Math Problem Statement
Develop a discrete probability distribution for x = Age.
What is the probability that a student's age is more than 20?
What is the mean age?
What is the standard deviation of the ages?
Suppose that we consider the distribution of ages to be approximately bell-shaped. We would expect 95% of the ages to lie between and years. (Remember to use non-rounded values in your calculations and only round your final answers.)
Solution
To approach this problem, let’s outline the steps:
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Discrete Probability Distribution: We need to know the specific age values and their corresponding probabilities. If you have any specific age data, we can use that. For now, I will assume we have data about student ages.
Let's suppose we have the following age distribution of students:
Age (x) Probability P(x) 18 0.20 19 0.25 20 0.30 21 0.15 22 0.10 -
Find the probability that a student's age is more than 20:
To find , we sum the probabilities for ages greater than 20:
= 0.15 + 0.10 = 0.25$$ -
Mean (Expected Value) of Age:
The mean (expected value) of the distribution can be found using the formula: Substituting the values from the table:
= 3.6 + 4.75 + 6.0 + 3.15 + 2.2 = 19.7$$ So, the mean age $$\mu = 19.7$$ years. -
Standard Deviation of Ages:
The standard deviation is calculated using the formula: First, calculate for each value of :
Age (x) 18 19 20 21 22 Now, sum up the last column: Finally, the standard deviation is:
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95% Confidence Interval (Assuming Bell-Shaped Distribution):
If the distribution is approximately bell-shaped (normal distribution), 95% of the data falls within .
Thus, we expect 95% of student ages to lie between 17.24 years and 22.16 years.
Summary:
- The probability that a student's age is more than 20 is 0.25.
- The mean age is 19.7 years.
- The standard deviation is approximately 1.23 years.
- We expect 95% of the ages to lie between 17.24 and 22.16 years.
Would you like more details or have any questions?
Here are some related questions:
- How would changing the probabilities of certain age groups affect the mean age?
- How would you calculate the probability for a student's age being between 19 and 21?
- How is the mean age impacted if the majority of students were slightly older?
- How would you compute the median of this age distribution?
- Can the skewness of the distribution impact the mean or standard deviation?
Tip: The standard deviation measures how spread out values are around the mean. The larger the standard deviation, the wider the spread of values.
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Math Problem Analysis
Mathematical Concepts
Probability Distribution
Expected Value
Standard Deviation
Normal Distribution
Formulas
Mean (Expected Value) formula: μ = Σ(x * P(x))
Standard Deviation formula: σ = √Σ((x - μ)^2 * P(x))
Probability P(x > 20): P(21) + P(22)
Confidence Interval for normal distribution: μ ± 2σ
Theorems
Law of Large Numbers
Empirical Rule (68-95-99.7 Rule) for normal distribution
Suitable Grade Level
Grade 11-12, College Introductory
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