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Subject 7. The standard normal distribution.

This subject is for the following CFA Learning Outcome Statement(s):

l. determine the probability that a normally distributed random variable lies inside a given interval;
m. define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution;

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Practice Question 1

True or False? If False, correct it.

A confidence interval estimate for a parameter is used to eliminate the element of chance from estimation.

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Practice Question 2

An investor estimated the mean return of this portfolio is 12% and the standard deviation is 16%. What is the 95% confidence interval for the mean return on this portfolio?

A. [-19.36%, 43.36%]
B. [-14.32%, 18.32%]
C. [-5.42%, 18.58%]

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Practice Question 3

An analyst wants to calculate a 95% confidence interval for the weighted-average of the projected mean return of a portfolio he manages. His latest calculations show that this portfolio has a weighted-average return of 8% and a variance of 36. What is the 95% confidence interval?

A. [-62.56%, 78.56%]
B. [-3.76%, 19.76%]
C. [-1.87%, 17.87%]

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Practice Question 4

For a data collection, if a score has a z-score of 1.5 then

A. the score is 1.5 standard deviations above the mean
B. the score is better than 15% of the scores in the data collection
C. the score is 1.5 standard deviations below the mean

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Practice Question 5

For a set of data with mean 70 and standard deviation 10, the z-score for 62 is

A. 0.8
B. -0.8
C. -8

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Practice Question 6

Which of the following statements is incorrect?

A. A standard normal random variable has a variance of 1.0, and is centered at zero.
B. A standard normal random variable has a mode at 1.0.
C. Any normal random variable can be converted to a standard normal.

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Practice Question 7

A computed z-value of -1.75 tells us

A. the value -1.75 has a standard normal distribution
B. the mean of the distribution is -1.75
C. the standard deviation of the distribution is -1.75
D. none of the above

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Practice Question 8

The daily sales at a certain cafe follow a normal distribution with a mean of $1,060 and a standard deviation of $310. What is the z-value associated with daily sales of $1,711?

A. -2.10
B. 2.10
C. 36.97

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Practice Question 9

Suppose a set of data has a mean of 52 and a standard deviation of 6. According to Chebyshev's Theorem, at least what percentage of the data lies between 40 and 64? If the data is normally distributed, about what percentage of the data lies between 40 and 64 according to the Empirical Rule?

A. 0%, 68%
B. 75%, 95%
C. 94%, 100%

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Practice Question 10

As part of a promotion for a new type of cracker, free trial samples are offered to shoppers in a local supermarket. The probability that a shopper will buy a packet of crackers after testing the free sample is 0.20. Different shoppers can be regarded as independent trials. Let X be the number among the next 100 shoppers who buy a packet of the crackers after tasting a free sample, then the probability that fewer than 30 buy a packet after tasting a free sample is approximately

A. 0.20
B. 0.9938
C. None of the above.

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Practice Question 11

The lifetime of a 2-volt non-rechargeable battery in constant use has a normal distribution with a mean of 516 hours and a standard deviation of 20 hours. The proportion of batteries with lifetimes exceeding 520 hours is approximately:

A. 0.20
B. 0.5793
C. 0.4207

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Practice Question 12

The random variable X denotes the time taken for a computer link to be made between the terminal in an executive's office and the computer at a remote factory site. It is known that X has a normal distribution with mean 15 seconds and standard deviation 3 seconds. P(X > 20) has value (choose the closest option):

A. 0.048.
B. 0.952.
C. 1.67

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Practice Question 13

For x a random variable from a normal distribution with mean 40 and standard deviation 5, P(x > 20) = ____ (to the nearest 0.1%).

A. 0+
B. 95%
C. 1-

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Practice Question 14

For x a random variable from a normal distribution with mean 40 and standard deviation 5, P(28 < x < 38) = ____ (to nearest 0.1%).

A. 35.3%
B. 34.5%
C. 33.6%

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Practice Question 15

For the MRC test, with scores normally distribution with m = 70 and s = 5, the cut-off for the bottom 10% is _____.

A. 60
B. 68.72
C. 63.6

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Practice Question 16

The daily sales at a certain shop follow a normal distribution with a mean of $1,060 and a standard deviation of $310. What is the probability that the daily sales are less than $409?

A. 0.0179
B. 0.4821
C. 0.5179

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Practice Question 17

The daily sales at a certain cafe follow a normal distribution with a mean of $1,060 and a standard deviation of $310. What is the probability that the daily sales are between $471 and $1,401?

