seventy-five percent of the students graduating from high school in a small iowa farm town attend college…

Explanation:

Step1: Identify binomial parameters

$n = 35$, $p = 0.75$, $q = 1 - p = 0.25$

Step2: Determine the required number of successes

$k \geq 0.85 \times 35 = 29.75 \rightarrow k \geq 30$

Step3: State the binomial probability formula

$P(X = k) = \binom{n}{k} p^k q^{n-k}$

Step4: Calculate probability for $k=30$

$P(30) = \binom{35}{30} (0.75)^{30} (0.25)^5 \approx 0.05435$

Step5: Calculate probability for $k=31$

$P(31) = \binom{35}{31} (0.75)^{31} (0.25)^4 \approx 0.02630$

Step6: Calculate probability for $k=32$

$P(32) = \binom{35}{32} (0.75)^{32} (0.25)^3 \approx 0.00925$

Step7: Calculate probability for $k=33$

$P(33) = \binom{35}{33} (0.75)^{33} (0.25)^2 \approx 0.00231$

Step8: Calculate probability for $k=34$

$P(34) = \binom{35}{34} (0.75)^{34} (0.25)^1 \approx 0.00034$

Step9: Calculate probability for $k=35$

$P(35) = \binom{35}{35} (0.75)^{35} (0.25)^0 \approx 0.00003$

Step10: Sum the probabilities

$P(X \geq 30) = 0.05435 + 0.02630 + 0.00925 + 0.00231 + 0.00034 + 0.00003 = 0.09258$

Answer:

0.0926