Question

The following table contains summary statistics on house sales in Canberra (by region) for the last 3 months.

REGION	Mean House price	Number of houses sold
Belconnen	$755481	210
Gungahlin	$815826	228
Inner North	$1180106	75
Inner South	$1489622	42
Molonglo Valley	$948222	28
Woden Valley	$1065248	95
Weston Creek	$802488	60
Tuggeranong	$723422	228

The regions of Gungahlin and Molonglo Valley were first settled in 1991 and 2010 respectively. All other regions were settled prior to 1975. How do these two relatively newer regions compare to the other older regions of Canberra on house prices? You are going to use the summary statistics in the table above to assess whether there is sufficient evidence of a difference in the mean house price between the newer area of Canberra (Gungahlin and Molonglo Valley) and the older area of Canberra (Inner North; Inner South; Belconnen; Woden Valley; Weston Creek; Tuggeranong).
  

Note, for this question you may assume that:

1. the sample sizes in the new and old areas of Canberra (as defined above) are respectively large enough such that a t-test (of some sort) can be validly applied.
2. house prices are independent (that is, the sold price of one house is not affected by the sold price of any other house).
3. the houses that were sold in the last 3 months constitute a simple unbiased random sample of the houses within

(d) The sample standard deviation of house prices in newer and older areas of Canberra are estimated to be $251250 and $493750 respectively. Will your test be a pooled-variance t-test or separate-variance t-test for the difference between two population means? Give a reason for your answer.  

(e) Calculate the value of your test statistic for this hypothesis test.

(Note: please show your working by providing the equations you used to calculate your test statistic).

(f) Assuming a significance level of   , what critical value(s) define your rejection region?

(Note: if your degrees of freedom is greater than 120, you may obtain your critical values from the standard normal distribution).

(g) Do you have statistical evidence to support your alternative hypothesis given the data? Why or why not? State your conclusion to the hypothesis test in the context of the question.

(H) Calculate the p-value of your test. (An exact value or interval range for the p-value may be given).

(Note: please also provide the mathematical expression you used to calculate your p-value).

Answer 1

mean house price of the newer area of Canberra (Gungahlin and Molonglo Valley) (x , say)

Region	mean house	number of house sold
Gungahlin	$815826	228
Molonglo Valley	$948222	28

combined mean of these newer areas are given by:-

and the mean house price of older area of Canberra (Inner North; Inner South; Belconnen; Woden Valley; Weston Creek; Tuggeranong) is given by (y , say)

combined means are given by:-

The provided sample means are shown below:

Also, the provided sample standard deviations are:

and the sample sizes are

The following null and alternative hypotheses need to be tested:

This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

Testing for Equality of Variances

A F-test is used to test for the equality of variances. The following F-ratio is obtained:

The critical values are and since F=0.259, then the null hypothesis of equal variances is rejected.

Based on the information provided, the significance level is , and the degrees of freedom are

Hence, it is found that the critical value for this two-tailed test is

The rejection region for this two-tailed test is

Since it is assumed that the population variances are unequal, the t-statistic is computed as follows:

Since it is observed that , it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is , and since , it is concluded that the null hypothesis is not rejected.

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean ​ is different that ​, at the 0.05 significance level.

hence there is no sufficient evidence of a difference in the mean house price between the newer area of Canberra (Gungahlin and Molonglo Valley) and the older area of Canberra (Inner North; Inner South; Belconnen; Woden Valley; Weston Creek; Tuggeranong).