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The z-test for proportions is a statistical test used to determine if there is a significant difference between two proportions. In this case, it helps assess whether the proportion of adults recognizing a company's logo has increased after a campaign. This test is especially useful when dealing with large samples.
To conduct the z-test for proportions, the following steps are involved:

• Identify the null and alternative hypotheses: Decide what you are testing. Usually, the null hypothesis (H_0) is that there is no effect or no difference, while the alternative (H_1) suggests a change or effect.


• Calculate the sample proportion: This is done by dividing the number of favorable outcomes by the total number of trials. In our example, 311 people recognized the logo out of 1200 surveyed. So, the sample proportion (\hat{p}) is 0.2592.


• Compute the z-statistic: Utilize the formula \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \], where \( p_0 \) is the null hypothesis proportion, and \( n \) is the sample size.

This statistic tells us how many standard deviations our sample proportion is from the assumed proportion. Depending on your z value and the level of significance, you'll decide whether to reject the null hypothesis.

Null and alternative hypotheses are the foundation for conducting hypothesis testing. A null hypothesis (H_0) is a statement that there is no effect or no difference in what you are testing. It's something to test against.
For the logo recognition example, the null hypothesis states that 23% of all adults recognize the logo, or \( p = 0.23 \).
The alternative hypothesis (H_1) is what we want to prove or test for. It suggests there is an effect or a difference. In this situation, it states that more than 23% of adults recognize the logo, \( p > 0.23 \).

• One-tailed vs. Two-tailed Tests: A one-tailed test checks for an increase or decrease only in one direction, while a two-tailed test assesses for changes in both directions. Here, we're conducting a one-tailed test to see if there's an increase.


• Rejection of H_0: If our test results are significant, meaning the calculated statistic falls beyond a critical value, we reject the null hypothesis in favor of the alternative.

Simply put, the null hypothesis serves as the default claim, while the alternative hypothesis stands as the new claim that needs evidence to be proven.

The level of significance defines the threshold for determining whether a hypothesis test result is statistically significant or not. It is denoted by \( \alpha \), and commonly set at values like 0.05, 0.01, or 0.10.
In the context of our example, a 1% level of significance means there's only a 1% risk of rejecting the null hypothesis when it is actually true.
This level implies a high degree of confidence in the result, as you are allowing for a very strict rejection criterion.

• Critical Value: For a one-tailed test at a 1% level of significance, the critical z-value is approximately 2.33. This is the cut-off point that determines whether you reject H_0.


• Interpretation: If your computed z-statistic exceeds this critical value, you consider your findings significant, indicating enough evidence to support the alternative hypothesis.

Thus, the level of significance is key in deciding whether to accept or reject the null hypothesis, based on the critical region determined by \( \alpha \). It provides a margin for error that you're willing to accept in your findings.