Hypothesis Testing for a Population Proportion

Reviewing from last lecture, you hypothesis testing is all about deciding between two hypothesis using data.

• There is the null hypothesis, which is the hypothesis that nothing interesting is happening, and...
• The alternative hypothesis, which suggests something unusual.

This requires that data comes from a random sample.

Test Statistics

• Z test
  • Standardized
• T test
  • Standardized
• F test
• \(\chi^2\) test

The formula to compute a test statistic is:

$$\frac{\text{Sample statistic - Null value}}{\text{(Null) Standard Error}}$$

The p-value

The p-value, between 0 and 1, is a conditional probability of seeing a test statistic as extreme, given that the null hypothesis is true.

Note that the p-value is not the probability that a null hypothesis is true, and not the probability that an alternative hypothesis is true.

The smaller the p-value, the stronger the evidence against the null hypothesis is.

Common Conventions

How small of a p-value is small enough? This is denoted by \(p \leq \alpha\), where \(\alpha\) is the significance level. When \(p \leq \alpha\), we say the result is statistically significant.

Common levels of significance:

• 0.01
• 0.05
• 0.10

You can either reject the null hypothesis, or fail to reject the null hypothesis. We say that the results are either significant or not significant at the level \(\alpha\).

One-Sided and Two-Sided

You can check for a significant deviation from a value in the following three ways:

• Checking that the proportion is less than the accepted value
  • One-tailed to the left
• Checking that the proportion is greater than the accepted value
  • One-tailed to the right
• Checking that the proportion deviates from the accepted value by enough
  • Two-tailed

The Test Statistic

You get a certain estimate of a population proportion \(p\), the sample proportion \(\hat{p}\). For a large sample size, the distribution of this value will be:

$$\sim N\left(p, \sqrt{\frac{p(1-p)}{n}}\right)$$

Therefore, the z-test statistic is:

$$z = \frac{x - \bar{x}}{\sigma} = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$$

And this will have approximately a \(N(0, 1)\) distribution. This again requires that \(np_0 \geq 10\) and \(n(1-p_0) \geq 10\).

Then, using a z-table, you can find the probability of getting that \(\hat{p}\) value.