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A one-sample z-test is a statistical method used to determine whether there is a significant difference between the sample proportion and a known population proportion.
This type of test is particularly useful when we want to compare the proportion of a specific outcome in a sample to a known value.
For instance, in the provided exercise, we want to compare the proportion of Lipitor users experiencing flu-like symptoms to the known proportion of 1.9% for users of competing drugs.

The formula for the test statistic in a one-sample z-test for proportions is as follows:
\[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\]

• \( \hat{p} \) is the sample proportion.
• \( p_0 \) is the population proportion (null hypothesis).
• \( n \) is the sample size.

This formula calculates the number of standard deviations the sample proportion is from the population proportion, allowing us to understand if the observed difference is statistically significant.

Proportion testing involves comparing the proportion of a specific outcome in a sample to a known value or another sample.
It's a common technique used when working with categorical data.
In the exercise, we test the proportion of Lipitor users with flu-like symptoms against the known 1.9%.

A few key points about proportion testing:

• It's usually conducted using a z-test when the sample size is large (n > 30).
• The sample proportion (\(\hat{p}\)) is calculated by dividing the number of successes by the total sample size.

Example calculation from the exercise: \[\hat{p} = \frac{19}{863} \approx 0.022\]
This calculation helps us understand the observed proportion of Lipitor users experiencing flu-like symptoms in our sample.

The significance level, denoted by \( \alpha \), is a threshold used to determine whether the test statistic is extreme enough to reject the null hypothesis.
It's often set at common values like 0.05, 0.01, or 0.10.

In the provided exercise, the significance level is set at \( \alpha = 0.01 \), meaning that we have a 1% risk of incorrectly rejecting the null hypothesis.

The critical value for a one-tailed test at \(\alpha = 0.01 \) from the z-distribution is roughly 2.33.
We compare our test statistic to this critical value:
\[\text{Critical value} = 2.33\]
If the test statistic is greater than 2.33, we reject the null hypothesis.

However, in our exercise:

• The test statistic was 0.664.
• Since 0.664 < 2.33, we fail to reject the null hypothesis.

This indicates that there's not enough evidence to conclude that more than 1.9% of Lipitor users experience flu-like symptoms at the \(\alpha = 0.01\) significance level.