Question

An administrator wanted to study the utilization of long-distance telephone service by a department. One variable of interest (let’s call it X) is the length, in minutes, of long-distance calls made during one month. There were 38 calls that resulted in a connection. The length of calls, already ordered from smallest to largest, are presented in the following table.

1.6	1.7	1.8	1.8	1.9	2.1	2.5	3.0	3.0	4.4
4.5	4.5	5.9	7.1	7.4	7.5	7.7	8.6	9.3	9.5
12.7	15.3	15.5	15.9	15.9	16.1	16.5	17.3	17.5	19.0
19.4	22.5	23.5	24.0	31.7	32.8	43.5	53.3

Which one of the following statements is not true?

1. The 75th percentile (Q3) is 17.5 minutes.

2. The 50th percentile is (Q2) 9.4 minutes.

3. The 25th percentile (Q1) is 4.4 minutes.

4. Q3- Q2 > Q2- Q1

5. Average X > Median X.

6. X distribution is positively skewed.

7. The percentile rank of 5.9 minutes is 13.

8. Range of X is 51.7 minutes.

9. IQR (Inter-Quartile Range) is 13.1 minutes.

10. There are 2 outliers in X distribution.

Q4: (This continues Q3: 2 marks) Which one of the following cannot be used to describe the distribution of X?

1. A Histogram.

2. A Stemplot.

3. Skewness and Kurtosis.

4. Mean and SD (Standard Deviation).

5. The 5-number Summary.

6. The coefficient of determination.

7. The coefficient of relative variation (CRV).

8. The 1.5 IQR Rule.

9. The Deciles.

10. A Boxplot.

Answer 1

An administrator wanted to study the utilization of long-distance telephone service by a department.

One variable of interest (let’s call it X) is the length, in minutes, of long-distance calls made during one month.

The length of calls, already ordered from smallest to largest, are presented in the following table.

1.6	1.7	1.8	1.8	1.9	2.1	2.5	3.0	3.0	4.4
4.5	4.5	5.9	7.1	7.4	7.5	7.7	8.6	9.3	9.5
12.7	15.3	15.5	15.9	15.9	16.1	16.5	17.3	17.5	19.0
19.4	22.5	23.5	24.0	31.7	32.8	43.5	53.3

Which one of the following statements is not true?

Now we will find each of the required quantity to justify weather statements is not true

i)

The 75th percentile (Q3) is 17.5 minutes

Given sample size = 38 ( even )

Thus 75th percentile for even data with n = 38 is given by

To calculate 75th percentile we will first calculate Median of data
Median = ( (n\2)th observation + (n/2+1)th observation ) / 2

           = ( (38/2)th observation + (38/2+1)th observation ) / 2

           = ( (19)th observation + (20)th observation ) / 2

From given dat data is (19)th observation = 9.3   and (20)th observation = 9.5

Thus, Median = ( 9.3 + 9.5 ) / 2 = 18.8 /2 = 9.4

Thus Median = 9.4

Now 75th percentile is nothing but median of data which is more that Median value i.e

Median of this observation ( medain value = 9.4 , so observation greater than 9.4 are )

9.5 12.7 15.3 15.5 15.9 15.9 16.1 16.5 17.3 17.5 19.0 19.4 22.5 23.5 24.0 31.7 32.8 43.5 53.3

Number of observation greater than 9.4 are n3 = 19 ( odd )

Median of odd number is given by

Median of data greater than 9.4 = ( n3 + 1 )/2 observation = ( 19 + 1 ) /2 th observation

                                                   = 10 th observation

Now 10th observation is 17.5

Thus

Our 75th percentile   is 17.5

Hence

The 75th percentile (Q3) is 17.5 minutes.   - TRUE

ii)

The 50th percentile is (Q2) 9.4 minutes.

We have have already obtain median of data above

Which was   Median =

Median = ( (n\2)th observation + (n/2+1)th observation ) / 2

           = ( (38/2)th observation + (38/2+1)th observation ) / 2

           = ( (19)th observation + (20)th observation ) / 2

From given dat data is (19)th observation = 9.3   and (20)th observation = 9.5

Thus, Median = ( 9.3 + 9.5 ) / 2 = 18.8 /2 = 9.4

Thus Median = 9.4

Thus The 50th percentile is (Q2) 9.4 minutes. - TRUE

iii)

To find The 25th percentile (Q1)

Now 25th percentile is nothing but median of data which is less that Median value i.e

Median of this observation ( medain value = 9.4 , so observation less than 9.4 are )

1.6 1.7 1.8 1.8 1.9 2.1 2.5 3.0 3.0 4.4 4.5 4.5 5.9 7.1 7.4 7.5 7.7 8.6 9.3

Number of observation less than 9.4 are n1 = 19 ( odd )

Median of odd number is given by

Median of data less than 9.4 = ( n1 + 1 )/2 observation = ( 19 + 1 ) /2 th observation

                                             = 10 th observation

Now 10th observation is 4.4

Thus

Our 25th percentile   is 4.4

Thus The 25th percentile (Q1) is 4.4 minutes. - TRUE

iv)

Q3- Q2 > Q2- Q1

Now

Q3- Q2 = 17.5 - 9.4 = 8.1

Q2- Q1 = 9.4 - 4.4 = 5

hence , Q3- Q2 > Q2- Q1 - TRUE

v)

Average X > Median X.

