Question

A statistic center compiles data on the length of stay by patients in​ short-term hospitals. A random sample of 2121 patients yielded the data on length of​ stay, in​ days, shown below. Complete parts​ (a) through​ (e) below. 24 4 5 18 4 14 55 23 23 7 10 5 9 1 20 25 5 18 19 14 2324 4 5 18 4 14 55 23 23 7 10 5 9 1 20 25 5 18 19 14 23.

a.

Obtain and interpret the quartiles.

Upper Q 1Q1	equals=	nothing
Upper Q 2Q2	equals=	nothing
Upper Q 3Q3	equals=	nothing

​(Type integers or​ decimals.)

Interpret the quartiles. Choose the correct answer below.

A.The quartiles suggest that​ 33% of the patients stayed less than

55

​days, 33% stayed between

55

and

2323

​days, and​ 33% stayed more than

2323

days.

B.The quartiles suggest that the average length of stay is

1414

days.

C.The quartiles suggest that all the patients stayed between

55

and

2323

days.

D.The quartiles suggest that​ 25% of the patients stayed less than

55

​days, 25% stayed between

55

and

1414

​days, 25% stayed between

1414

and

2323

​days, and​ 25% stayed more than

2323

days.

b.

Determine and interpret the interquartile range​ (IQR).

IQRequals=nothing

​(Simplify your answer. Type an integer or​ decimal.)

Interpret the IQR. Choose the correct answer below.

A.The data span roughly

1818

days.

B.The length of stay in the middle​ 50% of patients spans roughly

1818

days.

C.The approximate difference between each quartile is

1818.

D.The average of the first quartile and the third quartile is

1818.

c.

Find and interpret the​ five-number summary.

nothing   

nothing   

nothing   

nothing   

nothing

​(Type integers or decimals. Use ascending​ order.)

Interpret the​ five-number summary. Choose the correct answer below.

A.

The middle​ 50% has the greatest variation in the data set. The first and fourth quarters have the same variation.

B.The fourth quarter has the greatest variation. The

middle 50 %middle 50%

has the next largest variation and the

first quarterfirst quarter

has the least variation.

C.

There​ aren't as many patients in the fourth quarter as there are in the first quarter.

D.

The first quarter has the most patients recorded. The fourth quarter has the least amount of patients recorded.

d.

Identify potential​ outliers, if any. Choose the correct answer below​ and, if​ necessary, fill in the answer box to complete your choice.

A.The potential​ outlier(s) is(are)

nothing.

​(Type an integer or a decimal. Use a comma to separate answers as​ needed.)

B.

There are no potential outliers.

e.

Construct and interpret a boxplot. Choose the correct boxplot below.

A.

0102030405060

A boxplot has a horizontal axis labeled from 0 to 60 in increments of 10. There are vertical line segments drawn at the following values: 1, 5, 14, 23, 25. A box encloses the vertical line segments at 5, 14, and 23, and horizontal line segments extend outward from both sides of the box to the vertical line segments at 1 and 25. There is a plotted point at 55. All values are approximate.

B.

0102030405060

A boxplot has a horizontal axis labeled from 0 to 60 in increments of 10. There are vertical line segments drawn at the following values: 6, 10, 20, 29, 31. A box encloses the vertical line segments at 10, 20, and 29, and horizontal line segments extend outward from both sides of the box to the vertical line segments at 6 and 31. There are plotted points at 45 and 55. All values are approximate.

C.

0102030405060

A boxplot has a horizontal axis labeled from 0 to 60 in increments of 10. There are vertical line segments drawn at the following values: 2, 8, 21, 35, 38. A box encloses the vertical line segments at 8, 21, and 35, and horizontal line segments extend outward from both sides of the box to the vertical line segments at 2 and 38. All values are approximate.

Interpret the boxplot. Choose the correct answer below.

A.Most stays were approximately

2121

days. The longest stay was

37.537.5

and the shortest stay was

1.51.5.

B.The potential outlier is relatively close to the other data points. Most stays fall with the first quarter of the​ boxplot, between

7.57.5

and

2121.

C.The potential outlying observation falls far from the rest of the data. The other stays vary from

11

to

2525

days. The majority of the stays were between

55

and

2323

days.

D.The two potential outlying observations fall far from the rest of the data. The other observations vary from

6.056.05

to

31.2531.25

days. The majority of the stays were between

10.2510.25

and  

29.1529.15

days.

Answer 1

Answer: A statistic center compiles data on the length of stay by patients in​ short-term hospitals. A random sample of 2121 patients yielded the data on length of​ stay, in​ days, shown below.

24 4 5 18 4 14 55 23 23 7 10 5 9 1 20 25 5 18 19 14 23.

Solution:

To find the quartile first arrange the data in ascending order.

1,4,4,5,5,5,7,9,10,14,14,

18,18,19,20,23,23,23,24,25,55

a) Obtain and interpret the quartiles.

For Q1:

median of lower half

n = 10

n+1/2 = (10+1)/2 =11/2 = 5.5

Q1 = (5th + 6th value in ascending order) /2 = (5+5)/2 = 5

Q2 = 11th value in ascending order = 14

median of upper half

n+1/2 = (10+1)/2 = 11/2 = 5.5

Q3 = (5th and 6th value in ascending order) /2 = (23+23)/2 = 23

Therefore,

Q1 = 5

Q2 = 14

Q3 = 23

The option D is correct answer.

D.The quartiles suggest that​ 25% of the patients stayed less than 5 ​days, 25% stayed between 5 and 14 ​days, 25% stayed between 14 and 23 ​days, and​ 25% stayed more than 23 days.

b) Determine and interpret the interquartile range​ (IQR).

IQR = Q3- Q1

IQR = 23 - 5

IQR =18

Answer A. The data span roughly 18 days.

c) Find and interpret the​ five-number summary.

Five number summary:

Minimum, Q1, Q2, Q3, Maximum

1, 5, 14, 23, 55

The option B. is correct answer.

B.The fourth quarter has the greatest variation. The middle 50 % has the next largest variation and the first quarterfirst quarter has the least variation.

d) Identify potential​ outliers, if any.

The potential outlier is 55.

e) Construct and interpret a boxplot.

The option C. is correct answer.

C. The potential outlying observation falls far from the rest of the data. The other stays vary from 1 to 25 days, the majority of the stays were between 5 and 23 days.