Interest rates (in percent) on a certain AAA corporate bond over the last 20 mon

Question

Interest rates (in percent) on a certain AAA corporate bond over the last 20 months were as follows: 3.6, 4.5, 2.1, 3.4, 3.7, 1.8, 2.3, 1.5, 2.7, 2.1, 2.9, 3.6, 2.4, 1.9, 2.6, 1.5, 3.4, 2.3, 2.5, 2.1 Calculate the following descriptive statistics.

a.         Mean:              

(report your answer to 3 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 60th 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 2.75%:  

(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)

Explanation / Answer

a. Mean -= Sum of all observations / Number of observations

Mean : 2.645

b. Median : 2.45

c. Mode = Observation which has the maximum frequency.

Mode : 2.10

d. Standard deviation : 0.8172

e. Range = Maximum value - Minimum value

Range : 3

f. P(X < P60) = 0.60

P((X - mean ) / SD < (P60 - mean)/SD ) = 0.60

P( Z < (P60 - mean) / SD ) = 0.60

(P60 - mean) / SD ) = 0.253347

P60 = 2.85

60th percentile : 2.85

g. The interquartile range:

Let Q3 be third quartile and Q1 be first quartile

P(X < Q3) = 0.75

P((X - mean ) / SD < (Q3 - mean)/SD ) = 0.75

P( Z < (Q3 - mean) / SD ) = 0.75

(Q3 - mean) / SD ) = 0.6745

Q3 = 3.40

P(X < Q1) = 0.25

P((X - mean ) / SD < (Q1 - mean)/SD ) = 0.25

P( Z < (Q1- mean) / SD ) = 0.25

(Q1 - mean) / SD ) = - 0.6745

Q1 = 2.10

Third quartile - First quartile = 3.40 - 2.10 = 1.30

h. Z-score = (2.75 - mean) / SD

The z-score for a month where the interest rate is 2.75%: 0.13

i. Empirical rule states that 95% of the observations lie between 2 standard deviations within the mean

Upper limit = Mean + 2*Standard deviation = 2.645 + 2*0.8172 = 4.28

Lower limit = Mean - 2*Standard deviation = 2.645 - 2*0.8172 = 1.01

The interval is from 1.01% up to 4.28%

j. Coefficient of skewness : 0.72

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