One-sample proportion tests

1. One-sample proportion tests

Let’s return to thinking about testing proportions, as we did in Chapter 1.

2. Chapter 1 recap

The hypothesis tests in Chapter 1 measured whether or not an unknown population proportion was equal to some value. We used bootstrapping on the sample to estimate the standard error of the sample statistic. The standard error was then used to calculate a standardized test statistic, the z-score, which was used to get a p-value, so we could decide whether or not to reject the null hypothesis. A bootstrap distribution can be computationally intensive to calculate, so this time we'll instead calculate the test statistic without it.

3. Standardized test statistic for proportions

An unknown population parameter that is a proportion, or population proportion for short, is denoted p. The sample proportion is denoted p-hat, and the hypothesized value for the population proportion is denoted p-zero. As in Chapter 1, the standardized test statistic is a z-score. We calculate it by starting with the sample statistic, subtracting its mean, then dividing by its standard error. p-hat minus the mean of p-hat, divided by the standard error of p-hat. Recall from Sampling in Python that the mean of a sampling distribution of sample means, denoted by p-hat, is p, the population proportion. Under the null hypothesis, the unknown proportion p is assumed to be the hypothesized population proportion p-zero. The z-score is now p-hat minus p-zero, divided by the standard error of p-hat.

4. Simplifying the standard error calculations

For proportions, under H-naught, the standard error of p-hat equation can be simplified to p-zero times one minus p-zero, divided by the number of observations, then square-rooted. We can substitute this into our equation for the z-score. This is easier to calculate because it only uses p-hat and n, which we get from the sample, and p-zero, which we chose.

5. Why z instead of t?

We might wonder why we used a z-distribution here, but a t-distribution in Chapter 2. This is the test statistic equation for the two sample mean case. The standard deviation of the sample, s, is calculated from the sample mean, x-bar. That means that x-bar is used in the numerator to estimate the population mean, and in the denominator to estimate the population standard deviation. This dual usage increases the uncertainty in our estimate of the population parameter. Since t-distributions are effectively a normal distribution with fatter tails, we can use them to account for this extra uncertainty. In effect, the t-distribution provides extra caution against mistakenly rejecting the null hypothesis. For proportions, we only use p-hat in the numerator, thus avoiding the problem with uncertainty, and a z-distribution is fine.

6. Stack Overflow age categories

Returning to the Stack Overflow survey, let's hypothesize that half of the users in the population are under thirty and check for a difference. Let's set a significance level of point-zero-one. In the sample, just over half the users are under thirty.

7. Variables for z

Let's get the numbers needed for the z-score. p-hat is the proportion of sample rows where age_cat equals under thirty. p-zero is point-five according to the null hypothesis. n is the number of rows in the dataset.

8. Calculating the z-score

Inserting the values we calculated into the z-score equation yields a z-score of around three-point-four.

9. Calculating the p-value

For left-tailed alternative hypotheses, we transform the z-score into a p-value using norm-dot-cdf. For right-tailed alternative hypotheses, we subtract the norm-dot-cdf result from one. For two-tailed alternative hypotheses, we check whether the test statistic lies in either tail, so the p-value is the sum of these two values: one corresponding to the z-score and the other to its negative on the other side of the distribution. Since the normal distribution PDF is symmetric, this simplifies to twice the right-tailed p-value since the z-score is positive. Here, the p-value is less than the significance level of point-zero-one, so we reject the null hypothesis, concluding that the proportion of users under thirty is not equal to point-five.

10. Let's practice!

Let's try an example.

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