burning0spear

In a skewed distribution, what is the minimum amount of observations that will fall between +/- 1.5 standard deviations from the mean?

A) 44%.

B) 56%.

C) 95%.

---

Because the distribution is skewed, we must use Chebyshev’s Inequality, which states that the proportion of observations within k standard deviations of the mean is at least 1 – (1 / k2).
1 – (1 / 1.52) = 0.5555, or 56%.

burning0spear

Regardless of the shape of a distribution, according to Chebyshev’s Inequality, what is the minimum percentage of observations that will lie within +/– two standard deviations of the mean?

A) 68%.

B) 89%.

C) 75%.

---

According to Chebyshev’s Inequality, for any distribution, the minimum percentage of observations that lie within k standard deviations of the distribution mean is equal to:
1 – (1 / k2), with k equal to the number of standard deviations. If k = 2, then the percentage of distributions is equal to 1 – (1 / 4) = 75%.

burning0spear

In a skewed distribution, what is the minimum proportion of observations between +/- two standard deviations from the mean?

A) 95%.

B) 84%.

C) 75%.

---

For any distribution we can use Chebyshev’s Inequality, which states that the proportion of observations within k standard deviations of the mean is at least 1 – (1 / k2).
1 – (1 / 22) = 0.75, or 75%.
Note that for a normal distribution, 95% of observations will fall between +/- 2 standard deviations of the mean.

burning0spear

Assume a sample of beer prices is negatively skewed. Approximately what percentage of the distribution lies within plus or minus 2.40 standard deviations of the mean?

A) 95.5%.

B) 58.3%.

C) 82.6%.

---

Use Chebyshev’s Inequality to calculate this answer. Chebyshev’s Inequality states that for any set of observations, the proportion of observations that lie within k standard deviations of the mean is at least 1 – 1/k2. We can use Chebyshev’s Inequality to measure the minimum amount of dispersion whether the distribution is normal or skewed. Here, 1 – (1 / 2.42) = 1 − 0.17361 = 0.82639, or 82.6%.

burning0spear

Cameron Ryan wants to make an offer on the condominium he is renting. He takes a sample of prices of condominiums in his development that closed in the last five months. Sample prices are as follows (amounts are in thousands of dollars): $125, $175, $150, $155 and $135. The sample standard deviation is closest to:

A) 370.00.

B) 38.47.

C) 19.24.

---

Calculations are as follows:

• Sample mean = (125 + 175 + 150 + 155 + 135) / 5 = 148
• Sample Variance = [(125 – 148)2 + (175 – 148)2 + (150 – 148)2 + (155 – 148)2 + (135 – 148)2] / (5 – 1) = 1,480 / 4 = 370
• Sample Standard Deviation = 3701/2 = 19.24%.

burning0spear

There is a 40% chance that an investment will earn 10%, a 40% chance that the investment will earn 12.5%, and a 20% chance that the investment will earn 30%. What is the mean expected return and the standard deviation of expected returns, respectively?

A) 15.0%; 5.75%.

B) 15.0%; 7.58%.

C) 17.5%; 5.75%.

---

Mean = (0.4)(10) + (0.4)(12.5) + (0.2)(30) = 15%
Var = (0.4)(10 − 15)2 + (0.4)(12.5 − 15)2 + (0.2)(30 − 15)2 = 57.5
Standard deviation = √57.5 = 7.58

burning0spear

For the past three years, Acme Corp. has generated the following sample returns on equity (ROE): 4%, 10%, and 1%. What is the sample variance of the ROE over the last three years?

A) 21.0%.

B) 4.6%.

C) 21.0(%2).

---

[(4 − 5)2 + (10 − 5)2 + (1 − 5)2] / (3 − 1) = 21(%2).

burning0spear

When creating intervals around the mean to indicate the dispersion of outcomes, which of the following measures is the most useful? The:

A) variance.

B) median.

C) standard deviation.

---

The standard deviation is more useful than the variance because the standard deviation is in the same units as the mean. The median does not help in creating intervals around the mean.

burning0spear

The following data points are observed returns.  

4.2%, 6.8%, 7.0%, 10.9%, 11.6%, 14.4%, 17.0%, 19.0%, 22.5%, 28.1%

What return lies at the seventh decile (70% of returns lie below this return)?

A) 17.0%.

B) 18.4%.

C) 19.0%.

---

The formula for the seventh decile is Ly = (n + 1)(7 / 10) = 7.70 or between the seventh and eighth return from the left. The seventh return is 17%, while the eighth return is 19%. Interpolating, we find that the seventh decile is 17% + 0.7(19% – 17%) = 18.4%.

burning0spear

Consider the following set of stock returns: 12%, 23%, 27%, 10%, 7%, 20%,15%. The third quartile is:

A) 23%.

B) 21.5%.

C) 20.0%.

---

The third quartile is calculated as: Ly = (7 + 1) (75/100) = 6. When we order the observations in ascending order: 7%, 10%, 12%, 15%, 20%, 23%, 27%, “23%” is the sixth observation from the left.