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In hypothesis testing, the null hypothesis ( H_0 ) acts as a starting point. It is a statement saying there is no difference or effect in a given situation. For our scenario involving garbage disposal, the null hypothesis posits that the average amount of garbage discarded per household in Vermont matches the national average, which is 4.6 pounds per day. Essentially, it's suggesting that the Going Green campaign hasn't led to any reduction in waste compared to the national average. This hypothesis is typically what we aim to test against to see if there truly is a significant change or effect.

To test this, we gather data, calculate statistics, and use them to determine whether the data provide enough evidence to reject the null hypothesis. It's crucial to remember that rejecting the null doesn't prove that the alternative is true; rather, it indicates that there is enough evidence to suggest the null hypothesis is unlikely. In this case, if our calculations show a significant difference, it would imply the campaign might be making an impact.

Complementing the null hypothesis, the alternative hypothesis ( H_a ) is what researchers typically want to prove. It suggests that there is a statistically significant effect or difference. In this example, the alternative hypothesis argues that residents of the Vermont town discard less garbage on average than the national figure of 4.6 pounds. This is what the Going Green campaign is hoping to demonstrate through their efforts.

The alternative hypothesis is crucial because it frames what a significant finding would look like. By setting this as our hypothesis, we are effectively saying that any significant deviation we find would suggest a positive change, beneficial to the overarching goal of reducing waste. If the data show a statistically significant decrease in garbage disposal, we reject the null hypothesis, thereby supporting the alternative possibility that the campaign is effective.

The z-score is a key component in hypothesis testing, allowing us to determine how far away our sample mean is from the population mean under the null hypothesis. It's a standard score that shows the number of standard deviations a data point is from the mean, making it a great tool for comparison.

In our exercise, we use the z-score formula: \[Z = \frac{(\text{sample mean} - \text{population mean})}{\text{standard deviation}/\sqrt{\text{sample size}}}\]Applying this to our scenario, we input our sample mean of 3.89 pounds, population mean of 4.6 pounds, standard deviation of 1.322 pounds, and sample size of 18. This yields a z-score of -2.26.

Interpreting this value, we see that the sample mean is 2.26 standard deviations below the population mean. By comparing this with a critical value from the z-table (for a 0.05 significance level, which is generally -1.645 for one-tailed tests), we decide whether to reject the null hypothesis. In this case, since -2.26 is less than -1.645, there is sufficient evidence to reject the null hypothesis, pointing toward a successful outcome from the Going Green campaign.