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When we want to test a particular proportion in a population, especially using a sample, we use the z-test for proportions. This kind of test is particularly useful when the sample size is large, usually greater than 30. In our exercise, we're working with the proportion of overweight children.
To perform the z-test for proportions, you begin by calculating the sample proportion \[ \hat{p} = \frac{\text{number of successes}}{\text{sample size}} \] For our case, it was \( \hat{p} = \frac{13}{100} = 0.13 \).
After that, the test statistic \( z \) is computed using the formula:\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \] where \( p_0 \) is the population proportion stated in the null hypothesis. This helps us determine how far our sample proportion is from the assumed population mean in standard deviation units.

The null hypothesis (\( H_0 \)) serves as a starting assumption for any statistical test. It suggests that there is no effect or no difference, providing a baseline that we try to disprove with evidence.
In our particular study, the null hypothesis is that the proportion of overweight children is as previously reported, that is, 15%.
\( H_0: p = 0.15 \).
By assuming the null hypothesis is true initially, we're creating a point of comparison for our sample data. If our sample data significantly differs from this hypothesis, it suggests that our original assumption may not be correct.

The alternative hypothesis (\( H_a \)) proposes what we aim to support. It's the statement we consider if there is enough evidence to reject the null hypothesis.
In the exercise, we suspect that the actual proportion of overweight children might be lower than what was previously claimed.
This is expressed as \( H_a: p < 0.15 \). The direction of the inequality (less than) indicates that this is a left-tailed test. We are specifically testing whether the sample proportion is less than the assumed population proportion.

The level of significance, denoted by \( \alpha \), is a predetermined threshold used in hypothesis testing. It defines the probability of rejecting the null hypothesis when it is actually true, essentially determining the risk of making a Type I error.
For this exercise, the level of significance was set to 0.05 (\( \alpha = 0.05 \)). This means we are willing to accept a 5% chance that we may incorrectly reject the null hypothesis.
The chosen level of significance also helps us find the critical value for our test. This critical z-value is the boundary at which we would reject the null hypothesis.
In a left-tailed test, this critical value is around -1.645, which is used to compare against the test statistic to make our final decision.