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The standard error (SE) is a crucial concept in hypothesis testing as it measures the variability or dispersion of the sample proportion from the true population proportion. In simpler terms, it indicates how much the sample proportion you calculated is expected to vary from one sample to another.
Given a scenario where the true proportion, \( p \), is 0.50, and the sample size, \( n \), is 100, the formula for calculating the standard error is:

• \[ SE = \sqrt{\frac{p(1-p)}{n}} \]

Plugging in our values, we get:

• \[ SE = \sqrt{\frac{0.50(1-0.50)}{100}} = \sqrt{\frac{0.25}{100}} = \sqrt{0.0025} = 0.05 \]

Here, our SE is 0.05, representing the expected standard deviation of the sample proportion measurements from the true population proportion. A smaller SE suggests that the sample proportion is more closely likely to reflect the true population proportion, which is key in determining the reliability of our hypothesis test results.

The P-value is an important part of hypothesis testing because it helps us determine the strength of the evidence against the null hypothesis. The P-value represents the probability of obtaining a test result that is at least as extreme as the one observed, assuming that the null hypothesis is true.
In our context where the alternative hypothesis is \( H_a: p < 0.50 \), we deal with a left-tailed test. This means we are interested in finding the probability that the sample proportion is significantly less than the hypothesized population proportion.
With a test statistic \( z = -3.0 \), the P-value can be found using a standard normal distribution table or a statistical calculator. For this example, the P-value is approximately 0.00135.

This very low P-value indicates that there is only a 0.135% chance of observing a sample proportion of 0.35 or less if the true population proportion is 0.50. Since this probability is considerably lower than common significance levels like 0.05, it suggests that the observed sample proportion is statistically significant and provides strong evidence against the null hypothesis.

A test statistic is a standardized value used in statistical hypothesis testing to determine whether to reject the null hypothesis. It measures the degree of deviation of the sample result from the null hypothesis.
In standard problems involving proportions, the test statistic often used is the z-score. The formula for this z-score is:

• \[ z = \frac{\hat{p} - p}{SE} \]

Here, \( \hat{p} \) is the sample proportion, \( p \) is the hypothesized proportion from the null hypothesis, and \( SE \) is the standard error. In our example, we have a sample proportion \( \hat{p} = 0.35 \), a hypothesized proportion \( p = 0.50 \), and a standard error \( SE = 0.05 \).
Using the formula, we find:

• \[ z = \frac{0.35 - 0.50}{0.05} = \frac{-0.15}{0.05} = -3.0 \]

This calculated z-score of -3.0 tells us how far and in which direction our sample proportion deviates from the null hypothesis in units of standard deviation. A high absolute value of the test statistic often indicates that the sample result is unusual under the null hypothesis, leading us to question its validity.