cyber21

Monthly sales of hot water heaters are approximately normally distributed with a mean of 21 and a standard deviation of 5. What is the probabilility of selling 12 hot water heaters or less next month?

A) 1.80%.

B) 3.59%.

C) 96.41%.

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Z = (12 – 21) / 5 = -1.8
From the cumulative z-table, the probability of being more than 1.8 standard deviations below the mean, probability x < -1.8, is 3.59%.

cyber21

The standard normal distribution is most completely described as a:

A) distribution that exhibits zero skewness and no excess kurtosis.

B) symmetrical distribution with a mean equal to its median.

C) normal distribution with a mean of zero and a standard deviation of one.

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The standard normal distribution is defined as a normal distribution that has a mean of zero and a standard deviation of one. The other choices apply to any normal distribution.

JustasS

The average annual rainfall amount in Yucutat, Alaska, is normally distributed with a mean of 150 inches and a standard deviation of 20 inches. The 90% confidence interval for the annual rainfall in Yucutat is closest to:

A) 110 to 190 inches.

B) 137 to 163 inches.

C) 117 to 183 inches.

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The 90% confidence interval is µ ± 1.65 standard deviations. 150 − 1.65(20) = 117 and 150 + 1.65(20) = 183.

JustasS

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) 2.50%.

B) 15.87%.

C) 5.00%.

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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.

JustasS

An investment has a mean return of 15% and a standard deviation of returns equal to 10%. If returns are normally distributed, which of the following statements is least accurate? The probability of obtaining a return:

A) between 5% and 25% is 0.68.

B) greater than 25% is 0.32.

C) greater than 35% is 0.025.

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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%.

JustasS

A portfolio manager is looking at an investment that has an expected annual return of 10% with a standard deviation of annual returns of 5%. Assuming the returns are approximately normally distributed, the probability that the return will exceed 20% in any given year is closest to:

A) 2.28%.

B) 0.0%.

C) 4.56%.

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Given that the standard deviation is 5%, a 20% return is two standard deviations above the expected return of 10%. Assuming a normal distribution, the probability of getting a result more than two standard deviations above the expected return is 1 − Prob(Z ≤ 2) = 1 − 0.9772 = 0.0228 or 2.28% (from the Z table).

JustasS

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) 0.950.

B) 0.750.

C) 0.977.

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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.

JustasS

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) 34%.

B) 84%.

C) 98%.

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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%.

JustasS

The mean return of a portfolio is 20% and its standard deviation is 4%. The returns are normally distributed. Which of the following statements about this distribution are least accurate? The probability of receiving a return:

A) in excess of 16% is 0.16.

B) of less than 12% is 0.025.

C) between 12% and 28% is 0.95.

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The probability of receiving a return greater than 16% is calculated by adding the probability of a return between 16% and 20% (given a mean of 20% and a standard deviation of 4%, this interval is the left tail of one standard deviation from the mean, which includes 34% of the observations.) to the area from 20% and higher (which starts at the mean and increases to infinity and includes 50% of the observations.) The probability of a return greater than 16% is 34 + 50 = 84%.

Note: 0.16 is the probability of receiving a return less than 16%.

JustasS

For a normal distribution, what approximate percentage of the observations fall within ±3 standard deviation of the mean?

A) 66%.

B) 95%.

C) 99%.

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For normal distributions, approximately 99% of the observations fall within ±3 standard deviations of the mean.