# SPSS Term Project

The data for the first test to be conducted by our group consists of the prices of residential properties in various locations. The locations are Toronto, San Francisco and Montreal. The values of the samples are all represented in Canadian Dollars. The data taken are based on the residential property prices on January 8th 2012. Our group will execute a test to determine if there is a significant difference in the mean residential property prices for Toronto, San Francisco and Montreal.

Furthermore, if the tests conclude that there is a difference in mean prices, our group will indicate where the prices are higher or lower. Hypothesis Testing For this data set, our group has chosen to conduct a one way Analysis of Variance F test (one-way ANOVA F-test). A one-way ANOVA F-test is appropriate in this example since it is a hypothesis technique that is used to compare means from three or more populations. Since the data set reflects the mean prices of residential properties in Toronto, San Francisco and Montreal, a one way ANOVA F-test is sufficient.

By having at least three samples in the data, our group has eliminated the idea of testing the claim by using different tests, such as a “two sample T-test”, a “paired sample T-test” or a “two sample Z test. ” In order for a one way Analysis of Variance F test to be conducted, the following conditions must be met: (1) Each sample must be selected from a normal, or approximately normal, population. (2) The samples must be independent and randomly selected. (3) Each population must have the same variance.

Looking at the conditions stated above, all the samples provided by the Toronto Real Estate Board reflect data from that are randomly selected, which are independent of each other. That is, there is no correlation between the sample groups. Our group has constructed three box plots to test the normality of the sample values, one for each location. Similar to a t test, the F test is fairly non-sensitive to slight departure from normality. Since the box plots do not indicate extreme differences from a normal distribution, we can assume that the samples are selected from a normal population.

The third condition states that the variances of the sample groups are equal. Therefore, our group will conduct a Levene’s Test for Homogeneity of Variance using SPSS program to test whether the data set satisfies the third assumption. Results from Levene Test Null Hypothesis:? 12 = ? 22 = ? 32 Alternative Hypothesis:? 12, ? 22, ? 32are not all equal *Ho for this instance is the claim, since Ho is a statement of equality ?12 represents the variance for the population of residential properties in Toronto, ? 22 represents the variance for the population of residential properties in San Francisco and ?

32 represents the variance for the population of residential properties in Montreal. (? =0. 05) Using the data from SPSS output, the P-value (represented by “Sig. ” – Oneway DataSet 1\residential sales. sav) found on the first table – Test of Homogeneity of Variances is 0. 549. Since P-value > ? ; fail to reject Ho Therefore, at 5% level of significance, there is insufficient evidence to indicate that the claim that all the variances of the samples provided are equal is false. All the conditions are therefore satisfied, and our group can proceed with the one way analysis of variance F test.

Since all the conditions for a one way analysis of variance are satisfied, then the sampling distribution can now be approximated by the F distribution. Our group can now execute a one way Analysis of Variance F test by using a Post-Hoc Comparison Procedure to test the claim that “there is a significant difference in the mean residential property prices for Toronto, San Francisco and Montreal. ” Null Hypothesis:µ1 = µ2 = µ3 Alternative Hypothesis:At least one mean is different. *Ha for this instance is the claim, since Ha is a statement of inequality Parameters

µ1 represents the mean residential property price in Toronto. µ2 represents the mean residential property price in San Francisco, while µ3 represents the mean residential property price in Montreal. The null hypothesis suggests that there is no difference between the means of the three samples, while the claim in the alternative hypothesis suggests that at least one mean is different. Since no level of significance was given, we assume that: ? = 0. 05 Conclusion Using the data from SPSS output, the P-value (represented by “Sig. ” – One Way DataSet 1\residential sales. sav) found on the second table – ANOVA is 0.

140. Since P-Value > ? ; fail to reject Ho Therefore, at 5% level of significance, there is insufficient evidence to indicate that the claim that there is a significant difference in the mean residential property prices for Toronto, San Francisco and Montreal is true. *Full SPSS Output can be found in the appendix section of the report. Part B – Difference in Lot Sizes for Residential Properties in Toronto and Vancouver Introduction The data for the second test to be conducted by our group consists of lot sizes of the residential properties that are up for sale in Toronto and Vancouver.

The samples are represented in m2 (metres squared; area of the land in which the residential properties are built on). The data taken are based on the properties that are up for sale as of January 8th 2012. Our group will execute a test to determine if there is a significant difference in the lot sizes for the residential properties for sale in Toronto and Vancouver, as commissioned by the Toronto Real Estate Board. Hypothesis Testing For this data set, our group has chosen to conduct a two sample T-test.

A two sample T-test is appropriate in this case because of the attempt in determining the difference between two population means when the population standard deviations are unknown. Furthermore, the data given reflects independent samples. That is, the sample selected from the population in Toronto is not related to the sample from the population in Vancouver. In order for a two sample T-test for difference of means with small independent samples to be conducted, the following conditions must be met: (1) The samples must be randomly selected. (2) The samples must be independent.

(3) Each population must have a normal distribution with an unknown standard deviation. Since there is no correlation between the sample groups (Toronto and Vancouver lot sizes), a paired T-test cannot be conducted for this data set. Also, since there are exactly only two means that are being compared in the given case, and not means between three or more populations, a one-way analysis of variances test (one way ANOVA) cannot be used. Looking at the conditions stated above, the samples provided by the Toronto Real Estate Board are randomly selected and independent.

By checking the normality in each of the populations, our group constructed two separate box plots for Toronto and Vancouver respectively. There is no significant evidence to conclude that both the populations are not normally distributed since the box plots resemble a normal distribution. Having the conditions satisfied, our group can proceed to execute a two sample T-test for difference of means with small independent samples in testing the claim that “there is a significant difference in the lot sizes for the residential properties for sale in Toronto and Vancouver,” as commissioned by the Toronto Real Estate Board.