C955: Applied Probability and Statistics
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Free C955: Applied Probability and Statistics Questions
If the manufacturing process were to change, resulting in a mean weight of 52g and a standard deviation of 3g, what would be the new percentage of items rejected that weigh less than 49g or more than 55g?
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Approximately 10% (based on the new normal distribution)
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Approximately 5% (based on the new normal distribution)
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Approximately 15% (based on the new normal distribution)
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Approximately 2% (based on the new normal distribution)
Explanation
Explanation:
The rejection region is defined as weights less than 49g or greater than 55g. Using the new mean of 52g and standard deviation of 3g:
For 49g: z = (49 − 52) / 3 = −1
For 55g: z = (55 − 52) / 3 = 1
According to the 68-95-99.7 rule, approximately 68% of values lie within ±1 standard deviation from the mean. Therefore, about 32% lie outside this range. Since the rejection region includes both tails equally, the percentage rejected is 32% ÷ 2 = 16% for each tail. Actually, summing both tails gives approximately 32%. However, using standard normal tables, z = ±1 corresponds to roughly 15.87% in each tail combined, totaling approximately 16%, which rounds to 15% for practical purposes.
Correct Answer:
Approximately 15% (based on the new normal distribution)
Scores on a university exam are normally distributed with a mean of 68 and a standard deviation of 9. Using the 68-95-99.7 rule, what percentage of students score above 77.
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16%
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5%
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2.5%
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32%
Explanation
Explanation:
A score of 77 is one standard deviation above the mean (68 + 9 = 77). According to the 68-95-99.7 rule, approximately 68% of scores lie within one standard deviation of the mean. This leaves 32% of scores outside this range, with 16% above the mean plus one standard deviation. Therefore, about 16% of students score above 77.
Correct Answer:
16%
The average slice of bread has 75 calories, with a standard deviation of 5 calories.
What are the two values between which 95% of the data falls for the average calories in a slice of bread, assuming a normal distribution?
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72 calories and 78 calories
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70 calories and 80 calories
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68 calories and 82 calories
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65 calories and 85 calories
Explanation
Explanation
In a normal distribution, approximately 95% of the data falls within ±2 standard deviations from the mean.
Mean = 75, Standard deviation = 5
Lower limit = 75 − 2(5) = 75 − 10 = 65
Upper limit = 75 + 2(5) = 75 + 10 = 85
Thus, 95% of the data falls between 65 and 85 calories.
Correct Answer
D. 65 calories and 85 calories
At a hospital, 40% of patients are admitted for surgery, and 80% of patients recover within a week. Of patients admitted for surgery, 75% recover within a week.
Are the events "admitted for surgery" and "recover within a week" independent events?
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Yes, because 0.80 is not equal to 0.75
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No, because 0.80 is not equal to 0.75
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Yes, because 0.40 is not equal to 0.75
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No, because 0.40 is not equal to 0.75
Explanation
Explanation
Two events A and B are independent if P(B | A) = P(B). Here:
P(Recover | Surgery) = 0.75
P(Recover) = 0.80
Since 0.75 ≠ 0.80, the events are not independent.
Correct Answer
B. No, because 0.80 is not equal to 0.75
A team of scientists wants to understand the relationship between smoking and lung cancer. They collect data from 1,000 individuals, recording their smoking habits and whether they have been diagnosed with lung cancer. The scientists then analyze the data to find any correlations. Is this study observational or experimental?
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Experimental, because the researchers are testing the effects of smoking on lung cancer.
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Observational, because the study involves a large sample size.
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Observational, because the researchers are collecting data without manipulating any variables.
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Experimental, because the researchers are analyzing the data to find correlations.
Explanation
Explanation
Correct answer: (C) Observational, because the researchers are collecting data without manipulating any variables.
An observational study involves collecting and analyzing data without directly interfering with or assigning treatments to subjects. In this case, the scientists are simply recording existing smoking habits and lung cancer diagnoses without controlling or assigning smoking behavior. Option A is incorrect because no experimental manipulation or treatment assignment is taking place. Option B is incorrect because sample size does not determine whether a study is observational or experimental. Option D is incorrect because analyzing correlations does not make a study experimental; manipulation of variables is required for an experiment.
In a jar of 100 marbles, there are 30 red marbles, 20 blue marbles, and 50 green marbles.
What is the probability of not drawing a red marble?
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0.20
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0.30
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0.50
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0.70
Explanation
Explanation
The probability of not drawing a red marble is the number of non-red marbles divided by the total number of marbles.
Number of non-red marbles = 20 + 50 = 70
Total marbles = 100
Probability = 70 ÷ 100 = 0.70
Correct Answer
D. 0.70
A gardener records the length of a plant over a duration of 35 weeks. The scatterplot shows the data.
What is the relationship between length and time?
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There is a strong positive relationship between time and the height of the plant.
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There is a weak positive relationship between time and the height of the plant.
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There is a strong negative relationship between time and the height of the plant.
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There is a weak negative relationship between time and the height of the plant.
Explanation
Explanation:
As time increases, the length of the plant consistently increases. The points in the scatterplot show a clear upward trend with little deviation, indicating a strong positive relationship. Positive relationships occur when both variables increase together, and the consistency of the trend confirms it is strong.
Correct Answer:
There is a strong positive relationship between time and the height of the plant.
If five dice are thrown, what is the probability that the sum of the numbers appearing on them is 30? Assume the same conditions as before
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1/7776
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0
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1/6
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1/1296
Explanation
Explanation:
The maximum possible sum when throwing five six-sided dice is 6 × 5 = 30. There is exactly one outcome that achieves this total: all five dice must show 6. The total number of equally likely outcomes is 6⁵ = 7776. Therefore, the probability is 1 out of 7776.
Correct Answer:
1/7776
The initial fee to have a website set up using a specific server is $90, and it costs $46 per month to maintain the website on this server. This equation represents the total cost of setting up and maintaining the website, where x represents the number of months: y = 46x + 90.
Which graph represents this equation?
Explanation
Explanation
y = 46x + 90.
From the equation and the question the least price or the y intercept is 90, hence all the graphs that do not start at y = 90 are eliminated. After eliminating two of the options only one graph remains. After one month x will be 1 and y is expected to be;
y = (46 * 1) + 90.
y = 136.
This confirms that the graph is correct.
Correct Answer
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.
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