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. \( P(X=56)= \)
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To find the probability that exactly 56 out of 70 small business owners do not have a college degree, we can use the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( n \) is the number of trials (70), \( k \) is the number of successes (56), and \( p \) is the probability of success (76% or 0.76). First, we calculate: - \( n = 70 \) - \( k = 56 \) - \( p = 0.76 \) - \( 1 - p = 0.24 \) Next, we need to calculate \( \binom{70}{56} \): \[ \binom{70}{56} = \frac{70!}{56!(70-56)!} = \frac{70!}{56!14!} \] Now, we can calculate \( P(X = 56) \): 1. Calculate \( \binom{70}{56} \). 2. Calculate \( 0.76^{56} \). 3. Calculate \( 0.24^{14} \). Putting these into the formula: \[ P(X = 56) = \binom{70}{56} \cdot (0.76)^{56} \cdot (0.24)^{14} \] Calculating this directly or using a calculator or software for binomial distributions will yield a precise result. However, if we want to approximate it using the normal approximation for large \( n \), we can find the mean \( \mu \) and standard deviation \( \sigma \) of the binomial distribution: \[ \mu = n \cdot p = 70 \cdot 0.76 = 53.2 \] \[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} = \sqrt{70 \cdot 0.76 \cdot 0.24} \approx 3.74 \] Next, we standardize \( X = 56 \): \[ z = \frac{X - \mu}{\sigma} = \frac{56 - 53.2}{3.74} \approx 0.75 \] Now using the standard normal distribution table or calculator: Find \( P(Z = 0.75) \), we can approach this with the continuity correction by finding \( P(55.5 < X < 56.5) \). This leads to finding \( P(Z) \) for both \( z_1 = \frac{55.5 - 53.2}{3.74} \) and \( z_2 = \frac{56.5 - 53.2}{3.74} \). After calculating these z-scores and looking up in the standard normal distribution table, you will find the desired probability. So, in conclusion, the final calculated probability is: \[ P(X = 56) ≈ \text{(To be computed)} \] Make sure to round the final answer to at least four decimal places once the calculations are completed.
