Final answer:
To find the probability that more than 78% of the sample keypads pass inspection, we can use the normal distribution to approximate the probability. The probability is approximately 0.5000.
Step-by-step explanation:
To find the probability that more than 78% of the sample keypads pass inspection, we need to calculate the probability of having more than 78% of the 110 keypads pass inspection. Since 75% of the keypads pass inspection, we can use the normal distribution to approximate the probability. First, find the mean of the distribution, which is 110 * 0.75 = 82.5. Next, calculate the standard deviation using the formula: sqrt(n * p * (1 - p)), where n is the sample size and p is the proportion. In this case, the standard deviation is sqrt(110 * 0.75 * 0.25) = 5.692. Finally, we can calculate the z-score using the formula: (x - mean) / standard deviation, where x is the desired proportion. In this case, the z-score is (0.78 - 0.75) / 5.692 = 0.0053. We can then use a standard normal distribution table or calculator to find the corresponding probability, which is approximately 0.5000. Since we want the probability of having more than 78%, we subtract the probability from 1 to get 1 - 0.5000 = 0.5000.