1 IntroductionAccording to BBC News the main factor affecting the house prices is the location of the house. For instance ‘In London, the average property value has risen by nearly 70% in 10 years’ (Peachey and Stephenson, 2014). Therefore, analysing the relationship between house prices and various factors is such an interesting study because some factors are strongly related and affect the prices however, some of them slightly influence value of houses. The report includes hypothesis tests with explained particular aspects which show how different elements are related with each other in the house market. The overall research questions are stated below:What is the relation between housing prices, number of offers, type of neighbourhood, bricks and how affect each other? How do the house prices depend on various factors as square footage, type of neighbourhood and bricks?The report investigate diversified relations between variables in data set. The hypothesis stated involve tests, terms, concepts to provide enough evidence of the statistical significance.2 Data and Research Methods2.1 The SampleThe randomly chosen sample from the population of houses includes 128 observation. In the provided data set each observation lists the house’s price, number of offers, bedrooms, bathrooms, square footage, type of neighbourhood where the house is located and whether the house is made of bricks. These seven variable names are listed in row 1. The rectangular array of data which is cross-sectional contains numerical continuous variables as price and square footage, numerical discrete such as number of offers, bathrooms, bedrooms, categorical nominal coded numerically as type of neighbourhood and dummy variable as whether the house is made of bricks.Table 1.0 displays summaries of Bricks and Neighborhood, Figure 1.1, shows that the sample data contains 86 or 67% houses without bricks and 42 or 33% houses with bricks. The data contains 30% houses from more prestigious area of neighbourhood, 35% and 34% in the traditional area of neighbourhood as in the Figure 1.0.Table 1.0 Figure 1.1Table 1.3Therefore, according to the pivot table and column chart (Figure 1.4, 1.5) the largest number of offers (5 and 6 offers) are for houses in the neighbourhood 1 and 2 which are more traditional and cheaper in comparison to the prestigious neighbourhood area numer 3. We can summarize that people prefer more traditional area with lower prices. Table1.4 Figure 1.5 Moreover, from the excel file, ‘Data’ sheet, we see that data contains 56% of all houses with 2 bathrooms, 43% with 3 and 30% with 4 bathrooms. Additionally, 23% of houses from the sample, contains 2 bedrooms, 52% with 3, 23% with 4 and just 2% contains 5 bedrooms in the house.Figure 1.6 Shows correlations between variables. 2.2 Limitations The research reached its aim however, there appeared some inevitable limitations. First of all, the provided 128 sample size is not small but the study should have involved larger sample with more specifics about other factors which influence house prices. Secondly, there is no information about the date of the data which that is a strong limitation when the current assumptions are made on the out-of-date evidence. Additionally, in this study the certain level of subjectivity can be found due to the fact that the author conducted tests by herself.3 Findings 3.1 Difference in average offers between neighborhood 1 and 3 We investigate if average offers for houses in neighborhood 1 and 3 vary. We do Z-test for two means (Table 2.0), to analyse the relation between those mean values.Table 2.0Z-Test: Two Sample for Means H0- The mean of offers between houses in neighborhood 1 and 3 are equalH1- The mean of offers between houses in neighborhood 1 and 3 are not equalH0: ?N1= ?N3;H0: ?N1- ?N3=0H1: ?N1- ?N3?0Z value is greater than our z critical two-tail value. Therefore, we can reject the null hypothesis that average offers for houses in areas 1 and 3 are equal. Also the p-value equals 0.00 which is smaller than 0.05, it indicates that we do not reject the alternative hypothesis and based on those two samples we do not have enough evidence to summarize that number of offers in neighborhood 1 and 3 are equal.3.2 Difference in proportions of offers between neighborhood 1 and 3Therefore, we do the Z-test for proportions of offers between these two neighborhoods and examine the result.Table 2.1Z-Test: Two Sample for Proportions Data Hypothesized Difference 0Level of Significance 0.05Offers in area 1: Size of sample 37Proportion 0.36Offers in area 3: Size of sample 65Proportion 0.64Test statistic: z = (0.36 – 0.64) / 0.102666 -2.7273Two-tailed p-value 0.006386H0- The proportion of offers between houses in neighborhood 1 and 3 are equalH1- The proportion of offers between houses in neighborhood 1 and 3 are not equalH0: ?N1= ?N3;H0: ?N1- ?N3=0H1: ?N1- ?N3?0According to Table 2.1, p-value = 0.006386 < 0.05 = ? , H0 is rejected. At 5% level of significance level, there is sufficient evidence to reject the null hypothesis and conclude that on average, the proportions of offers between areas 1 and 3 are not equal.Figure 2.23.3 Difference in average house prices between neighborhood 1 and 3 According to excel sheet 'T-test' and Figure 2.3, 2.4 which indicate clearly greater average prices in area in neighborhood 3 than 2. We make a test to provide the evident proof for the assumed hypothesis.Figure 2.3 Table 2.4Row Labels Average of Price1 220309.093 318589.74Grand Total 266489.16 Table 2.5 t-Test: Two-Sample Assuming Unequal Variances H0- The mean price of houses between neighborhood 1 and 3 are equalH1- The mean price of houses between neighborhood 1 and 3 are not equalH0: µN1= µN3;H0: µN1- µN3=0H1: µN1- µN3?0The mean price in neighbourhood 1, less traditional neighborhood area equals 220309.09 and in the prestigious area 3, the average price is 318589.74 . The T-test statistic shows that the p-value- 0.00 is below 0.05. Thererefore, we reject the null hypothesis and summarize that with 95% confidence that mean house prices in two areas are not equal, a significant difference in mean price between areas 1 and 3 equals 98222.214.171.124 How house prices depend on square footage?We will investigate if there is a relation between the dependent variable- housing prices (Y) versus the explanatory- square footage (X). From Figure 2.6, we can recognize a linear, positive