C955: Applied Probability and Statistics
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Free C955: Applied Probability and Statistics Questions
A computer game randomly selects one of three prizes for players completing a level: 500 points, a power boost, or an extra life. The probability a player will receive 500 points is 5/12. The probability a player will receive a power boost is 1/3. What is the probability a player will receive an extra life?
Explanation
Correct answer
Explanation
In probability, all possible outcomes must sum to 1 (or 100%). Here, the three outcomes—500 points, power boost, and extra life—must add up to exactly 1. Given probabilities for 500 points () and power boost (), the sum of these two probabilities is . Converting to common denominators, becomes , and the total for the two known prizes is + = . Thus, the probability for the remaining outcome (extra life) is 1 − = , simplifying to .
A researcher has collected information on participants at a marathon and wants to display the ages of the runners:
16, 25, 36, 27, 48, 35, 26, 57, 45, 35, 66, 47, 28, 34, 25, 48
Which graph type should be used to display these data?
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Bar graph
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Stem-and-leaf plot
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Line graph
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Pie chart
Explanation
Correct answer
b. Stem-and-leaf plot
Explanation
The correct answer is option b, Stem-and-leaf plot, because it effectively organizes numerical data by displaying both distribution and individual data values. It clearly shows the frequency of ages while preserving original numerical data. A stem-and-leaf plot is ideal for relatively small datasets, allowing a clear visual representation of the spread and central tendency of the data.
A study was conducted on the possible relationship between a categorical variable and a quantitative variable. Which type of graphical display is appropriate for displaying this relationship?
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Scatterplot
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Decision table
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Side-by-side boxplot
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Two-way table
Explanation
Correct answer
c. Side-by-side boxplot
Explanation
The correct answer is option c, Side-by-side boxplot, because it effectively compares distributions of a quantitative variable across categories of a categorical variable. Box plots clearly show the median, quartiles, range, and potential outliers for each category. This makes a side-by-side boxplot an ideal graphical method for displaying and comparing the distributions of a quantitative variable between categories, providing clear insights into differences or similarities among the groups.
Describe how the Poisson distribution is used to model the number of flaws in a wire.
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The Poisson distribution is used to determine the total number of wires produced in a factory.
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The Poisson distribution calculates the average length of wire needed to produce a certain number of flaws.
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The Poisson distribution estimates the time taken to produce a wire with a specific number of flaws.
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The Poisson distribution models the number of flaws in a wire by providing the probability of a given number of events occurring in a fixed interval, based on the average rate of occurrence.
Explanation
Explanation:
The Poisson distribution is appropriate for modeling the number of rare, independent events occurring in a fixed interval of space or time when the average rate of occurrence is known. Flaws in a wire can be considered random defects that happen independently along a length of wire. Using the Poisson distribution, one can calculate the probability of observing a specific number of flaws in a fixed length of wire if the average flaw rate per unit length is known.
Correct Answer:
The Poisson distribution models the number of flaws in a wire by providing the probability of a given number of events occurring in a fixed interval, based on the average rate of occurrence.
A person has a 0.5 probability of taking a train to work, a 0.3 probability of carpooling, and a 0.2 probability of walking. The probability of being late is 0.2 when taking the train, 0.1 when carpooling, and 0.05 when walking. What is the probability of taking the train and being on time, or walking and being on time?
- 0.3
- 0.4
- 0.45
- 0.59
Explanation
Explanation
Correct answer: (D) 0.59
Train and on time:
0.5 × (1 − 0.2) = 0.5 × 0.8 = 0.40
Walking and on time:
0.2 × (1 − 0.05) = 0.2 × 0.95 = 0.19
Add them:
0.40 + 0.19 = 0.59
A sample of 100 houses were surveyed in each of two states: Georgia and Kansas. Square footage of the houses are summarized by state in the following side-by-side box plots. Which of the statements about house sizes is valid?
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There were approximately twice as many houses smaller than 2800 sq. ft. in Georgia than in Kansas
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There were approximately twice as many houses larger than 2800 sq. ft. in Kansas than in Georgia
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More than half of all houses were above 2800 sq. ft.
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All houses in Georgia were less than 2800 sq. ft.
Explanation
Correct answer
a. There were approximately twice as many houses smaller than 2800 sq. ft. in Georgia than in Kansas
Explanation
The correct answer is option a, as indicated by the box plots. The line in the box plot represents the median, with half of the houses above and half below that point. Observing both box plots, we can see the median for Georgia is clearly below 2800 sq. ft., indicating that roughly half of Georgia houses are smaller than 2800 sq. ft., while significantly fewer Kansas houses fall below this mark. Thus, approximately twice as many Georgia houses compared to Kansas houses have square footage below 2800 sq. ft.
What is the range of this dataset: 2, 4, 6, 8, 10?
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The range is 4.
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The range is 6.
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The range is 10.
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The range is 8.
Explanation
Explanation:
The range of a dataset is calculated as the maximum value minus the minimum value.
Here, the maximum value = 10 and the minimum value = 2.
Range = 10 − 2 = 8
Correct Answer:
D. The range is 8.
A single ball is drawn from an opaque bag that contains red, blue, and green balls. The probability of drawing a red ball is 0.3, and the probability of drawing a blue ball is 0.6. What is the probability of drawing a green ball?
- 0.1
- 0.3
- 0.5
- 0.9
Explanation
Explanation
Correct answer: (A) 0.1
The total probability of all possible outcomes must equal 1.
So, P(green) = 1 − (0.3 + 0.6) = 1 − 0.9 = 0.1.
A research group is trying to find the relationship between the number of years a person has been smoking and their lung capacity. The data recorded is plotted below.
The regression equation (y = 100 - 2.5x) represents the lung capacity, (y) in percent, after a person has been smoking for (x) years.
What does the slope represent in the regression equation?
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The lung capacity decreases by 100% for every 2.5 years of smoking.
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The lung capacity increases by 100% for every 2.5 years of smoking.
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The lung capacity decreases by 2.5% for every year of smoking.
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The lung capacity increases by 2.5% for every year of smoking.
Explanation
Explanation:
In a regression equation (y = mx + b), the slope (m) indicates the change in the dependent variable (y) for each one-unit increase in the independent variable (x). Here, the slope is -2.5, meaning lung capacity decreases by 2.5% for each additional year of smoking.
Correct Answer:
The lung capacity decreases by 2.5% for every year of smoking.
If the number of white balls in the bag is increased to 15 while the number of red balls remains the same, what is the new probability of drawing a white ball?
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5/9
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15/27
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5/7
Explanation
Explanation:
The original total and number of red balls are not explicitly stated, but the key is that increasing white balls to 15 while keeping red balls the same increases the total number of balls. The probability of drawing a white ball is the number of white balls divided by the total. Among the options, 5/7 correctly represents a scenario where the proportion of white balls is higher than before, consistent with adding more white balls while keeping red constant.
Correct Answer:
5/7
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