1. Probability that a randomly selected worker had not claimed bias is 0.652. The probability of a white worker who didn't claim bias is 70%.The probability of a black worker who didn't claim bias is 60%.
The probability of other workers who didn't claim bias is 100%.Thus, the probability that a worker was white is 0.6, the probability that a worker was black is 0.3, and the probability that a worker was of another race is 0.1.
The probability that a randomly selected worker didn't claim bias is calculated as follows:0.6 × 0.7 (for whites) + 0.3 × 0.6 (for blacks) + 0.1 × 1 (for others) = 0.652 or 65.2%.
2. The probability that the worker is white, given that a randomly selected worker had claimed bias is 0.315. Let A be the event that the worker claimed bias, and B be the event that the worker is white.
Then, the probability that the worker is white, given that the worker claimed bias is:P(B|A) = P(A|B) × P(B) / P(A), whereP(A|B) = probability that the worker claimed bias, given that the worker is white = 0.3,P(B) = probability that the worker is white = 0.6, andP(A) = probability that the worker claimed bias0.6 × 0.3 / (0.6 × 0.3 + 0.3 × 0.4 + 0.1 × 0) = 0.315.
3. The probability that the worker is white, given that a randomly selected worker had not claimed bias is 0.457. Let A be the event that the worker didn't claim bias, and B be the event that the worker is white.
Then, the probability that the worker is white, given that the worker didn't claim bias is:P(B|A) = P(A|B) × P(B) / P(A), whereP(A|B) = probability that the worker didn't claim bias, given that the worker is white = 0.7,P(B) = probability that the worker is white = 0.6, andP(A) = probability that the worker didn't claim bias0.6 × 0.7 / (0.6 × 0.7 + 0.3 × 0.6 + 0.1 × 1) = 0.457. 4. Probability that a randomly selected worker is white and had claimed bias is 0.18. Let A be the event that the worker claimed bias, and B be the event that the worker is white.
Then, the probability that the worker is white and claimed bias is:P(A and B) = P(A|B) × P(B) = 0.3 × 0.6 = 0.18. 5. The probability that the worker had not claimed bias, given that the worker was not white is 0.731. Let A be the event that the worker had not claimed bias, and B be the event that the worker is not white.
Then, the probability that the worker had not claimed bias, given that the worker was not white is:P(A|B) = P(B|A) × P(A) / P(B), whereP(B|A) = probability that the worker is not white, given that the worker didn't claim bias = (0.3 × 0.4 + 0.1 × 0) / 0.348 = 0.34,P(A) = probability that the worker didn't claim bias = 0.652, andP(B) = probability that the worker is not white = 0.4 + 0.1 = 0.5.P(A|B) = 0.34 × 0.652 / 0.5 = 0.731.
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The rate of return of a Fortune 500 company over the past 15 years are: 3.17%, 4.43%, 5.93%, 5.43%, 7.29%, 8.21%, 6.23%, 5.23%,
4.34%, 6.68%, 7.14%, -5.56%, -5.23%, -5.73%, -10.34%
1. Compute the arithmetic mean rate of return per year.
2. Compute the geometric mean rate of return per year for the first four years.
3. Construct a boxplot for the rate of return. What is the shape of the distribution for the rate of return?
1. The arithmetic mean rate of return per year is approximately 3.96%. 2. The geometric mean rate of return per year for the first four years is approximately 1.06%. 3. The shape of the distribution for the rate of return is negatively skewed.
1. To compute the arithmetic mean rate of return per year, we sum up the rates of return for the 15 years and divide by the number of years.
Arithmetic mean = (3.17% + 4.43% + 5.93% + 5.43% + 7.29% + 8.21% + 6.23% + 5.23% + 4.34% + 6.68% + 7.14% - 5.56% - 5.23% - 5.73% - 10.34%) / 15
= 59.45% / 15
= 3.9633%
Therefore, the arithmetic mean rate of return per year is approximately 3.96%.
2. To compute the geometric mean rate of return per year for the first four years, we multiply the individual rates of return and then take the fourth root.
Geometric mean = (1 + 0.0317) × (1 + 0.0443) × (1 + 0.0593) × (1 + 0.0543)^(1/4)
= (1.0317 × 1.0443 × 1.0593 × 1.0543)^(1/4)
= 1.0425^(1/4)
= 1.0106 - 1
= 1.06%
Therefore, the geometric mean rate of return per year for the first four years is approximately 1.06%.
3. To construct a boxplot for the rate of return, we need to determine the five-number summary: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.
The boxplot provides a visual representation of the distribution and identifies outliers.
The five-number summary is as follows:
Minimum: -10.34%
Q1: -5.73%
Median: 4.34%
Q3: 6.23%
Maximum: 8.21%
The boxplot will show a box with the median (Q2) as a line inside it, with the lower end of the box at Q1 and the upper end at Q3. The whiskers extend from the box to the minimum and maximum values, respectively. Any data points beyond the whiskers are considered outliers.
Based on the given data, the shape of the distribution for the rate of return is negatively skewed. This is evident from the fact that the mean is lower than the median, and the presence of negative returns pulls the overall distribution towards the left.
The outliers, represented by the minimum and maximum values, also contribute to the asymmetry of the distribution.
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