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When conducting a hypothesis test involving proportions, it's essential to first define the concept of population proportion. A population proportion is a fraction of the population that possesses a particular characteristic. In our example, this refers to the proportion of Canadian children who receive antibiotics in their first year of life.
The symbol used for population proportion is often \(p\), representing the hypothesized value in the hypothesis tests. In the context of our exercise, the null hypothesis proposes that the population proportion is 0.7, which means 70% of children receive antibiotics during infancy. Meanwhile, in our sample, we observed that 438 out of 616 children received antibiotics, giving a sample proportion (\(\hat{p}\)) of approximately 0.71. This sample proportion is computed as follows:

• \(\hat{p} = \frac{438}{616} \approx 0.71\).

By comparing these proportions, you'll go on to use hypothesis testing to decide if the observed proportion differs significantly from the hypothesized proportion.

The test statistic in a hypothesis test provides a standardized measure of the difference between the observed sample proportion and the hypothesized population proportion. It essentially offers a mathematical way to see if the data collected could have happened by random chance.
In the case of testing a population proportion, the test statistic is calculated using the formula:

• \[Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\]

where:

• \(\hat{p}\) is the sample proportion (0.71 in our case),
• \(p_0\) is the null hypothesis proportion (0.7 here),
• \(n\) is the sample size (616 here).

Plugging in the values, the Z-score quantifies how far away our sample proportion is from the hypothesized value under the assumption that the null hypothesis is true.
This test statistic is crucial, as it forms the basis for finding the p-value in the next step.

The p-value is a probability that helps us make informed decisions in hypothesis testing. It represents the likelihood of observing our data, or something more extreme, assuming the null hypothesis is true.
To find the p-value, you'll use the Z-score calculated in the previous step. Since we are dealing with a right-tailed test (our alternative hypothesis is that more than 70% of children receive antibiotics), the p-value indicates the area to the right of the calculated Z-score in the standard normal distribution.
A small p-value suggests that the observed data are unlikely under the null hypothesis. Conversely, a large p-value would imply that the data do not provide strong evidence against the null hypothesis.
Once you've determined the p-value using statistical software or a Z-distribution table, it will directly inform whether the null hypothesis should be rejected or not.

The significance level, often denoted as \(\alpha\), is a threshold set before conducting a hypothesis test. It represents the probability of wrongly rejecting the null hypothesis if it is true.
In many studies, a common choice for the significance level is \(\alpha = 0.05\), which equates to a 5% risk of concluding that a difference exists when there is none. Simply put, if your p-value is less than or equal to this significance level, you'll reject the null hypothesis, concluding that there is enough evidence to suggest a significant difference or effect.
In the context of our antibiotic study, a significance level of 5% means if the p-value calculated from the Z-score is 0.05 or less, we will reject the null hypothesis. This would suggest that there is evidence that more than 70% of infants receive antibiotics in their first year.
Understanding and setting the appropriate significance level is crucial for interpreting results and supporting decisions made from hypothesis tests.