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

Solution Steps

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

\[ \mu = n \cdot p \]

where:

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

Substituting the values:

\[ \mu = 82 \cdot 0.39 = 31.98 \]

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

\[ \sigma^2 = n \cdot p \cdot q \]

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

\[ q = 1 - 0.39 = 0.61 \]

Now substituting the values into the variance formula:

\[ \sigma^2 = 82 \cdot 0.39 \cdot 0.61 = 19.508 \]

The standard deviation is the square root of the variance:

\[ \sigma = \sqrt{n \cdot p \cdot q} = \sqrt{19.508} \approx 4.417 \]

Final Answer