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A proportion is a type of ratio that compares a part to a whole. In hypothesis testing, proportions are used to understand the fraction or percentage of a population that displays a certain characteristic. For instance, in our drug test example, we're interested in the proportion of patients who experience nausea after taking a new drug.
The proportion represents the ratio of the number of favorable outcomes (e.g., patients experiencing nausea) to the total number of observations or trials (e.g., total patients tested). In our example, the sample proportion, denoted as \( \hat{p} \), is calculated using the formula:

• \( \hat{p} = \frac{\text{Number of patients experiencing nausea}}{\text{Total number of patients in the sample}} \)
• \( \hat{p} = \frac{25}{300} = 0.0833 \).

This tells us that approximately 8.33% of the sample population experienced nausea.

The significance level, denoted by \( \alpha \), is a critical component in hypothesis testing. It is the probability of rejecting the null hypothesis when it is actually true, essentially representing the risk of making a Type I error. In simpler terms, it is the threshold we set for deciding whether an observed effect is statistically significant.
For the drug example, we use a significance level of \( \alpha = 0.05 \). This means we have a 5% risk of concluding that the proportion of patients experiencing nausea is different from the claimed \(10\%\), even if it isn't true. It helps us determine the critical value against which we'll compare our test statistic.

• If our test statistic is more extreme than the critical value, we reject the null hypothesis.
• Otherwise, we do not reject the null hypothesis.

In our case, the critical z-value is approximately -1.645 for a one-tailed test at the 0.05 significance level.

Standard deviation measures the amount of variation or dispersion in a set of values. When calculating proportions, the standard deviation is crucial for assessing how much the sample proportion might differ from the population proportion.
To compute the standard deviation for a sample proportion, we use the formula:

• \( \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \)

In the context of our drug test experiment, \( p \) is the hypothesized population proportion (0.10) and \( n \) is the sample size (300). Plugging these numbers into the formula gives us:

• \( \sigma_{\hat{p}} = \sqrt{\frac{0.10 \times 0.90}{300}} \approx 0.0173 \).

This tells us that the standard deviation of our sample proportion is approximately 0.0173, providing insight into the expected variability of our sample proportion around the hypothesized population proportion.

The null hypothesis, often referred to as \( H_0 \), is a crucial starting point for hypothesis testing. It represents the statement we aim to test, usually positing that there is no effect or no difference. In many cases, as with the drug test example, the null hypothesis serves as a baseline claim that we want to challenge.
For the manufacturer's claim about the drug, the null hypothesis \( H_0 \) is:

• \( p = 0.10 \)

This proposes that the true proportion of patients experiencing nausea is exactly 10%. The alternative hypothesis \( H_1 \) here is:\( p < 0.10 \). This outlines that the proportion is actually less than 10%, supporting the manufacturer's claim.
In hypothesis testing, the objective is to gather enough evidence to either reject the null hypothesis or fail to reject it, based on the sample data. In this case, since the test statistic did not pass the critical threshold, we failed to reject \( H_0 \), indicating that we did not find significant evidence against the assumption that 10% of patients experience nausea.