AIM 1: Describe the key properties of the normal distribution, the standard normal distribution.

1、Which of the following statements about the normal probability distribution is most accurate?

A) The normal curve is asymmetrical about its mean.

B) The standardized normal distribution has a mean of zero and a standard deviation of 10.

C) Five percent of the normal curve probability falls more than outside two standard deviations from the mean. 

D) Sixty-eight percent of the area under the normal curve falls between 0 and +1 standard deviations above the mean.

The correct answer is C

The normal curve is symmetrical with a mean of zero and a standard deviation of 1 with 34% of the area under the normal curve falling between 0 and +1 standard deviation above the mean. Ninety-five percent of the normal curve is within two standard deviations of the mean, so five percent of the normal curve falls outside two standard deviations from the mean.

2、A food retailer has determined that the mean household income of her customers is $47,500 with a standard deviation of $12,500. She is trying to justify carrying a line of luxury food items that would appeal to households with incomes greater than $60,000. Based on her information and assuming that household incomes are normally distributed, what percentage of households in her customer base has incomes of $60,000 or more?

A) 5.00%.

B) 2.50%.

C) 34.13%.

D) 15.87%.

The correct answer is D

Z = ($60,000 – $47,500) / $12,500 = 1.0

From the table of areas under the normal curve, 84.13% of observations lie to the left of +1 standard deviation of the mean. So, 100% – 84.13% = 15.87% with incomes of $60,000 or more.

3、A client will move his investment account unless the portfolio manager earns at least a 10% rate of return on his account. The rate of return for the portfolio that the portfolio manager has chosen has a normal probability distribution with an expected return of 19% and a standard deviation of 4.5%. What is the probability that the portfolio manager will keep this account?

A) 0.750.

B) 0.950.

C) 0.977.

D) 1.000.

The correct answer is C

Since we are only concerned with values that are below a 10% return this is a 1 tailed test to the left of the mean on the normal curve. With μ = 19 and σ = 4.5, P(X ≥ 10) = P(X ≥ μ ? 2σ) therefore looking up -2 on the cumulative Z table gives us a value of 0.0228, meaning that (1 ? 0.0228) = 97.72% of the area under the normal curve is above a Z score of -2. Since the Z score of -2 corresponds with the lower level 10% rate of return of the portfolio this means that there is a 97.72% probability that the portfolio will earn at least a 10% rate of return.

4、Standardizing a normally distributed random variable requires the:

A) mean and the standard deviation.

B) mean, variance and skewness.

C) natural logarithm of X.

D) variance and kurtosis.

The correct answer is A

All that is necessary is to know the mean and the variance. Subtracting the mean from the random variable and dividing the difference by the standard deviation standardizes the variable.

5、An investment has a mean return of 15% and a standard deviation of returns equal to 10%. Which of the following statements is least accurate? The probability of obtaining a return:

A) greater than 35% is 0.025.

B) less than 5% is 0.16.

C) greater than 25% is 0.32.

D) between 5% and 25% is 0.68.

The correct answer is C

Sixty-eight percent of all observations fall within +/- one standard deviation of the mean of a normal distribution. Given a mean of 15 and a standard deviation of 10, the probability of having an actual observation fall within one standard deviation, between 5 and 25, is 68%. The probability of an observation greater than 25 is half of the remaining 32%, or 16%. This is the same probability as an observation less than 5. Because 95% of all observations will fall within 20 of the mean, the probability of an actual observation being greater than 35 is half of the remaining 5%, or 2.5%.

6、A group of investors wants to be sure to always earn at least a 5% rate of return on their investments. They are looking at an investment that has a normally distributed probability distribution with an expected rate of return of 10% and a standard deviation of 5%. The probability of meeting or exceeding the investors' desired return in any given year is closest to:

A) 98%.

B) 34%.

C) 84%.

D) 50%.

The correct answer is C

The mean is 10% and the standard deviation is 5%. You want to know the probability of a return 5% or better. 10% - 5% = 5% , so 5% is one standard deviation less than the mean. Thirty-four percent of the observations are between the mean and one standard deviation on the down side. Fifty percent of the observations are greater than the mean. So the probability of a return 5% or higher is 34% + 50% = 84%.

