Assume a study of 500 randomly selected school bus routes revealed 480 arrived on time. Is it significant for a school bus to arrive late?

To determine the significance of a school bus arriving late, we can perform a hypothesis test on the data. The hypothesis test will help us determine if the difference between the observed proportion (480/500) and a specified proportion (e.g., 0.90, if we consider arriving on time as 90% of the routes) is statistically significant.

Here's how we can perform the hypothesis test:

1. Formulate the null hypothesis (H0) and alternative hypothesis (Ha):
- Null hypothesis (H0): The proportion of school bus routes arriving on time is equal to the specified proportion (e.g., 0.90).
- Alternative hypothesis (Ha): The proportion of school bus routes arriving on time is different from the specified proportion.

2. Choose an appropriate significance level (alpha) to help make a decision. The most common significance levels are 0.05 (5%) or 0.01 (1%).

3. Calculate the test statistic:
- The test statistic used for comparing proportions is the z-test.
- Formula for the z-test: z = (p - P0) / √(P0 * (1 - P0) / n)
where p is the sample proportion, P0 is the specified proportion, and n is the sample size.

4. Determine the critical value for the chosen significance level:
- The critical value represents the cutoff point beyond which we reject the null hypothesis. It is obtained from the standard normal distribution table or a calculator.

5. Compare the test statistic to the critical value:
- If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a significant difference.
- If the absolute value of the test statistic is less than or equal to the critical value, we fail to reject the null hypothesis.

6. Calculate the p-value (optional):
- The p-value is the probability of obtaining a test statistic as extreme as the one observed (or more extreme), assuming the null hypothesis is true.
- If the p-value is less than the chosen significance level, we reject the null hypothesis.

By performing these steps, you can determine if it is statistically significant for a school bus to arrive late based on the given data.