What are the key findings of the charity's claim and the watchdog group's investigation?
The charity claims that 0.75 of donations go to communities where the average family income is less than $20,000 a year. The watchdog group conducted a random sample of 60 communities receiving funds from the charity to determine if the proportion differs from 0.75 at a significance level of 0.05.
Hypothesis Testing of Proportions
Calculating Z-Score and P-Value
To determine if the proportion of donations going to communities with an average family income of less than $20,000 a year significantly differs from 0.75, hypothesis testing of proportions is used. In this scenario, the null hypothesis states that the proportion equals 0.75.
In this specific case, with missing data from Sheet 8 of the Excel file, we cannot perform the exact calculations. However, the general process involves finding the sample proportion (p hat), standard deviation of the sampling distribution (sigma), and the level of significance (alpha), which is typically set at 0.05.
The z-score is then calculated using the formula: z = (p ^ - p) / sigma. The p-value represents the probability of observing the sample data given that the null hypothesis is true. If the p-value is less than the significance level (0.05), the null hypothesis is rejected.
The significance level indicates the threshold for accepting or rejecting the null hypothesis. A p-value lower than the significance level signifies that there is enough evidence to suggest a significant difference in the proportion of donations going to communities with lower average family incomes.
It's essential to understand that the z-score measures the standard deviations the sample proportion is from the population proportion, while the p-value represents the likelihood of observing the sample data purely by chance.
For a deeper understanding of hypothesis testing, you can explore more resources on statistical analysis and hypothesis testing techniques. Reach out to data analysts or statistical experts for detailed assistance in conducting such tests.