7、A study of hedge fund investors found that their household incomes are normally distributed with a mean of $280,000 and a standard deviation of $40,000.

The percentage of hedge fund investors that have incomes greater than $350,000 is closest to:

A) 5.0%.

B) 25.0%.

C) 3.0%.

D) 4.0%.

The correct answer is D

z = (350,000 – 280,000)/40,000 = 1.75. Using the z-table, F(1.75) = 0.9599. So, the percentage greater than $350,000 = (1 – 0.9599) = 4.0%.

The percentage of hedge fund investors with income less than $180,000 is closest to:

A) 1.15%.

B) 0.62%.

C) 2.50%.

D) 6.48%.

The correct answer is B

z = (180,000 – 280,000)/40,000 = –2.50. Using the z-table, F(–2.50) = (1 – 0.9938) = 0.62%.

8、Approximately 50 percent of all observations for a normally distributed random variable fall in the interval: fficeffice" />

A) 

B) 

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D)  [upload=

The correct answer is B

9、For a normal distribution, what approximate percentage of the observations falls within ± 2 standard deviations of the mean?

A) 95%.

B) 99%.

C) 92%. 

D) 90%.

The correct answer is A

For normal distributions, approximately 95 percent of the observations fall within ± 2 standard deviations of the mean.

10、Given the probabilities N(–0.5) = 0.3085, N(0.75) = 0.7734, and N(1.50) = 0.9332 from a z-table, the probability of 0.2266 corresponds to:

A) N(–0.25). 

B) N(–0.75). 

C) N(0.25). 

D) N(0.50).

The correct answer is B

This problem is checking your knowledge of a normal distribution and gives you more information than you need to answer the question. The area of a normal distribution is 1, with two symmetric halves that equal 0.5 each. N(0.75) means that the area to the left of 0.75 on the positive portion of the curve is 0.7734. This means that the area to the right of 0.75 is (1.0 – 0.7734) = 0.2266. Since the halves of a normal distribution are symmetrical, that means the area to the left of (–0.75) is also 0.2266.

11、The return on a portfolio is normally distributed with a mean return of 8 percent and a standard deviation of 18 percent. Which of the following is closest to the probability that the return on the portfolio will be between -27.3 percent and 37.7 percent?

A) 92.5%.

B) 68.0%.

C) 81.5%.

D) 96.5%.

The correct answer is A

12、The distribution of annual returns for a bond portfolio is approximately normal with an expected value of $120 million and a standard deviation of $20 million. Which of the following is closest to the probability that the value of the portfolio one year from today will be between $110 million and $170 million?

A) 74%.

B) 58%.

C) 42%.

D) 66%.

The correct answer is D

Calculating z-values, z1 = (110 ? 120) / 20 = ?0.5. z2 = (170 ? 120) / 20 = 2.5. Using the z-table, P(?0.5) = (1 ? 0.6915) = 0.3085. P(2.5) = 0.9938. P(?0.5 < X < 2.5) = 0.9938 ? 0.3085 = 0.6853. Note that on the exam, you will not have access to z-tables, so you would have to reason this one out using the normal distribution approximations. You know that the probability within +/? 1 standard deviation of the mean is approximately 68%, meaning that the area within ?1 standard deviation of the mean is 34%. Since ?0.5 is half of ?1, the area under ?0.5 to 0 standard deviations under the mean is approximately 34% / 2 = 17%. The probability under +/? 2 standard deviations of the mean is approximately 99%. The value $170 is mid way between +2 and +3 standard deviations, so the probability between these values must be (99% / 2) = 2%. The value from 0 to 2.5 standard deviations must therefore be (99% / 2) ? (2% / 2) = 48.5%. Adding these values gives us an approximate probability of (48.5% + 17%) = 65.5%.

13、The annual returns for a portfolio are normally distributed with an expected value of ￡50 million and a standard deviation of ￡25 million. What is the probability that the value of the portfolio one year from today will be between ￡91.13 million and ￡108.25 million?

A) 0.025.

B) 0.040.

C) 0.075.

D) 0.090.

