Question: An advertising agency is interested in determining the proportion of city residents that pass by a certain street corner at least once a week. The agency is planning to conduct a telephone survey of randomly selected city residents to ask them whether they pass that corner at least once a week. The information collected will help them decide on the effectiveness of an expensive billboard they are considering placing at that corner. The agency would like to have an accurate measurement with a margin of error of at most 1%.

1. Assuming a 99% confidence level, what sample size should the agency collect? Show the steps of the calculation.
2. Since the cost of the study increases as the sample size increases, the CFO of the agency would like to understand options that will require smaller sample sizes. Explain clearly what requirements will need to be changed (and in which direction – i.e., increase/decrease) in order to reduce the sample size.
3. If it is known from historic studies that at most 35% of the residents cross that intersection, what will be the sample size recommendation for a 99% confidence interval with 1% margin of error?
4. Below is a contingency table between the region and the number of property crime report volume (number of property crime reports categorized to "at most 600", "600 to 1200", and "1200+"). Use a chi-square test for independence to see whether there is a relationship between the region and property crime report volume. Report the statistic, the p-value, and the conclusion.

   	At Most 600 Reports	Between 600 and 1200 Reports	More Than 1200 Reports
   North Central	69	66	106
   North East	53	41	55
   South	51	43	140
   West	10	13	53

   (fo - fe)²/fe	At most 600 reports	Between 600 and 1200 reports	More than 1200 reports
   North Central	0.5706	1.7399	2.0683
   North East	5.0657	1.1455	5.4967
   South	1.6922	2.4223	3.9656
   West	4.9016	1.2467	5.5201

   
   The calculations required are shown below:
   
   
   
   
   
   Hence, the value of Chi-Square statistics is
   
   
   The critical Chi-Square value for and degrees of freedom is . Since > , then we reject the null hypothesis, which means that we have enough evidence to reject the null hypothesis of independence.
5. Find 95% confidence intervals for the average number of motor vehicles stolen, by each country-division and by total.
6. Describe any patterns observed from the confidence intervals in part b.
7. Is there a significant difference between the average amount of property damage between city agencies and county agencies? Conduct the appropriate hypothesis test and report the p-value.
8. Interpret the finding from part d.

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