Question

In Florida, real estate agents refer to homes having a swimming pool as pool homes. In this case, Sunshine Pools Inc. markets and installs pools throughout the state of Florida. The company wishes to estimate the percentage of a pool's cost that can be recouped by a buyer when he or she sells the home. For instance, if a homeowner buys a pool for which the current purchase price is $30,000 and then sells the home in the current real estate market for $20,000 more than the homeowner would get if the home did not have a pool, the homeowner has recouped (20,000/30,000) × 100% = 66.67% of the pool's cost.

To make this estimate, the company randomly selects 80 homes from all of the homes sold in a Florida city (over the last six months) having a size between 2,000 and 3,500 square feet. For each sampled home, the following data are collected: Price - selling price (in thousands of dollars); Size - square footage; Bathrooms - the number of bathrooms; Rating - a niceness rating (expressed as an integer from 1 to 7 and assigned by a real estate agent); and Pool - whether or not the home has a pool (1 = yes, 0 = no).

1. Your results in (4) show different mean selling prices for homes with a pool and homes that do not have a pool. Can we conclude that the mean selling price for homes without a pool is significantly lower than the mean selling price for homes with a pool.

2. What about differences in average price among homes with different number of bathrooms? Can we conclude that the differences are significant? If so, which types of bedrooms have significant mean differences?

3. The president of the company told a potential buyer that on average, homes in the city sell for more than 258,000. Is there evidence to support this claim?

4. (a) Is a home having a pool independent of the number of bathrooms?

(b) What proportion of homes with a pool have 2, 2.5, 3, 3.5, and 4 bathrooms?

5. The president of the company would like to estimate the percentage of homes with a pool for the entire city using the sample data he has. At the 95% confidence level, provide an interval estimate, with an explanation, for the president.

6. Create a correlation matrix of the following metrics: Price, Size, Bathrooms, and Rating. Comment on the results.

Data Set:

Home	Price	Size	Bathrooms	Rating	Pool
1	260.9	2666	2.5	7	0
2	337.3	3418	3.5	6	1
3	268.4	2945	2.0	5	1
4	242.2	2942	2.5	3	1
5	255.2	2798	3.0	3	1
6	205.7	2210	2.5	2	0
7	249.5	2209	2.0	7	0
8	193.6	2465	2.5	1	0
9	242.7	2955	2.0	4	1
10	244.5	2722	2.5	5	0
11	184.2	2590	2.5	1	0
12	325.7	3138	3.5	7	1
13	266.1	2713	2.0	7	0
14	166.0	2284	2.5	2	0
15	330.7	3140	3.5	6	1
16	289.1	3205	2.5	3	1
17	268.8	2721	2.5	6	1
18	276.7	3245	2.5	2	1
19	222.4	2464	3.0	3	1
20	241.5	2993	2.5	1	0
21	307.9	2647	3.5	6	1
22	223.5	2670	2.5	4	0
23	231.1	2895	2.5	3	0
24	216.5	2643	2.5	3	0
25	205.5	2915	2.0	1	0
26	258.3	2800	3.5	2	1
27	227.6	2557	2.5	3	1
28	255.4	2805	2.0	3	1
29	235.7	2878	2.5	4	0
30	285.1	2795	3.0	7	1
31	284.8	2748	2.5	7	1
32	193.7	2256	2.5	2	0
33	247.5	2659	2.5	2	1
34	274.8	3241	3.5	4	1
35	264.4	3166	3.0	3	1
36	204.1	2466	2.0	4	0
37	273.9	2945	2.5	5	1
38	238.5	2727	3.0	1	1
39	274.4	3141	4.0	4	1
40	259.6	2552	2.0	7	1
41	285.6	2761	3.0	6	1
42	216.1	2880	2.5	2	0
43	261.3	3426	3.0	1	1
44	236.4	2895	2.5	2	1
45	267.5	2726	3.0	7	0
46	220.2	2930	2.5	2	0
47	300.1	3013	2.5	6	1
48	260.0	2675	2.0	6	0
49	277.5	2874	3.5	6	1
50	274.9	2765	2.5	4	1
51	259.8	3020	3.5	2	1
52	235.0	2887	2.5	1	1
53	191.4	2032	2.0	3	0
54	228.5	2698	2.5	4	0
55	266.6	2847	3.0	2	1
56	233.0	2639	3.0	3	0
57	343.4	3431	4.0	5	1
58	334.0	3485	3.5	5	1
59	289.7	2991	2.5	6	1
60	228.4	2482	2.5	2	0
61	233.4	2712	2.5	1	1
62	275.7	3103	2.5	2	1
63	290.8	3124	2.5	3	1
64	230.8	2906	2.5	2	0
65	310.1	3398	4.0	4	1
66	247.9	3028	3.0	4	0
67	249.9	2761	2.0	5	0
68	220.5	2842	3.0	3	0
69	226.2	2666	2.5	6	0
70	313.7	2744	2.5	7	1
71	210.1	2508	2.5	4	0
72	244.9	2480	2.5	5	0
73	235.8	2986	2.5	4	0
74	263.2	2753	2.5	7	0
75	280.2	2522	2.5	6	1
76	290.8	2808	2.5	7	1
77	235.4	2616	2.5	3	0
78	190.3	2603	2.5	2	0
79	234.4	2804	2.5	4	0
80	238.7	2851	2.5	5	0