The correct answer is B

Calculate the standardized variable corresponding to the outcomes. Z1 = (91.13 ? 50) / 25 = 1.645, and Z2 = (108.25 ? 50) / 25 = 2.33. The cumulative normal distribution gives cumulative probabilities of F(1.645) = 0.95 and F(2.33) = 0.99. The probability that the outcome will lie between Z1 and Z2 is the difference: 0.99 ? 0.95 = 0.04. Note that even though you will not have a z-table on the exam, the probability values for 1.645 and 2.33 are commonly used values that you should have memorized.

14、The probability of returns less than ?10%, assuming a normal distribution with expected return of 6.5% and standard deviation of 10%, is:

A) less than 2.5%.

B) approximately 10%.

C) not defined with only this information.

D) approximately 5%.

The correct answer is D

?10 is 16.5% below the mean return, (16.5 / 10) = 1.65 standard deviations, which leaves approximately 5% of the possible outcomes in the left tail below ?10%.

15、The probability that an observation lies within three standard deviations of the mean for any probability distribution is at least:

A) 75%.

B) 54%.

C) 89%.

D) 99%.

The correct answer is C

One can use Chebyshev's Inequality to calculate this proportion. 1 ? (1/32) = 89%.

AIM 2: Explain the process of sampling from a normal population, and concepts of random sampling, random variables, independently and identically distributed variables, standard error.

1、An equity analyst needs to select a representative sample of manufacturing stocks. Starting with the population of all publicly traded manufacturing stocks, she classifies each stock into one of the 20 industry groups that form the Index of Industrial Production for the manufacturing industry. She then selects a number of stocks from each industry based on its weight in the index. The sampling method the analyst is using is best characterized as:

A) data mining.

B) nonrandom sampling.

C) simple random sampling.

D) stratified random sampling.

The correct answer is D

In stratified random sampling, a researcher classifies a population into smaller groups based on one or more characteristics, takes a simple random sample from each subgroup based on the size of the subgroup, and pools the results.

2、An analyst divides the population of U.S. stocks into 10 equally sized sub-samples based on market value of equity. Then he takes a random sample of 50 from each of the 10 sub-samples and pools the data to create a sample of 500. This is an example of:

A) simple random sampling.

B) complex normal sampling.

C) stratified random sampling.

D) systematic cross-sectional sampling.

The correct answer is C

In stratified random sampling we first divide the population into subgroups, called strata, based on some classification scheme. Then we randomly select a sample from each stratum and pool the results. The size of the samples from each strata is based on the relative size of the strata relative to the population. Simple random sampling is a method of selecting a sample in such a way that each item or person in the population being studied has the same (non-zero) likelihood of being included in the sample.

3、Joseph Lu calculated the average return on equity for a sample of 64 companies. The sample average is 0.14 and the sample standard deviation is 0.16. The standard error of the mean is closest to:

A) 0.1600.

B) 0.0025.

C) 0.0200.

D) 0.0500.

The correct answer is C

The standard error of the mean = σ/√n = 0.16/√64 = 0.02.

4、From a population of 5,000 observations, a sample of n = 100 is selected. Calculate the standard error of the sample mean if the population standard deviation is 50.

A) 4.48.

B) 5.00.

C) 4.00.

D) 50.00.

The correct answer is B

The standard error of the sample mean equals the standard deviation of the population divided by the square root of the sample size: 50 / 1001/2 = 5.

5、A population has a mean of 20,000 and a standard deviation of 1,000. Samples of size n = 2,500 are taken from this population. What is the standard error of the sample mean?

A) 0.04.

B) 20.00.

C) 400.00.

D) 8.00.

The correct answer is B

The standard error of the sample mean is estimated by dividing the standard deviation of the sample by the square root of the sample size: sx = s / n1/2 = 1000 / (2500)1/2 = 1000 / 50 = 20.

6、A population’s mean is 30 and the mean of a sample of size 100 is 28.5. The variance of the sample is 25. What is the standard error of the sample mean?

A) 0.05.

B) 0.50.

C) 0.25.

D) 2.50.

The correct answer is B

7、The population mean for equity returns is 14 percent with a standard deviation of 2 percent. If a random sample of 49 returns is drawn, what is the standard error of the sample mean?

