Question

In: Statistics and Probability

Note: Please write down the formulas clearly. Thank you The office occupancy rates were reported for...

Note: Please write down the formulas clearly. Thank you

The office occupancy rates were reported for four California metropolitan areas.
Do the following data suggest that the office vacancies were independent of the metropolitan area?
Run a hypothesis test at alpha of 0.05. What is your conclusion?


Observed Frequencies
Occupancy Status/Metropolitan Area	Los Angeles	San Diego	San Francisco	San Jose	Total
Occupied	160	116	192	174	642
Vacant	40	34	33	26	133
Total	200	150	225	200	775

Expert Solution

Null hypothesis : H_o : Office vacancies were independent of the metropolitan area

Alternate Hypothesis : H_a : Office vacancies were not independent of the metropolitan area

Test Statistic :

O : Observed Frequency

E: Expected Frequency

Expected Frequencies
Occupancy Status/Metropolitan Area	Los Angeles	San Diego	San Francisco	San Jose	Total
Occupied	(642*200)/775	(642*150)/775	(642*225)/775	(642*200)/775	642
Vacant	(133*200)/775	(133*150)/775	(133*225)/775	(133*200)/775	133
Total	200	150	225	200	775

Expected Frequencies
Occupancy Status/Metropolitan Area	Los Angeles	San Diego	San Francisco	San Jose	Total
Occupied	165.6774	124.2581	186.3871	165.6774	642
Vacant	34.3226	25.7419	38.6129	34.3226	133
Total	200	150	225	200	775

O	E	O-E	(O-E)²	(O-E)²/E
160	165.6774	-5.6774	32.2331	0.1946
40	34.3226	5.6774	32.2331	0.9391
116	124.2581	-8.2581	68.1956	0.5488
34	25.7419	8.2581	68.1956	2.6492
192	186.3871	5.6129	31.5047	0.1690
33	38.6129	-5.6129	31.5047	0.8159
174	165.6774	8.3226	69.2653	0.4181
26	34.3226	-8.3226	69.2653	2.0181
			Total	7.7528

Test Statistic :

Degrees of freedom = (number of rows -1)*(Number of columns - 1) = (2-1)*(4-1) = 3

for 3 degrees of freedom ,

As p-value : 0.0514 > level of significance : : 0.05 ; Fail to reject the null hypothesis.

These is not sufficient for the data to suggest that the Office vacancies were not independent of the metropolitan area.

p-value is computed using the excel function : CHISQ.DIST.RT

CHISQ.DIST.RT function

Returns the right-tailed probability of the chi-squared distribution.

The χ2 distribution is associated with a χ2 test. Use the χ2 test to compare observed and expected values. For example, a genetic experiment might hypothesize that the next generation of plants will exhibit a certain set of colors. By comparing the observed results with the expected ones, you can decide whether your original hypothesis is valid.

Syntax

CHISQ.DIST.RT(x,deg_freedom)

The CHISQ.DIST.RT function syntax has the following arguments:

X Required. The value at which you want to evaluate the distribution.
Deg_freedom Required. The number of degrees of freedom.

orchestra answered 1 year ago

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