91Ó°ÊÓ

In hypothesis testing, defining the null and alternative hypotheses forms the foundation of your analysis. The null hypothesis, denoted as \(H_0\), is a statement that signifies no effect or no difference. For our exercise, it claims that 40% of people prefer national brand coffee (\(H_0: p = 0.40\)).
On the other hand, the alternative hypothesis, denoted as \(H_1\), suggests that there is a significant effect or difference. In this context, it claims that the percentage of people who prefer national brand coffee is not 40% (\(H_1: p eq 0.40\)).
This step sets the stage for testing, helping us determine if the collected data provides enough evidence to support a significant difference from what was previously assumed.

Calculating the z-score is crucial to understanding how far your sample's observation is from the assumed population mean in terms of standard deviations. It helps identify if there's a meaningful difference between the sample proportion and the hypothesized proportion.
The formula for the z-score is: \[ Z = \frac{(\hat{p} - p)}{\sqrt{ \frac{p(1-p)}{n} }} \]

• \(\hat{p}\) is the sample proportion, calculated as the number of successes divided by the sample size.
• \(p\) is the hypothesized population proportion.
• \(n\) is the sample size.

By plugging in the numbers, we find \(Z \approx -1.63\). This tells us how many standard deviations our sample proportion is away from the assumed 40%.

The critical value in hypothesis testing represents the threshold beyond which we will reject the null hypothesis. It depends on the chosen level of significance (\(\alpha\)) and whether the test is one-tailed or two-tailed.
In our case, with a \(1\%\) significance level and a two-tailed test, the critical values are approximately \(\pm 2.58\). These come from the standard normal distribution table.
Since our calculated z-score \(-1.63\) falls within the range between \(-2.58\) and \(2.58\), we do not have enough statistical evidence to reject the null hypothesis. This means that we cannot conclude that the preference for national brand coffee is significantly different from 40%.

The p-value helps us understand the strength of the results in hypothesis testing. It indicates the probability of observing a test statistic as extreme as, or more extreme than, the observed one assuming the null hypothesis is true.
For the given z-score \(-1.63\), the p-value is found to be \(0.0516\). Since the test is two-tailed, we multiply this by 2 to get \(0.1032\).
If this p-value is less than our level of significance (\(\alpha = 0.01\)), we would reject the null hypothesis. Here, \(0.1032 > 0.01\), indicating insufficient evidence to reject the null hypothesis. This aligns with our findings using the critical value approach.