Question

Problem 3 – Warwick rental property price Warwick is a suburb in the Southern Downs Region of Queensland. Warwick has a population of 12,223 people and 32.29% of its occupants live in rental accommodation. A real estate office claims that the average rental price in Warwick is less than $400 per week. To test if this belief is correct, a random sample of 96 rental properties is selected. The mean of the weekly rental price computed from the sample is $385. Assume that the population standard deviation of weekly rental price is $100.

a) You were recently hired as a junior data analyst working for the real estate office. Assist the office in performing a hypothesis test at a 3% level of significance to check whether the claim made is justified. Display the six steps process (involving drawing ​the rejection region/s and determining the critical value/s for the decision rule) in performing the test. ​

​ b) Calculate the p-value of the test above. Display working. State the decision rule of the test should you want to use the p-value method hypothesis testing. ​ ​

​ c) This hypothesis test is conducted on the basis that the sampling distribution of the sample mean is approximately normally distributed. Specify the required condition to ensure this. ​ ​ ​ d) Identify which one of these two types of error (Type I or Type II) you could make when drawing the conclusion in part a). Briefly explain your selection.

Answer 1

We are given that the population standard deviation is , n=96 and .

(a).

1. Null Hypothesis: The average rental price in Warwick is $400.

2. Alternate Hypothesis: The average rental price in Warwick is less than $400.

3. Level of significance:

4. We know that , and n=96.

5. Test Statistic:

will have a standard Normal distribution.

For our data,

  

6. The critical region:

Since we have an one sided alternative, the region of reject at 3% level of significance is:

  .  

7. Decision: Since the Calculated value of Z is not in the rejection region,(z=-1.4697>-1.8808) we fail to reject the null hypothesis. Hence we conclude that there is not enough evidence in the real estate office's claim that the rental price in Warwick is less than $400.

b) The p-value can be obtained from a table or EXCEL function NORM.S.DIST(-1.4697,true) which is 0.0708. Since the p-value(0.0708)>0.03, we fail to reject the null hypothesis.

c) Apart from Normality, we also assume that the samples are drawn independently of each other. Also since the sample size=96 which is large, we can conveniently assume by invoking Central Limit Theorem that the sample comes from a Normal population.

d). The p-value is the Probability of Type I error and we could make Type I error when drawing conclusion in part a).