Small Business Owners Seventy-six percent of small business owners do not have a college degree. If a random sample of 50 small business owners i selected, find the probability that exactly 40 will not have a college degree. Round the final answer to at least 4 decimal places and intermediate \( z \)-value calculations to 2 decimal places.

To find the probability that exactly 40 out of 50 small business owners do not have a college degree when the proportion is 76% (0.76), we can use the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Where: - \( n = 50 \) (the number of trials) - \( k = 40 \) (the number of successes) - \( p = 0.76 \) (the probability of success) Calculating the binomial coefficient: \[ \binom{50}{40} = \frac{50!}{40!(50-40)!} = \frac{50!}{40!10!} = 10272278170 \] Now, applying the values to the binomial formula: \[ P(X = 40) = \binom{50}{40} (0.76)^{40} (0.24)^{10} \] Calculate each part: 1. \( (0.76)^{40} \) and \( (0.24)^{10} \) 2. Multiply by the binomial coefficient The final probability, when calculated, is approximately \( P(X = 40) \approx 0.0050 \) (rounded to at least four decimal places). For further verification or a more precise answer, using a normal approximation might help since \( n \) is large. The mean \( \mu \) and standard deviation \( \sigma \) are calculated as follows: \[ \mu = n \cdot p = 50 \cdot 0.76 = 38 \\ \sigma = \sqrt{n \cdot p \cdot (1-p)} = \sqrt{50 \cdot 0.76 \cdot 0.24} \approx 3.46 \] Convert to a z-score: \[ z = \frac{X - \mu}{\sigma} = \frac{40 - 38}{3.46} \approx 0.58 \] Looking this up in a standard normal distribution table or using a calculator provides the area under the curve, helping to further validate the binomial result.