Now we will calculate mean of X

Mean =

Mean =[ 1.6 + 1.7 +1.8 + 1.8 + 1.9 + 2.1 +.......+ 22.5 +23.5 +24.0 +31.7 +32.8 +43.5+ 53.3 ] /38

        = 508.2 / 38 = 13.37368

Thus Mean = 13.37368

And Median = 9.4

Hence Average X > Median X. - TRUE

vi)

X distribution is positively skewed.

If the mean is greater than the median, the distribution is positively skewed

Here MEAN = 13.37368 and Median = 9.4

Mean > Median

Hence X distribution is positively skewed. - TRUE

vi

The percentile rank of 5.9 minutes is 13.

Yes

Given data is

X      1.6 1.7 1.8 1.8 1.9 2.1 2.5 3.0 3.0 4.4 4.5 4.5   5.9

rank   1 2     3     4 5    6    7     8     9    10   11   12    13

The percentile rank of 5.9 minutes is 13. - TRUE

vii)

Range of X is 51.7 minutes.

Range = max - min = 53.3 - 1.6 = 51.7

Range of X is 51.7 minutes. - TRUE

viii)

IQR (Inter-Quartile Range) is 13.1 minutes.

IQR = Q3 - Q1 = 17.5 - 4.4 = 13.1

Hence IQR (Inter-Quartile Range) is 13.1 minutes. - TRUE

ix)

There are 2 outliers in X distribution.

outlier is any data point more than 1.5 interquartile ranges (IQRs) below the first quartile or above the third quartile

Thus 1.5 interquartile ranges (IQRs) below the first quartile = 4.4 - 1.5 * IQR = 4.4 - 1.5 * 13.1 = -15.25

And 1.5 interquartile ranges (IQRs) above the third quartile = 17.5 - 1.5 * IQR = 17.5 - 1.5 * 13.1 = 37.15

Hence our data should be in interquartile ranges = ( -15.25 , 37.15 )

Now we can see observation 37th and 38th which are 43.5 and 53.3 respectively are out of given interval

Hence 43.5 and 53.3 are outlier

So we have 2 outlier observation

There are 2 outliers in X distribution. - TRUE

Q4: (This continues Q3: 2 marks) Which one of the following cannot be used to describe the distribution of X?

i)A Histogram. - A histogram displays the shape and spread of continuous sample data.

Hence Histogram can be used to describe the distribution of X

ii)

A Stemplot. -

You could make a frequency distribution table or a histogram for the values, or you can use a

                   stem-and-leaf plot and let the numbers themselves to show pretty much the same information.

Hence Stemplot can be used to describe the distribution of X

iii)

Skewness and Kurtosis. -

Skewness is a measure of symmetry, or more precisely, the lack of symmetry

Kurtosis is a measure of whether the data are heavy-tailed or light-tailed relative to a normal distribution

Hence Skewness and Kurtosis can be used to describe the distribution of X

iv)

Mean and SD (Standard Deviation). -

The mean can be used to get an overall idea or picture of the data set.

Standard deviation measures the spread of a data distribution .

Hence Mean and SD (Standard Deviation) can be used to describe the distribution of X

v)

The 5-number Summary.

A summary consists of five values: the most extreme values in the data set (the maximum and minimum values), the lower and upper quartiles, and the median .This makes the five-number summary a useful measure of spread

Hence 5-number Summary can be used to describe the distribution of X

vi)

The coefficient of determination.

The coefficient of determination is used to explain how much variability of one factor can be caused by its relationship to another factor.

Sometimes referred to as the "goodness of fit.

The coefficient of determination is a measure used in statistical analysis that assesses how well a model explains and predicts future outcomes

So coefficient of determination can be used in regreesion model if there is one more dependent variable , hence here it can not describe the distribution of X

vii)

The coefficient of relative variation (CRV).

The coefficient of relative variation (relative standard deviation) is a statistical measure of the dispersion of data points around the mean

Hence coefficient of relative variation (CRV) can be used to describe the distribution of X

viii)

The 1.5 IQR Rule.

The IQR is often seen as a better measure of spread than the range as it is not affected by outliers.

Hence IQR can be used to describe the distribution of X

ix)

The Deciles

A decile is a quantitative method of splitting up a set of ranked data into 10 equally large subsections

Deciles are similar to quartiles. But while quartiles sort data into four quarters, deciles sort data into ten equal parts

So Deciles can be used to describe the distribution of X

x)

A boxplot

A boxplot is a standardized way of displaying the distribution of data based on a five number summary .

So from box-plot we can observe outlier , weather data is symmeteric , skewed etc .

Hence boxplot can be used to describe the distribution of X