Answer 1

1.

The mean selling prices for homes with a pool and for homes that do not have a pool are different. According to the summary statistics, average price for homes with pool is 276.056 and average price for homes without pool is 226.9.

Pool presence	Average of Price
0 - No	226.9
1 - Yes	276.055814

To find significant difference between mean selling prices, t-test is used. T-test is used here since the population has exactly two values, which is 0(where homes do not have pool) and 1(where homes have pool).

Assuming null hypothesis to be H₀ : mean selling price for homes without a pool is not significantly lower than the mean selling price for homes with a pool

Alternative hypothesis H₁ : mean selling price for homes without a pool is significantly lower than the mean selling price for homes with a pool

P-value is calculated using RStudio or using other analytical tools. It can also be found out from p-value tables.

P-value is = 9.54846771943487E-126 . Now when alpha is 0.05 and p-value is less than alpha, nullhypothesis is rejected.

Hence, we can conclude that the mean selling price for homes without a pool is significantly lower than the mean selling price for homes with a pool.

2. What about differences in average price among homes with different number of bathrooms? Can we conclude that the differences are significant? If so, which types of bedrooms have significant mean differences?

Let null hypothesis be H₀ : There is no significant differences in average prices among homes with different no. of bathrooms.

Alternative hypothesis be H1 : There is significant differences in average prices among homes with different no. of bathrooms.

Now for testing significant differences in mean, we use t test. Assuming the distribution to be normal and a confidence level of 95%, the p-value has been found out to be 6.64343313452496E-111. P-value was found out using Rstudio commands. This can also be found out from t-test calculation tables.

Since p-value is less than alpha(0.05), null hypothesis is rejected.

This means there is significant differences in average prices among homes with different no. of bathrooms.

Bathrooms Average of Price 2 241.1454545 2.5 242.9155556 3 254 3.5 300.6666667 4 309.3

Significant mean difference is found in homes that have 4 and 2 bathrooms

3. The president of the company told a potential buyer that on average, homes in the city sell for more than 258,000. Is there evidence to support this claim?

The average of all the 80 sample homes = $ 253,321.25

The president quoting $ 258,000 as the average price of a home is acceptable. Even though there is a difference of $ 4,678.75 between the actual average of the samples and the price quoted by the president, it is variable because the calculated value is only based on the random 80 samples.

4. (a) Is a home having a pool independent of the number of bathrooms?

No, homes having a pool cannot be independent of the number of bathrooms.

(b) What proportion of homes with a pool have 2, 2.5, 3, 3.5, and 4 bathrooms?

No.of Bathrooms	Count of homes with the specified no. of Bathrooms	Percentages
2 2.5 3 3.5 4	4 19 8 9 3	13.75 56.25 15 11.25 3.75
Total:	43 out of 80 homes have bathrooms	100