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Lexington Housing Prices

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INTRODUCTION In the town of Lexington, Massachusetts, the real estate company VALMAX has received many complaints from two sets of customers: the first set believes that they were advised to post asking-prices for their houses that were too low compared to other brokers’ listings while the second set believes that they were advised to accept selling-prices for houses that they purchased that were too high relative to the selling-prices for houses brokered by other realtors. Based on the given data for both VALMAX and the other brokers (will henceforth be called “OTHERS”), it is decided that the clients’ complaints are justified. In order to properly come up with this conclusion, the correct data must first be chosen, the null and alternative hypotheses must be stated, the degrees of freedom will be calculated, the rejection region will be defined, and ultimately, the test statistic will be calculated and interpreted. After this, there are three methods for coming up with the end conclusion, the first being qualitative and the last two being quantitative.
ANALYSIS
Choosing the Data:
Three sets of data are given for both VALMAX and OTHERS for which we assume to be normally distributed: asking prices, selling prices, and the differences between the two. Either the asking/selling prices can be used or the differences data can be used, but not both. The reason why we have chosen to not use the asking and selling price data is because we cannot distinguish between the buying and selling agents. Also, we would have to assume that the asking and selling prices of VALMAX and OTHERS are in similar neighborhoods. Because there are two sets of discontent customers, the data for asking and selling prices would have to be combined in some fashion as to account for both sets of complaints. For these reasons, we have chosen to only use the difference data.
Creating the Null and Alternative Hypotheses: VALMAX’s clients are complaining that asking prices are too low and selling prices are too high. When combined, they are cumulatively complaining that the difference between the asking and selling price is too low. Therefore, the null hypothesis states that VALMAX’s difference is greater than or equal to the OTHERS difference. This indicates that VALMAX is not at fault and is not cheating their clients. The alternative hypothesis states that VALMAX’s difference is less than OTHERS difference and is therefore at fault and their clients’ complaints are justified (Exhibit 1).
Degrees of Freedom and Defined Rejection Region: To calculate the degrees of freedom, the number of samples taken from VALMAX is added to the number of samples taken from OTHERS and then the number 2 is subtracted. This calculates to 77. Because this number is greater than or equal to 30, we use the z table to define the rejection region. Using an alpha of 0.05 (confidence level of 95%), the z value comes out to -1.645. Our alternative hypothesis states that we are looking at a left-sided, one-tailed rejection area (Exhibit 2).
Calculating and Interpreting the Test Statistic: The average and standard deviation are first calculated on Excel for both VALMAX and OTHERS. Using the formula for the hypothesis test for two means, the test statistic is calculated out as -2.0604 (Exhibit 3). As mentioned earlier, there are three ways to now come up with a conclusion. The first qualitative way is to draw the normal distribution curves for the differences for both VALMAX and OTHERS and then look at the overlap. In this example, there is very little overlap, so this signifies that the two are not similar and therefore, the differences between them are significant (Exhibit 4). The second method is to compare the test statistic to the z value defining the rejection region. Because the z value is -1.645 and the test statistic is -2.0604, this value falls inside the rejection region and we can conclude that we have enough information to reject the null hypothesis and therefore, accept the alternative hypothesis. The third method is to compare the p value to alpha. The p value is determined by using the test statistic to find the correlating p value from the z table. This p value comes out to 0.0197. Because the p value is less than alpha, the null hypothesis can be rejected (Exhibit 5). The evidence proves that VALMAX is at fault and their clients’ complaints are justified; therefore, VALMAX should restructure their housing prices and decide how to move forth in a fair way.
Errors and Adjusting the Confidence Level: We began this assessment using 0.05 as our alpha with a 95% confidence level, but this can be changed depending on what error we decide is the most detrimental to this situation. First, let us begin by defining the type I and type II errors in this case: Type I: saying VALMAX is at fault when they actually are not (wrongfully rejecting null) Type II: saying VALMAX is not at fault when they actually are (wrongfully accepting null)
Both errors are extremely costly in this instance. If a type I error occurs, VALMAX might change its price restructuring and end up losing money and possibly damaging the company. Also, management might wrongfully punish its associates even though the associates have done nothing wrong. In the case of a type II error, clients might take the matter into their own hands and hire an external agent to view the data and discover that VALMAX has been cheating when even VALMAX was unaware. This could lead to several lawsuits and ultimately, a severe loss of customers and vitality of the business. At the very minimum, VALMAX would not restructure their company because they are unaware of the underlying blame. If we try to eliminate type I errors and thereby increase the confidence level to 99%, type II errors are now more likely to occur. The z value for rejection becomes -2.326 and suddenly the decision is to accept the null hypothesis. This is exactly what happens with a beta error: the null is wrongfully accepted when it should be rejected. If on the other hand we decrease the confidence level to 90%, type II errors will decrease, but type I errors will increase. The z value for rejection becomes -1.282. This would cause us to reject the null hypothesis when actually we should be accepting it

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