Question

Blocking refers to the idea that the variability in a variable can be reduced by segmenting the data by some other variable. The data in the accompanying table represent the recumbent length​ (in centimeters) of a sample of 10 males and 10 females who are 40 months of age. Complete parts​ (a) through​ (d).

Blocking refers to the idea that the variability in a variable can be reduced by segmenting the data by some other variable. The data in the accompanying table represent the recumbent length​ (in centimeters) of a sample of 10 males and 10 females who are 40 months of age. Complete parts​ (a) through​ (d).

Males   Females
104.0 102.7
93.7   100.4
98.3   102.8
86.2   98.1
90.7   95.4
94.4   100.8
97.6   96.3
100.6   105.0
103.0 106.5
100.9   114.5

​(a) Determine the standard deviation of recumbent length for all 20 observations.

cm ​(Round to two decimal places as​ needed.)

​(b) Determine the standard deviation of recumbent length for the males.

.......cm ​(Round to two decimal places as​ needed.)

​(c) Determine the standard deviation of recumbent length for the females.

...........cm ​(Round to two decimal places as​ needed.)

​(d) What effect does blocking by gender have on the standard deviation of recumbent length for each​ gender?

...........cm ​(Round to two decimal places as​ needed.)

2, Suppose that a customer is purchasing a car. He conducts an experiment in which he puts 10 gallons of gas in the car and drives it until it runs out of gas. He conducts this experiment 15 times on each car and records the number of miles driven.

Car 1	Car 2
220	239
212	246
226	219
215	238
248	251
273	155
261	230
258	274
255	154
243	291
250	273
247	319
298	311
255	291
293	285

Median for Car 1

M = mi​ / 10 gal

​(Type an integer or decimal rounded to one decimal place as​needed.)

Median for Car 2

Me =   mi​ / 10 gal

​(Type an integer or decimal rounded to one decimal place as​needed.)

Answer 1

Q1

Formula - $SD(\sigma ) = \sqrt[2]({(\sum (X- \bar{X})^{2})}/(N-1))$

where,

SD is standard deviation

X is sample data given

Xbar is Mean of sample data

N is Number of sample data

1(a) - Standard Deviation for all 20 observations - Answer is 6.12 (Solution attached)

ALL 20 OBSERVATIONS
S No	X	X - Xbar	(X-Xbar)2
1	104	4.405	19.40403
2	93.7	-5.895	34.75103
3	98.3	-1.295	1.677025
4	86.2	-13.395	179.426
5	90.7	-8.895	79.12102
6	94.4	-5.195	26.98802
7	97.6	-1.995	3.980025
8	100.6	1.005	1.010025
9	103	3.405	11.59403
10	100.9	1.305	1.703025
11	102.7	3.105	9.641025
12	100.4	0.805	0.648025
13	102.8	3.205	10.27203
14	98.1	-1.495	2.235025
15	95.4	-4.195	17.59802
16	100.8	1.205	1.452025
17	96.3	-3.295	10.85703
18	105	5.405	29.21403
19	106.5	6.905	47.67903
20	114.5	14.905	222.159
Xbar(Mean)	99.595
∑(X-Xbar)2	711.4095
S.D. (as per formula)	SQRT(711.4095/19)	6.12

1(b) - Standard Deviation for Males observations - Answer is 5.67 (Solution Attached)

MALES
S No	X	X - Xbar	(X-Xbar)2
1	104	7.06	49.8436
2	93.7	-3.24	10.4976
3	98.3	1.36	1.8496
4	86.2	-10.74	115.3476
5	90.7	-6.24	38.9376
6	94.4	-2.54	6.4516
7	97.6	0.66	0.4356
8	100.6	3.66	13.3956
9	103	6.06	36.7236
10	100.9	3.96	15.6816
Xbar(Mean)	96.94
∑(X-Xbar)2	289.164
S.D. (as per formula)	SQRT(289.164/9)	5.67

1(c) - Standard Deviation for Females observations - Answer is 5.59 (Solution Attached)

FEMALES
S No	X	X - Xbar	(X-Xbar)2
1	102.7	0.45	0.2025
2	100.4	-1.85	3.4225
3	102.8	0.55	0.3025
4	98.1	-4.15	17.2225
5	95.4	-6.85	46.9225
6	100.8	-1.45	2.1025
7	96.3	-5.95	35.4025
8	105	2.75	7.5625
9	106.5	4.25	18.0625
10	114.5	12.25	150.0625
Xbar(Mean)	102.25
∑(X-Xbar)2	281.265
S.D. (as per formula)	SQRT(281.265/9)	5.59

1(d) - Answer is - The standard deviation is lower for each individual group than it is for the genders combined.

Q2

For median, first step is always to sort given sample data in ascending order

CAR 1	Ascending		CAR 2	Ascending
220	212		239	154
212	215		246	155
226	220		219	219
215	226		238	230
248	243		251	238
273	247		155	239
261	248		230	246
258	250		274	251
255	255		154	273
243	255		291	274
250	258		273	285
247	261		319	291
298	273		311	291
255	293		291	311
293	298		285	319

Formula for Median - when n (sample size) is odd number.

Median = (n+1)/2 th term

Formula for Median - when n (sample size) is even number.

Median: [(n/2) th term + {(n/2)+1} th term ] / 2

In our case, n is 15 which is odd number, ​

So,

Median for Car 1 = (15+1)/2 th term i.e. 8th term.

Median for Car 1 = 250

Median for Car 2 = (15+1)/2 th term i.e. 8th term.

Median for Car 2 = 251