Question

Question Set 2: Two Independent Means Answer the following questions using the NYC2br.MTW file. You can find this dataset in this assignment in Canvas (i.e., where you downloaded this document and where you’ll upload your completed lab). Data were collected from a random sample of two-bedroom apartments posted on Apartments.com in Manhattan and Brooklyn.

A. What is one type of graph that could be used to compare the monthly rental rates of these two-bedroom apartments in Manhattan and Brooklyn? Explain why this is an appropriate graph. [10 points]

B. Using Minitab Express, Construct the graph you described in part A to compare the Manhattan and Brooklyn apartments in this sample. [10 points]

C. Use the five-step hypothesis testing procedure given below to determine if the mean monthly rental rates are different in the populations of all Manhattan and Brooklyn two-bedroom apartments. If assumptions are met, use a t distribution to approximate the sampling distribution. You should not need to do any hand calculations. Use Minitab Express and remember to include all relevant output. [30 points]

Step 1: Check assumptions and write hypotheses

Step 2: Calculate the test statistic

Step 3: Determine the p value

Step 4: Decide to reject or fail to reject the null hypothesis

Step 5: State a real-world conclusion

NYC2br.MTW file. - Data Set

Area Rent Address

Manhattan	5495	56 W 91st ST
Manhattan	2200	10 W 107th ST
Manhattan	3386	181 W 126th ST
Manhattan	2383	236 W 135th ST
Manhattan	2550	3 W 137th ST
Manhattan	2600	533 W 142nd ST
Manhattan	2600	260 W 171st ST
Manhattan	2150	518 W 204th ST
Manhattan	3200	680 Fort Washington AVE
Manhattan	2200	69 Cooper ST
Manhattan	2199	24 Thayer ST
Manhattan	2950	45 Tiemann PL
Manhattan	2780	510 E 117th ST
Manhattan	2695	314 E 106th ST
Manhattan	2900	320 E 93rd ST
Manhattan	5900	58 W 58th ST
Manhattan	25000	160 Central Park S
Manhattan	15000	30 E 62nd ST
Manhattan	4650	220 E 54th ST
Manhattan	2175	431 E 9th ST
Manhattan	7950	18 W 48th ST
Manhattan	2675	534 W 47th ST
Manhattan	3507	331 W 52nd ST
Manhattan	5195	236 E 47th ST
Manhattan	7750	445 W 35th ST
Manhattan	6883	1050 6th AVE
Manhattan	13754	7 W 21st ST
Manhattan	3995	172 Spring ST
Manhattan	5750	133 2nd AVE
Manhattan	6000	364 E 10 ST
Manhattan	3900	216 Centre ST
Manhattan	3400	167 Mott ST
Manhattan	16000	20 Greene ST
Manhattan	3200	174 Canal ST
Manhattan	4000	196 Stanton ST
Manhattan	8500	46 Warren ST
Manhattan	5950	108 South ST
Manhattan	3600	75 Wall ST
Manhattan	6598	44 Trinity PL
Manhattan	7995	377 Rector PL
Brooklyn	6508	41 River TER
Brooklyn	3300	76 Franklin ST
Brooklyn	3150	80 Meserole ST
Brooklyn	4225	224 Wythe AVE
Brooklyn	3256	228 Manhattan AVE
Brooklyn	2500	1421 Dekalb AVE
Brooklyn	2900	381 Myrtle AVE
Brooklyn	4350	9 Old Fulton ST
Brooklyn	5500	365 Bridge ST
Brooklyn	3900	117 Congress ST
Brooklyn	2600	224 Sachett ST
Brooklyn	4358	280 Ashland PL
Brooklyn	3125	229 5th AVE
Brooklyn	3900	753 Carroll ST
Brooklyn	3450	497 Saint Marks AVE
Brooklyn	2000	544 Franklin AVE
Brooklyn	2575	237 Troy AVE
Brooklyn	2300	223 Rockaway AVE
Brooklyn	2150	231 Amboy ST
Brooklyn	2000	789 Belmont AVE
Brooklyn	1800	542 E 93rd ST
Brooklyn	4213	125 Parkside AVE
Brooklyn	12500	3 Pierrepont PT
Brooklyn	5000	135 Willow ST
Brooklyn	2500	224 22nd ST
Brooklyn	2400	2025 Dorchester RD
Brooklyn	1900	7301 4th AVE
Brooklyn	2000	2071 E 61st ST
Brooklyn	1925	1063 E 2nd ST
Brooklyn	2300	2031 W 6th ST
Brooklyn	2250	9747 Shore RD
Brooklyn	3500	155 Oceana DR
Brooklyn	2759	3510 Neptune AVE
Brooklyn	2200	2832 Bragg ST
Brooklyn	1950	1780 W 3rd ST
Brooklyn	2400	9602 4th AVE
Brooklyn	3500	26 Bay Ridge AVE
Brooklyn	2400	1519 New York AVE
Brooklyn	3304	941 Washington AVE
Brooklyn	3300	412 Herkimer ST
Brooklyn	5203	593 Baltic ST
Brooklyn	7500	78 Amity ST
Brooklyn	2625	692 Chauncey ST
Brooklyn	15000	260 Park AVE
Brooklyn	8750	100 Jay ST
Brooklyn	6455	475 Clermont AVE
Brooklyn	5775	300 Ashland PL

Answer 1

Result:

MINITAB used.

A. What is one type of graph that could be used to compare the monthly rental rates of these two-bedroom apartments in Manhattan and Brooklyn? Explain why this is an appropriate graph. [10 points]

Box plot graph can be used to compare the monthly rental rates of two-bedroom apartments in Manhattan and Brooklyn. since rent data may be in skewed distribution, median and quartiles may be best describe the data.

B. Using Minitab Express, Construct the graph you described in part A to compare the Manhattan and Brooklyn apartments in this sample. [10 points]

Step 1: Check assumptions and write hypotheses

Test and CI for Two Variances: Rent vs Area

Method

σ₁: standard deviation of Rent when Area = Brooklyn

σ₂: standard deviation of Rent when Area = Manhattan

Ratio: σ₁/σ₂

The Bonett and Levene's methods are valid for any continuous distribution.

Descriptive Statistics

Area	N	StDev	Variance	95% CI for σ²
Brooklyn	47	2631.489	6924734.700	(2634006.857, 1.98238E+07)
Manhattan	40	4642.421	2.15521E+07	(7617068.821, 6.74261E+07)

Ratio of Variances

Estimated Ratio	95% CI for Ratio using Bonett	95% CI for Ratio using Levene
0.321303	(0.056, 1.911)	(0.078, 1.206)

Test

Null hypothesis	H₀: σ₁² / σ₂² = 1
Alternative hypothesis	H₁: σ₁² / σ₂² ≠ 1
Significance level	α = 0.05

Method	Test Statistic	DF1	DF2	P-Value
Bonett	*			0.166
Levene	3.04	1	85	0.085

Levene test shows that equality of variance assumption is not violated. KS test for normality shows that there is some violation of normality.

Step 2: Calculate the test statistic

Even though data are not normal, sample sizes are large( > 30), we use t test to compare the means.

Test statistic = -2.06

Step 3: Determine the p value

P value = 0.042

Step 4: Decide to reject or fail to reject the null hypothesis

Reject the null hypothesis

Step 5: State a real-world conclusion

We conclude that there is significant difference in the monthly rental rates of these two-bedroom apartments in Manhattan and Brooklyn.