Phoenix Marketing International identified Bridgeport, Connecticut, Los Alamos, New Mexico, Naples, Florida and Washington D. C. as the four U. S. cities with the highest percentage of millionaires. Data consistent with that study show the following number of millionaires for samples of individuals from each of the four cities.

City
Millionaire Bridgeport Los Alamos Naples Washington DC
YES 44 35 36 34
NO 456 265 364 366
a) What is the estimate of percentage of millionaires in each of these cities?

b) Using a 0.05 level of significance, test for the equality of the population proportion of millionaires for these four cities. What is the p-value and what is your conclusion?

Part a: The percentages for Bridgeport, Los Alamos, Naples and Washington DC are 8.80%, 11.67%,9.0% and 8.5% respectively

Part b: The value of  p for the χ²value of 2.43 and degree of freedom 3, is 0.4783. As the value is greater than the value of the level of significance, thus the null Hypothesis is valid i.e. the population proportion of the millionaires in the four cities is same.

Explanation:

Part a :

The estimate of percentage of millionaire in the cities are given as below:

                                                      Millionaire

City                                    Yes             No          Total         Percentage

Bridgeport                         44              456          500        44/500*100=8.80%

Los Alamos                       35              265          300        35/300*100=11.67%

Naples                               36              364          400        36/400*100=9.00%

Washington DC                34              366          400        34/400*100=8.50%

Part b:

Step 1: State the Null Hypothesis as

H0: p1=p2=p3=p4

where p1,p2,p3 and p4 are the population proportion of Bridgeport, Los Alamos, Naples and Washington DC respectively.

H1: p1≠p2≠p3≠p4

Step 2: Calculate the expected frequency, difference of expected frequency and the value of χ² which is as in the attached tables. The value of χ² is 2.43.

As the degree of freedoms are 4-1=3 so for the χ²value of 2.43 and degree of freedom 3, the p value is given as 0.4783.

As the value of the p is greater than the level of significance which is given as 0.05 thus the null hypothesis is valid and this indicates that the population proportion of the millionaires in the four cities is equal.