Small Business Owners Seventy-six percent of small business owners do not have a college degree. If a random sample of 70 small business owners is selected, find the probability that exactly 56 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. \( P(X=56)=0.0860 \)

To find the probability that exactly 56 small business owners out of 70 do not have a college degree, we can use the binomial distribution formula. Given: - \( n = 70 \) (total number of small business owners) - \( p = 0.76 \) (probability of a small business owner not having a college degree) - \( X = 56 \) (number of small business owners not having a college degree) The binomial distribution formula is: \[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \] Substitute the given values into the formula: \[ P(X=56) = \binom{70}{56} (0.76)^{56} (1-0.76)^{70-56} \] Now, we can calculate the probability using the binomial distribution formula. Calculate the value by following steps: - step0: Calculate: \( { }_{70}C_{56}\times 0.76^{56}\left(1-0.76\right)^{14}\) - step1: Subtract the numbers: \( { }_{70}C_{56}\times 0.76^{56}\times 0.24^{14}\) - step2: Expand the expression: \(\frac{70!}{56!\times \left(70-56\right)!}\times 0.76^{56}\times 0.24^{14}\) - step3: Subtract the numbers: \(\frac{70!}{56!\times 14!}\times 0.76^{56}\times 0.24^{14}\) - step4: Multiply the terms: \(\frac{70!}{87178291200\times 56!}\times 0.76^{56}\times 0.24^{14}\) - step5: Divide the terms: \(193253756909160\times 0.76^{56}\times 0.24^{14}\) - step6: Convert the expressions: \(193253756909160\left(\frac{19}{25}\right)^{56}\times 0.24^{14}\) - step7: Convert the expressions: \(193253756909160\left(\frac{19}{25}\right)^{56}\left(\frac{6}{25}\right)^{14}\) - step8: Simplify: \(193253756909160\times \frac{19^{56}}{25^{56}}\times \left(\frac{6}{25}\right)^{14}\) - step9: Multiply the terms: \(\frac{38650751381832\times 19^{56}}{5^{111}}\times \left(\frac{6}{25}\right)^{14}\) - step10: Evaluate the power: \(\frac{38650751381832\times 19^{56}}{5^{111}}\times \frac{6^{14}}{25^{14}}\) - step11: Multiply the fractions: \(\frac{38650751381832\times 19^{56}\times 6^{14}}{5^{111}\times 25^{14}}\) - step12: Multiply: \(\frac{38650751381832\times 19^{56}\times 6^{14}}{5^{139}}\) The probability that exactly 56 small business owners out of 70 do not have a college degree is approximately 0.0860.