Question

Sports Facilities. A researcher investigated whether there has been a suburban-to-urban shift in the location of major sport facilities. In 1985, 40% of all major sport facilities were located downtown, 30% in central city, and 30% m suburban areas. The new study showed that, of all the major sports franchises, 58 were downtown, 26 in a central city, and suburban area. Test whether the population proportions have changed, using level of significance a -0.05. 6 29 in a

Answer 1

Total frequency = 58 + 26 + 29 = 113

Therefore using the given proportions, the expected frequencies here are computed as:

	Downturn	Central City	Suburban Areas
Observed Frequency	58	26	29
Expected Frequency	0.4*113 = 45.2	0.3*113 = 33.9	0.3*113 = 33.9

The chi square test statistic for the goodness of fit here is computed as:

$\small \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

$\small \chi^2 = \frac{12.8^2}{45.2} + \frac{7.9^2}{33.9} + \frac{4.9^2}{33.9}$

$\small \chi^2 = 6.1740$

For n - 1 = 2 degrees of freedom, we compute the p-value here from the chi square distribution tables here as:

$\small p = P(\chi^2_2 > 6.1740) = 0.0456$

Now as the p-value here is 0.0456 < 0.05 which is the level of significance, therefore the test is significant and we can reject the null hypothesis here and conclude that the population proportions have changed. ( at the 5% level of significance )