Questions: For this binomial distribution, what is the Expected Value for the number of successes when sample size is 82 with a probability of 0.39 ? Add your answer integer, decimat, or Enotation allowed

For this binomial distribution, what is the Expected Value for the number of successes when sample size is 82 with a probability of 0.39 ?

Add your answer
integer, decimat, or Enotation allowed
Transcript text: For this binomial distribution, what is the Expected Value for the number of successes when sample size is 82 with a probability of 0.39 ? Add your answer integer, decimat, or Enotation allowed
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Solution

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Solution Steps

Step 1: Calculate the Mean

For a binomial distribution, the expected value (mean) is calculated using the formula:

μ=np \mu = n \cdot p

where:

  • n=82 n = 82 (the number of trials),
  • p=0.39 p = 0.39 (the probability of success).

Substituting the values:

μ=820.39=31.98 \mu = 82 \cdot 0.39 = 31.98

Step 2: Calculate the Variance

The variance of a binomial distribution is given by the formula:

σ2=npq \sigma^2 = n \cdot p \cdot q

where q=1p q = 1 - p . Thus, we have:

q=10.39=0.61 q = 1 - 0.39 = 0.61

Now substituting the values into the variance formula:

σ2=820.390.61=19.508 \sigma^2 = 82 \cdot 0.39 \cdot 0.61 = 19.508

Step 3: Calculate the Standard Deviation

The standard deviation is the square root of the variance:

σ=npq=19.5084.417 \sigma = \sqrt{n \cdot p \cdot q} = \sqrt{19.508} \approx 4.417

Final Answer

The expected value (mean) for the number of successes in this binomial distribution is:

31.98 \boxed{31.98}

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