relationship between variables. It is not a perfect relationship, the points tend to rise from bottom left- 1500 square footage to top right- approximately 3000 square footage outside from the mean regression line. The scatterplot indicates that the price is dependent on the square footage, the bigger number of square footage of the house cause larger value of the house. Analysing the relationship between the house price and square footage of the house and see how strong it is, we will use the simple linear regression test as shown below in Figure 2.7.Figure 2.6 Table 2.7Summary output. Table 2.8 Equation on Simple Linear Regression: Y = a + b1X1+…+uY= 8896.466+ 118.15XHypothesis:H0: There is no linear relationship between house prices and square footage of houses (the slope is zero)H1: There is a linear relationship between house prices and square footage of houses (the slope is not zero)H0: ?1=0H1: ?1?0Therefore, Y equals the predicted price. From the Figure 2.8 the slope, 118.15, indicates that the selling price index tends to increase by about 118.15 for each one-unit increase in the square footage index. Moreover, if two houses are compared, where the second has one square footage more than the first house, the predicted price index for the second house is 118.15 larger than the selling price index for the first house. The intercept, 8896.47, shows the least squares line that enables predict the house price of Y values for the range of observed X, square footage values. The slope, 118.15, does not equal 0.00, the p-value= 0 is below 0.05, it indicates that we reject null hypothesis with 95% of confidence that there is a linear relationship between house prices and square footage.We see from Figure 2.7 the standard error equals 37250.17 is small compared to standard deviation which is 45178.52,the R Square equals 0.33 which is close to zero rather than 1, it indicates that square footage is not a main factor which explain changes in house prices and there is a small correlation, we will add another explanatory variable (bricks) which make.3.5 How house prices depends on square footage and bricks?Furthermore, we will make the multiple regression test by adding another variable- bricks to the previous simple regression to check how the house prices depend on square footage and bricks and how strong the relation is.General Multiple Regression Equation:Y = a + b1X1 + b2X2+....+BKXKMultiple Linear Regression:Y= Price+ Square footage + BricksEstimated Regression LinePredicted Price =-18888.58 +132.12Sq Ft+ 46890.19Houses with bricksTable 2.9 Regression Statistics Figure 3.0 Anova table Figure 3.1 H0: All coefficients of the explanatory variable are zeroH1: At last one of the coefficients is not zeroH0: ?2= ?3=0H1: ?2? ?3?0Interpretation:According to Figure 2.9, 47% of Variation in Overhead is explained by Square footage and Houses with bricks, the prediction of the housing price from the multiple regression is not accurate. Figure 3.1 indicates when Square footage increase by 1 unit then the house price increase by 132.12, house price with bricks is higher by 46890.192 than price of house without bricks(dummy variables);Therefore, form Figure 2.7 and 2.9 we see a clear increase in Adjusted R Square from the simple linear regression which is 0.32 to higher- 0.47 closer to 1, the multiple regression, this shows that the variation in house prices is explained better after adding another variable. As a result of adding an extra variable- bricks, the Adjusted R Square increases, 47% of changes in house prices is being explained by two factors: square footage and bricks. P-value= 0.00 less than 0.05 means that we do not reject the alternative hypothesis and conclude with 95% of confidence that at least one of the coefficients is not zero.3.6 Variance in houses is made of bricks and withoutWe make a f-test to prove that the value of houses with bricks is greater than without.H0: The variability of house prices with bricks is equal or less than that of houses without bricksH1: The variability of house prices with bricks is greater than that of houses without bricks H0: ?wB-?nB?0H1: ?wB-?nB>0Figure 3.2 Count of Price StdDev of Price Average of Price Row Labels No Yes No Yes No Yes1 37 7 31547 31000 217168 2369143 23 16 27048 32591 296461 350400Grand Total 60 23 48903 61943 247563 315861Table 3.3 F-Test Two-Sample for Variances We can assume, Figure 3.3 that there is variance between house prices with bricks and without bricks, 2879263391.41?2041098790.70. Additionally, the F Critical value= 1.53 is higher than F-value which is 1.41 so that we cannot reject the null hypothesis and we run t-test assuming unequal variances below.3.7 Difference in mean house prices with and without bricksFigure 3.4 t-Test: Two-Sample Assuming Unequal Variances H0: The mean of house prices with bricks is equal or less than that of houses without bricksH1: The mean of house prices with bricks is greater than that of houses without bricks H0: µwB-µnB?0H1: µwB-µnB>0T= 5.37According to the Figure 3.4 the t-test statistic is, t= 5.37 which is greater than Critical one-tail value= 1.67, what is more the P- value= 0.00 less than 0.05. Therefore, the mean house prices with bricks-295538.10>243916.28- mean house prices without bricks. We reject the null hypothesis, there is enough evidence to conclude with 95% of confidence that the alternative hypothesis that mean house prices with bricks are greater than without bricks is statistically significant.4 Conclusions and RecommendationsDifferent types of tests showed how various factors affect the value of houses and how relations vary between different variables. That was an engaging process of investigation from provided data. There are main factors which affect house prices: neighbourhood, whether the house is made from bricks and the size- square footage. The area of houses is the most dominant factor, the research shows that houses in more traditional area are cheaper and there is greater number of offer what means higher demand. As the research shows, it is recommended to notice that some potential factors are strongly related to the house prices. Furthermore, customers are advised to take into consideration factors, previously mentioned, as the area- the location of the property and the square footage of the house because those are the most strongly related components when buying a house.