Question

Create a hypothesis test question and show all the steps to solve it. Example: You found the mean for each quantitative variable. So, one at a time, is there significant evidence that the population maintenance cost for buses is more than###? and Find the regression equation for the 2 variables and explain what it is you found. Create a hypothesis test question for price and Size.

record	Agent	Price	Size	Bedrooms	Baths	Pool (yes is 1)	Garage (Yes is 1)	Days	Township	Mortgage type	Years	FICO	Default (Yes is 1)
1	Marty	206424	1820	2	1.5	1	1	33	2	Fixed	2	824	0
2	Rose	346150	3010	3	2	0	0	36	4	Fixed	9	820	0
3	Carter	372360	3210	4	3	0	1	21	2	Fixed	18	819	0
4	Peterson	310622	3330	3	2.5	1	0	26	3	Fixed	17	817	0
5	Carter	496100	4510	6	4.5	0	1	13	4	Fixed	17	816	0
6	Peterson	294086	3440	4	3	1	1	31	4	Fixed	19	813	0
7	Carter	228810	2630	4	2.5	0	1	39	4	Adjustable	10	813	0
8	Isaacs	384420	4470	5	3.5	0	1	26	2	Fixed	6	812	0
9	Peterson	416120	4040	5	3.5	0	1	26	4	Fixed	3	810	0
10	Isaacs	487494	4380	6	4	1	1	32	3	Fixed	6	808	0
11	Rose	448800	5280	6	4	0	1	35	4	Fixed	8	806	1
12	Peterson	388960	4420	4	3	0	1	50	2	Adjustable	9	805	1
13	Marty	335610	2970	3	2.5	0	1	25	3	Adjustable	9	801	1
14	Rose	276000	2300	2	1.5	0	0	34	1	Fixed	20	798	0
15	Rose	346421	2970	4	3	1	1	17	3	Adjustable	10	795	0
16	Isaacs	453913	3660	6	4	1	1	12	3	Fixed	18	792	0
17	Carter	376146	3290	5	3.5	1	1	28	2	Adjustable	9	792	1
18	Peterson	694430	5900	5	3.5	1	1	36	3	Adjustable	10	788	0
19	Rose	251269	2050	3	2	1	1	38	3	Fixed	16	786	0
20	Rose	547596	4920	6	4.5	1	1	37	5	Fixed	2	785	0
21	Marty	214910	1950	2	1.5	1	0	20	4	Fixed	6	784	0
22	Rose	188799	1950	2	1.5	1	0	52	1	Fixed	10	782	0
23	Carter	459950	4680	4	3	1	1	31	4	Fixed	8	781	0
24	Isaacs	264160	2540	3	2.5	0	1	40	1	Fixed	18	780	0
25	Carter	393557	3180	4	3	1	1	54	1	Fixed	20	776	0
26	Isaacs	478675	4660	5	3.5	1	1	26	5	Adjustable	9	773	0
27	Carter	384020	4220	5	3.5	0	1	23	4	Adjustable	9	772	1
28	Marty	313200	3600	4	3	0	1	31	3	Fixed	19	772	0
29	Isaacs	274482	2990	3	2	1	0	37	3	Fixed	5	769	0
30	Marty	167962	1920	2	1.5	1	1	31	5	Fixed	6	769	0

Answer 1

We will use R in this case to solve the hypothesis and regression problem.

Now, the mean of the price is 360048.2, we can find it using the following code:

> data = read.table(file = 'clipboard',sep = '\t',header=TRUE)
> mean(data$Price)
[1] 360048.2

Now, we will propose the following null and alternative hypothesis:

$\\ H_0: \mu \le 350000 \\ H_a: \mu > 350000$

Note that the test is a right tailed test. We will use the following statistic:

$T = \frac{\mu-350000}{S/\sqrt{n-1}} \sim t_{n-1}$

Now, the standard deviation of the price is 1178732.5. Hence the observed statistic value becomes:

$t = \frac{10048.2\cdot(29)^{1/2}}{117832.5} \approx 0.459$

Now,the p-value in this case would be:

$= P(t_{29} > 0.459) \approx 0.325$

Hence, at 95% confidence level, we can say that we fail to reject the null hypothesis.

The following code is used for calculation:

> mean(data$Price)
[1] 360048.2
> sd(data$Price)
[1] 117832.5
> pt(0.459,29,lower.tail=FALSE)
[1] 0.324828

Now, we will analyze the regression between price and space. The analysis in R is given below:

> model = lm(Price~Size,data=data)
> summary(model)

Call:
lm(formula = Price ~ Size, data = data)

Residuals:
   Min     1Q Median     3Q    Max 
-88801 -38584   3610  34405  95796 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 17838.578  29916.627   0.596    0.556    
Size           98.440      8.221  11.975 1.57e-12 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 48470 on 28 degrees of freedom
Multiple R-squared:  0.8366,    Adjusted R-squared:  0.8308 
F-statistic: 143.4 on 1 and 28 DF,  p-value: 1.566e-12

From the summary of the result, we can see that the linear regression equation between price and size is

$Price = 17838.578+98.440\cdot Size$

and the t test for coefficient of Size returns a value lesser than 0.05. Hence, we can say that the slope is non zero. Also the R squared statistic implies that the regression model explains 83.66% of the variation in Price, thus making the model good enough