined in opposition to the null hypothesis (\( H_0 \)), which typically suggests no effect or no difference. An alternative hypothesis is formulated based on what we suspect might be true.

For example, let's go back to our coin tossing: the null hypothesis might assert that the coin is fair (\( H_0: p = 0.5 \)), and the alternative hypothesis might claim that the coin is biased towards heads (\( H_a: p > 0.5 \)). We use evidence from the sample to support the alternative hypothesis, looking for a small p-value to reject the null. In our original exercise, the alternative hypothesis suggests that the true proportion is greater than 0.5, inspired by the sample statistic (\( \bar{p} = 0.58 \)).

When we have a significant result, it strengthens the claim of our alternative hypothesis; however, it's essential to avoid overstating the conclusion. A result aligning with \( H_a \) simply diminishes the credibility of \( H_0 \) but doesn't prove \( H_a \) is correct definitively.

Sample Size Effect

Sample size plays a pivotal role in statistical inference. The larger the sample size, the more confident we are about our estimates of the population's characteristics. This confidence comes from the Law of Large Numbers, which states that as a sample size grows, the sample mean gets closer to the population mean.

When dealing with hypothesis testing and p-values, sample size can dramatically impact results. Larger samples reduce the margin of error and therefore can detect smaller differences from the null hypothesis value. This can lead to a smaller p-value and stronger evidence against the null hypothesis. Conversely, small samples may be more susceptible to random chance, potentially leading to less reliable results.

Referring to our original exercise, when we compare the p-values obtained from different sample sizes, a larger sample size providing the same sample statistic (\( \bar{p} \)) will usually give us more robust evidence for the alternative hypothesis. So, when using sample sizes of 50 and 500, with the same observed proportion (\( \bar{p} = 0.58 \)), the larger sample size would generate a p-value that more convincingly argues against the null hypothesis.