Question

Below, n is the sample size, p is the population proportion, and X is the number of successes in the sample. Find the probability.

n = 78, p = 0.43

P(30)

a. 0.122
b. 0.200
c. 0.305
d. 0.421

Answer 1

For the given values of sample size (n = 78) and population proportion (p = 0.43), the probability P(30) is approximately 0.305. (option c)

Step-by-step explanation:

To calculate the probability P(X = 30) in a binomial distribution with parameters n = 78 and p = 0.43, we use the binomial probability formula:

`\[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^((n-k)) \]`

In this context, k represents the number of successes (30). Substituting the provided values into the formula:

`\[ P(30) = \binom{78}{30} \cdot (0.43)^(30) \cdot (1-0.43)^(48) \]`

Evaluating this expression results in a probability of approximately 0.305.

The use of subscript/superscript style enhances the clarity of mathematical notation, allowing for precise representation of combinations
`(\( \binom{n}{k} \))` and exponents in the explanation.(option c)