Math Problem Statement
Assume that a procedure yields a binomial distribution with n=1301 trials and the probability of success for one trial is p=72%.
Find the mean for this binomial distribution. (Round answer to one decimal place.) μ=
Find the standard deviation for this distribution. (Round answer to two decimal places.) σ=
Use the range rule of thumb to find the minimum usual value μ–2σ and the maximum usual value μ+2σ. Enter answer as an interval using square-brackets only with whole numbers. usual values =
Solution
To solve this problem, we will use the formulas for the mean and standard deviation of a binomial distribution. The mean and standard deviation for a binomial distribution are given by:
- Mean
- Standard deviation
Where:
- (the number of trials)
- (the probability of success)
Step 1: Find the Mean ()
So, the mean is approximately:
Step 2: Find the Standard Deviation ()
So, the standard deviation is approximately:
Step 3: Find the Usual Range Using the Range Rule of Thumb
The usual values are calculated using:
For the minimum usual value:
For the maximum usual value:
Thus, the usual values fall within the interval:
Final Answers:
- Mean (): 936.7
- Standard deviation (): 16.2
- Usual values interval: [904, 969]
Let me know if you'd like more details or have any questions! Also, here are some related questions:
- What is the variance of this binomial distribution?
- How would you calculate the probability of getting exactly 900 successes in this binomial distribution?
- How can the binomial distribution be approximated by a normal distribution in this case?
- What is the probability of getting fewer than 800 successes?
- How does increasing or decreasing the number of trials affect the standard deviation of a binomial distribution?
Tip: The mean of a binomial distribution is helpful for determining the expected number of successes in a given number of trials.
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Math Problem Analysis
Mathematical Concepts
Binomial Distribution
Probability
Standard Deviation
Mean
Formulas
Mean: μ = n × p
Standard Deviation: σ = √(n × p × (1 - p))
Usual Values: μ - 2σ, μ + 2σ
Theorems
Range Rule of Thumb for Usual Values
Suitable Grade Level
Grades 11-12
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