Question

1. The ages of the Supreme Court Justices in December 2016 are below
   

61

80

68

83

78

66

62

56

1. Find the mean 8.3125, median 67and mode {56, 61, 62, 66, 68, 78, 80, 83}of the data set. Which best represents the center of the data.
2. Find the range 27, variance 86.1875, and standard deviation 9.283 of the data set.
3. Find the quartiles, Q1, Q2 and Q3 = Q1: 61.5, Q2: 67, Q3: 79
4. Find the interquartile range 17.5
5. Identify any outliers, if there are any. No outliers
6. Construct a stem and leaf plot.
7. Construct a box plot.
8. If selected at random, what is the probability that the Supreme Court Justices are less than 79 years old?

Answer 1

Solution:- Given that 61,80,68,83,78,66,62,56

a. The mean = sum of terms/number of terms
= 558/8
= 69.25

Median = 67
The median is the middle number in a sorted list of numbers. So, to find the median, we need to place the numbers in value order and find the middle number.

Ordering the data from least to greatest, we get:

56 61 62 66 68 78 80 83   

As you can see, we do not have just one middle number but we have a pair of middle numbers, so the median is the average of these two numbers:

Median = (66+68)/2 = 67
mode : no mode

Explanation:-
The mode of a set of data is the value in the set that occurs most often.

Ordering the data from least to greatest, we get:

56 61 62 66 68 78 80 83   

Since each value occurs only once, there is no mode for this data set.

b.

The range = 27

The range is the difference between the highest and lowest values in the data set.

Ordering the data from least to greatest, we get:

56   61   62   66   68   78   80   83   

The lowest value is 56.

The highest value is 83.

The range = 83 - 56 = 27.

Variance = 98.5

Standard deviation = 9.9247

c. Q1 = 61.5

  

Explanation

The first quartile (or lower quartile or 25th percentile) is the median of the bottom half of the numbers. So, to find the first quartile, we need to place the numbers in value order and find the bottom half.

56   61   62   66   68   78   80   83   

So, the bottom half is

56   61   62   66   

The median of these numbers is 61.5.

Q2 = 67

Q3 = 79

Explanation

The third quartile (or upper quartile or 75th percentile) is the median of the upper half of the numbers. So, to find the third quartile, we need to place the numbers in value order and find the upper half.

56   61   62   66   68   78   80   83   

So, the upper half is

68   78   80   83   

The median of these numbers is 79.

d.  The interquartile range of the data set is 17.5.

Explanation

The interquartile range is the difference between the third and first quartiles.

The third quartile is 79.

The first quartile is 61.5.

The interquartile range = 79 - 61.5 = 17.5.

e. No outliers.

f. stem and leaf plot :

  

stem	leaf
5	6
6	1 2 6 8
7	8
8	0 3

g. box plot :

h. P(X < 79) = P(X-mean)/sd) < (79-69.25)/9.9247)

= P(Z < 0.9824)

= 0.8365