Question

The interest rates (in percent) on a Huntington Bank automobile loan over the last 15 months were as follows:

1.8, 2.3, 1.6, 2.7, 2.1, 2.9, 3.6, 2.4, 1.9, 2.6, 1.4, 3.4, 2.3, 2.5, 2.2

Calculate the following descriptive statistics for these data. NOTE: Double-check your data entry. A substantial number of points will be lost if you are not working with the correct data values!!!

a. Mean: (report your answer to 2 decimal places, using conventional rounding rules)

b. Median: (report your answer to 2 decimal places, using conventional rounding rules)

c. Mode: (report your answer to 2 decimal places, using conventional rounding rules)

d. Standard deviation: (report your answer to 4 decimal places, using conventional rounding rules)

e. Range: (report your answer to 2 decimal places, using conventional rounding rules)

f. The 35th percentile: (report your answer to 2 decimal places, using conventional rounding rules)

g. The interquartile range: (report your answer to 2 decimal places, using conventional rounding rules)

h. The z-score for a month where the interest rate is 3.64%: (report your answer to 2 decimal places, using conventional rounding rules)

i. Use the Empirical Rule to establish an interval which includes about 95% of the observations: The interval is from % up to % (report your answers to 2 decimal places, using conventional rounding rules)

j. Determine the coefficient of skewness for these data using Pearson’s method. (round your answer to 2 decimal places, using conventional rounding rules)

Answer 1

Here n = 15

a. Mean

The formula of mean is,

Mean = 2.38

b. Median

Median is nothing but the middlemost observation.

First, arrange the observations in increasing order.

The observations in increasing order are,

1.4, 1.6, 1.8, 1.9, 2.1, 2.2, 2.3, 2.3, 2.4, 2.5, 2.6, 2.7, 2.9, 3.4, 3.6

Out of 15 observations, eighth observation is nothing but middle

So the eighth observation from arranged observation is, 2.3

Median = 2.3

c. Mode

The mode is nothing but the most repeated observation.

In all the 15 observations, 2.3 is repeated 2 times so 2.3 is mode.

Mode = 2.3

d. Standard deviation

The formula of standard deviation is,

Standard deviation = 0.5901

e. Range

The range is nothing but the difference between the smallest and largest observation

Largest observation = 3.6

Smallest observation = 1.4

Range = 3.6 - 1.4 = 2.2

f. 35th percentile

35th percentile (35 * n / 100 )th observation = 5.25th observation from arranged observation

Rounded to the 6th observation that is 2.2

35^th percentile = 2.2

g. Interquartile range:

The formula of an interquartile range is Q3 - Q1

Q3 is the third quartile and Q1 is the first quartile.

First divide the data set in two equal parts and the median of first-half data set is Q1 and median of second data set is nothing but Q3.

There are total 15 observations, so first 7 are in first-half then median and the remaining 7 are in the second half.

The first 7 observations are, 1.4, 1.6, 1.8, 1.9, 2.1, 2.2, 2.3

The median of this data set is 1.9 that is 4th observation.

Q1 = 1.9

The second-half data set is, 2.4, 2.5, 2.6, 2.7, 2.9, 3.4 and 3.6

The median of this data set is 2.7 that is 4th observation.

Q3 = 2.7

Interquartile range = Q3 - Q1 = 2.7 - 1.9 = 0.8

h. z score for 3.64

The formula of z score is,

z-score for 3.64 is 2.14

i. Interval which includes 95% of the observation using Empirical rule.

According to the empirical rule, 95% of the observations fall within 2 standard deviations from the mean.

The interval is from 1.20% to 3.56%

j. Coefficient of skewness

The formula of the coefficient of skewness using the Pearson method is,

Coefficient of skewness = 0.14