Question

Although painful experiences are induced in social rituals in many parts of the world, little is known about the social effects of pain. Will sharing painful experiences in a small group lead to greater bonding of group members than sharing a similar non‑painful experience? Fifty‑four university students in South Wales were divided at random into a pain group containing 2727 students, with the remaining students in the no‑pain group.

Pain was induced by two tasks. In the first task, students submerged their hands in freezing water for as long as possible, moving metal balls at the bottom of the vessel into a submerged container; in the second task, students performed a standing wall squat with back straight and knees at 9090 degrees for as long as possible. The no‑pain group completed the first task using room temperature water for 9090 seconds and the second task by balancing on one foot for 6060 seconds, changing feet if necessary.

In both the pain and no‑pain settings, the students completed the tasks in small groups, which typically consisted of four students and contained similar levels of group interaction. Afterward, each student completed a questionnaire to create a bonding score based on answers to questions such as “I feel the participants in this study have a lot in common” or “I feel I can trust the other participants.” The table contains the bonding scores of the two groups.

No‑pain group	3.43	4.86	1.71	1.71	3.86	3.14	4.14	3.14	4.43	3.71
	3.00	3.14	4.14	4.29	2.43	2.71	4.43	3.43	1.29	1.29
	3.00	3.00	2.86	2.14	4.71	1.00	3.71
Pain group	4.71	4.86	4.14	1.29	2.29	4.43	3.57	4.43	3.57	3.43
	4.14	3.86	4.57	4.57	4.29	1.43	4.29	3.57	3.57	3.43
	2.29	4.00	4.43	4.71	4.71	2.14	3.57

(a) Find the five‑number summaries for the pain and no‑pain groups. (Enter your answers rounded to two decimal places.)

No‑pain group Min =

No‑pain group ?1=

No‑pain group Median =

No‑pain group ?3=

No‑pain group Max =

pain group Min =

pain group ?1=

pain group Median =

pain group ?3=

pain group Max =

(b) Construct a comparative boxplot for the two groups and choose the correct image from the choices.

(c) Which group tends to have higher bonding scores?

no‑pain group

pain group

Compare the variability in the two groups.

The no‑pain group tends to have less variable bonding scores.

The variability in the two groups is similar.

The pain group tends to have less variable bonding scores.

Choose the correct statement about outliers.

Neither group contains one or more clear outliers.

Only the no‑pain group contains one or more clear outliers.

Only the pain group contains one or more clear outliers.

Both groups contain one or more clear outliers.

A scale is tested by repeatedly weighing a standard 9.0 kg weight. The weights for 8 measurements are

8.9,8.6,8.0,8.2,8.3,8.5,8.2,9.3

Mean: ________ kg

Answer 1

Fifty‑four university students in South Wales were divided at random into a pain group containing 27 students, with the remaining students in the no‑pain group.

The table contains the bonding scores of the two groups are as follows

No‑pain group	3.43	4.86	1.71	1.71	3.86	3.14	4.14	3.14	4.43	3.71
	3.14	4.14	4.29	2.43	2.71	4.43	3.43	1.29	1.29	3.00
	3.00	2.86	2.14	4.71	1.00	3.71	3.00
Pain group	4.71	4.86	4.14	1.29	2.29	4.43	3.57	4.43	3.57	3.43
	3.86	4.57	4.57	4.29	1.43	4.29	3.57	3.57	3.43	4.14
	4.00	4.43	4.71	4.71	2.14	3.57	2.29

(a) Find the five‑number summaries for the pain and no‑pain groups. (Enter your answers rounded to two decimal places.)

So we will five number summary i.e Min , ?1 , Median , ?3 , Max for both groups .

For No Pain group , first we will arrange data in assending order

No Pain :-

1.00 1.29 1.29 1.71 1.71 2.14 2.43 2.71 2.86 3.00 3.00 3.00 3.14 3.14 3.14
3.43 3.43 3.71 3.71 3.86 4.14 4.14 4.29 4.43 4.43 4.71 4.86

Thus

Min : - 1

Max : - 4.86

Now we will find Median , 1st Quartille Q1 and 3rd quartile Q3 for No Pain Group .

Number of observation = 27 ( odd )

Thus

Median = ( n+1 )/2 th obsⁿ

               = ( 27 + 1 )/2 th obsⁿ = ( 28 )/2 th obsⁿ

               = 14 th obsⁿ

               = 3.14

Hence Median = 3.14

Now Q1 is nothing but median of numbers less than 3.14 (14 th obs ) i.e True median.

