Final answer:
To determine if the claim made by the health center is supported by the sample of patients referred by the local hospital, a 90% confidence interval needs to be calculated based on the sample data. If the confidence interval includes the claimed proportion of referred patients, it supports the claim made by the health center. Otherwise, the claim is rejected.
Step-by-step explanation:
This question is asking whether the claim made by the health center is supported by the sample of patients referred by the local hospital. The sample consists of 500 patients, and 125 were referred by the local hospital. To determine if the claim is supported, we need to calculate a Confidence Interval based on the sample data.
Since the Confidence Interval is based on a 90% confidence level, meaning we are 90% confident that the true proportion of referred patients falls within the Confidence Interval, we can conclude that the sample supports the claim made by the health center if the Confidence Interval includes the claimed proportion of 22%. Otherwise, we would reject the claim.
To calculate the Confidence Interval, we first need to determine the standard error, which is the standard deviation divided by the square root of the sample size. The standard deviation can be calculated using the formula:
Standard Deviation = sqrt(p * (1-p) / n)
where p is the proportion referred by the local hospital in the sample, and n is the sample size
Next, we calculate the Margin of Error, which is the critical value multiplied by the standard error. The critical value will be determined using a Z-table for the desired confidence level (in this case, 90%).
Finally, we calculate the Confidence Interval by subtracting the Margin of Error from the sample proportion (125/500) to get the lower bound, and adding the Margin of Error to the sample proportion to get the upper bound of the Confidence Interval.