A) 0.29%.

B) 0.04%.

C) 2.00%.

D) 7.00%. 

The correct answer is A

8、If n is large and the population standard deviation is unknown, the standard error of the sampling distribution of the sample mean is equal to the:

A) population standard deviation divided by the sample size. 

B) sample standard deviation divided by the square root of the sample size.

C) population standard deviation multiplied by the sample size.

D) sample standard deviation divided by the sample size. 

The correct answer is

The formula for the standard error when the population standard deviation is unknown is:

AIM 3: Define the central limit theorem.

1、The central limit theorem concerns the sampling distribution of the:

A) sample mean.

B) population mean.

C) sample standard deviation.

D) population standard deviation.

The correct answer is A

The central limit theorem tells us that for a population with a mean m and a finite variance σ2, the sampling distribution of the sample means of all possible samples of size n will approach a normal distribution with a mean equal to m and a variance equal to σ2 / n as n gets large.

2、The central limit theorem states that, for any distribution, as n gets larger, the sampling distribution:

A) becomes larger.

B) becomes smaller.

C) approaches the mean.

D) approaches a normal distribution.

The correct answer is D

As n gets larger, the variance of the distribution of sample means is reduced, and the distribution of sample means approximates a normal distribution.

3、To apply the central limit theorem to the sampling distribution of the sample mean, the sample is usually considered to be large if n is greater than:

A) 15.

B) 20.

C) 30.

The correct answer is C

A sample size is considered sufficiently large if it is larger than or equal to 30.

AIM 4: Describe the key properties of the t-distribution, chi-square distribution, and F-distribution.

1、Which statement best describes the properties of Student’s t-distribution? The t-distribution is:

A) symmetrical, defined by a single parameter and is less peaked than the normal distribution.

B) symmetrical, defined by two parameters and is less peaked than the normal distribution.

C) skewed, defined by a single parameter and is more peaked than the normal distribution.

D) skewed, defined by a single parameter and is less peaked than the normal distribution

The correct answer is A

The t-distribution is symmetrical like the normal distribution but unlike the normal distribution is defined by a single parameter known as the degrees of freedom and is less peaked than the normal distribution.

2、Which one of the following statements about the t-distribution is most accurate?

A) The t-distribution is the appropriate distribution to use when constructing confidence intervals based on large samples.

B) The t-distribution approaches the standard normal distribution as the number of degrees of freedom becomes large.

C) Compared to the normal distribution, the t-distribution is more peaked with more area under the tails.

D) Compared to the normal distribution, the t-distribution is flatter with less area under the tails.

The correct answer is B

As the number of degrees of freedom grows, the t-distribution approaches the shape of the standard normal distribution. Compared to the normal distribution, the t-distribution is less peaked with more area under the tails. When choosing a distribution, three factors must be considered: sample size, whether population variance is known, and if the distribution is normal.

3、When is the t-distribution the appropriate distribution to use? The t-distribution is the appropriate distribution to use when constructing confidence intervals based on:

A) small samples from populations with known variance that are at least approximately normal.

B) large samples from populations with known variance that are nonnormal.

C) large samples from populations with known variance that are at least approximately normal.

D) small samples from populations with unknown variance that are at least approximately normal

The correct answer is D

The t-distribution is the appropriate distribution to use when constructing confidence intervals based on small samples from populations with unknown variance that are either normal or approximately normal.

AIM 5: Explain the degree of freedom

1、With 60 observations, what is the appropriate number of degrees of freedom to use when carrying out a statistical test on the mean of a population?

A) 61.

B) 60.

C) 58.

D) 59.

The correct answer is D

When performing a statistical test on the mean of a population based on a sample of size n, the number of degrees of freedom is n – 1 since once the mean is estimated from a sample there are only n – 1 observations that are free to vary. In this case the appropriate number of degrees of freedom to use is 60 – 1 = 59.

2、Which one of the following distributions is described entirely by the degrees of freedom?

A) Student’s t-distribution.

B) Normal distribution.

C) Binomial distribution.

D) Lognormal distribution.

The correct answer is A

Student’s t-distribution is defined by a single parameter known as the degrees of freedom

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