A. 0.4713
B. 0.3643
C. 0.8356

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Practice Question 18

The amount of fill in a certain brand of bottled water is normally distributed with a mean of 16 ounces and a standard deviation of .25 ounce. What is the probability that a bottle has more than 16.25 ounces?

A. .1587
B. .3413
C. .6587

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Practice Question 19

A portfolio with 10 stocks has a weighted average return of 5.5% and a variance of 25. What is the 85% confidence interval for the weighted average?

A. [-35.625%, 46.625%]
B. [-30.5%, 41.5%]
C. [-1.7%, 12.7%]

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Practice Question 20

A standard normal distribution has:

A. the mean equal to the variance
B. a mean of 1 and a variance of 1
C. a mean of 0 and a variance of 1

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Practice Question 21

A bell-shaped, symmetrical frequency distribution has a mean of 45. If 95% of the observations on the variable fall between 30 and 60, the standard deviation of the variable is:

A. 5.00
B. 7.50
C. 15.0

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Practice Question 22

The distribution of the annual incomes of a group of middle management employees approximated a normal distribution with a mean of $37,200 and a standard deviation of $800. About 68 percent of the incomes lie between what two incomes?

A. $34,800 and $39,600
B. $36,400 and $38,000
C. $35,600 and $38,800

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Practice Question 23

The lifetime of a 2-volt non-rechargeable battery in constant use has a normal distribution with a mean of 516 hours and a standard deviation of 20 hours. Ninety percent of all batteries have a lifetime less than

A. 517.28 hours.
B. 536.00 hours.
C. 541.60 hours.

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Practice Question 24

Daily travel expenses for H&J Employees are normally distributed with a mean of $250 and a standard deviation of $38. Q1 for this distribution is ______.

A. $224.54
B. $275.46
C. $288.00

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Practice Question 25

The distribution of lifetimes for a certain type of light bulb is normally distributed with a mean of 1000 hours and a standard deviation of 100 hours. Find the 33rd percentile of the distribution of lifetimes.

A. 560
B. 1044
C. 956

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Practice Question 26

Suppose you were told that scores on an examination were converted to standard scores with a mean = 500, range of 800, and a standard deviation of 100. A person with a score of 600 has performed better than what percent of the persons taking the test?

A. 57 percent
B. 97.5 percent
C. 84 percent

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Practice Question 27

Consider a normal distribution with MU = 67 and SIGMA2 = 144. If each score is raised by 7 points, what percentage of the new scores is less than 74?

A. 72%
B. 88%
C. 50%

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Practice Question 28

The seasonal output of a new experimental strain of pepper plants was carefully weighed. The mean weight per plant is 15.0 pounds, and the standard deviation of the normally distributed weights is 1.75 pounds. Of the 200 plants in the experiment, how many produced peppers weighing between 13 and 16 pounds?

A. 118
B. 100
C. 197

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Practice Question 29

A bell-shaped, symmetrical frequency distribution has a mean of 10. If 16% of the observations in the distribution are negative, what is the coefficient of variation of X?

A. 0.1
B. 1.0
C. 10.0

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Practice Question 30

What is the area under the normal curve between z = 1.0 and z = 2.0?

A. 0.4772
B. 0.1359
C. 0.7408

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Practice Question 31

A study of a company's practice regarding the payment of invoices revealed that on the average an invoice was paid 20 days after it was received. The standard deviation equaled five days. Assuming that the distribution is normal, what percent of the invoices were paid within 15 days of receipt?

A. 34.13%
B. 37.91%
C. 15.87%

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Practice Question 32

We would like to refer all probability statements to one set of normal probability values. The distribution hat fills this role is known as the

A. binomial distribution.
B. standard normal distribution.
C. continuous uniform distribution.

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Practice Question 33

An analyst who manages an equity portfolio forecasts a portfolio return of 10% and estimates a standard deviation of annual return of 18%. What is the probability that portfolio return will be between 10% and 20%?

A. 18.7%.
B. 19.67%.
C. 21.2%.

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Practice Question 34

What is the probability that a value of 30 or more will be observed from a normal distribution with a mean of 15 and standard deviation of 25?

A. 50%
B. 27.4%
C. 60%

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Practice Question 35

A standard normal distribution (Z-distribution) has:

A. mean 1 and standard deviation 0.
B. mean 1 and standard deviation 1.
C. mean 0 and standard deviation 1.

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Practice Question 36

An analyst determines that approximately 99% of the observations of daily sales for a company are within the interval from $230,000 to $480,000 and that daily sales for the company are normally distributed. The standard deviation of daily sales (in $) for the company is closest to:

A. 41,667.
B. 62,500.
C. 83,333.

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