Thus Median of these numbers

1.00 1.29 1.29 1.71 1.71 2.14 2.43 2.71 2.86 3.00 3.00 3.00 3.14 3.14

Number of observation = 14 (even )

Q1 = [ ( n/2 ) th obsⁿ + ( n/2+1 ) th obsⁿ ] / 2

Q1 = [ ( 14/2 ) th obsⁿ + ( 14/2+1 ) th obsⁿ ] / 2

Q1 = [ 7 th obsⁿ + 8 th obsⁿ ] / 2

Q1 = [ 2.43 + 2.71 ] / 2

Q1 = 2.57

Now Q3 is nothing but median of numbers greater than 3.14 (14 th obs ) i.e True median.

Thus Median of these numbers

3.14 3.14 3.43 3.43 3.71 3.71 3.86 4.14 4.14 4.29 4.43 4.43 4.71 4.86

Number of observation = 14 (even )

Thus

Q3 = [ ( n/2 ) th obsⁿ + ( n/2+1 ) th obsⁿ ] / 2

Q3 = [ ( 14/2 ) th obsⁿ + ( 14/2+1 ) th obsⁿ ] / 2

Q3 = [ 7 th obsⁿ + 8 th obsⁿ ] / 2

Q3 = [ 3.86 + 4.14 ] / 2

Q3 = 4.00

Hence

Min = 1.00    ,   Max = 4.86

Q1 = 2.57 , Median = 3.14 and   Q3 = 4.00

Also From R-Software we have 5-number summary for No Pain as follow

> summary(x)               # No Pain Group
   Min. 1st Qu. Median    Mean 3rd Qu.    Max.
1.000   2.570   3.140   3.137   4.000   4.860

Similarly we will find 5-number summary for Pain group using R-Software ( if manually calculation also required for Pain Group can ask for that in comment box }

> summary(y)                # Pain Group
   Min. 1st Qu. Median    Mean 3rd Qu.    Max.
1.290   3.500   4.000   3.714   4.430   4.860

Thus

No‑pain group Min = 1.00

No‑pain group ?1 = 2.57

No‑pain group Median = 3.14

No‑pain group ?3 = 4.00

No‑pain group Max = 4.86

pain group Min = 1.29

pain group ?1 = 3.50

pain group Median = 4.00

pain group ?3 = 4.43

pain group Max = 4.86

(b) Construct a comparative boxplot for the two groups and choose the correct image from the choices.

We will use R-Software to draw box-plot , similar plot can be drawn mannually .

For No Pain Group

Pain Group :-

Multiple Boxplots for comparision :-

R-Code

> boxplot(x,y,at = c(1,2),names = c("No Pain", "Pain" ), col = c(2,7))

(c) Which group tends to have higher bonding scores?

In the above Boxplot and 5-number summary , we can see that Pain group have higher bonding scores as Pain Group have Q1 , Median and Q3 greater than No-Pain group , which suggest that bonding scores are higher in Pain Group .

So Correct option is

pain group

Q . Compare the variability in the two groups.

From boxplot we can see that No Pain Group have higher spread ( denoted by red region for No-Pain in above plots ) in data than Pain group , which implies that no Pain Group have higher variability than Pain group i.e pain group have less variability .

Thus Correct option is

The pain group tends to have less variable bonding scores.

Q. Choose the correct statement about outliers.

In above given box-plot for Pain group we observe two outliers ( denoted by dots in lower part ) which are 1.29 & 1.43

No outlier for No -Pain Group

Also 1.5IQR range for two groups are as follow

• For No-Pain Group

Interval = { Q1 - 1.5 ( Q3-Q1) , Q3 + 1.5 ( Q3-Q1) }

              = { 2.57-1.5*(4.00-2.57) , 4.00+1.5*(4-2.57) }
              = { 0.425 , 6.145 }

So For No-Pain Group any number outside these 1.5IQR { 0.425 , 6.145 } will be consider as outlier.

We note that there are no number outside this limit so , no outlier for No Pain Group.

• For Pain Group

Interval = { Q1 - 1.5 ( Q3-Q1) , Q3 + 1.5 ( Q3-Q1) }

              = { 3.5-1.5*(4.43-3.5) , 4.43 + 1.5*(4.43-3.5) }
              = { 2.105 ,   5.825 }

So For Pain Group any number outside these 1.5IQR { 2.105 ,   5.825 } will be consider as outlier.

We note that there are two observation which are 1.29 & 1.49 outside this limit so , two outlier for Pain Group.

Hence Correct option are

Only the pain group contains one or more clear outliers.

A scale is tested by repeatedly weighing a standard 9.0 kg weight. The weights for 8 measurements are

8.9,8.6,8.0,8.2,8.3,8.5,8.2,9.3

To find mean

Formula :-

Mean = = (8.9 + 8.6 + 8.0 + 8.2 + 8.3 + 8.5 + 8.2 + 9.3) / 8

                            = 68 / 8 = 8.5

Mean: __8.5__ kg