Question

The time for oil to percolate to all parts of an engine can be treated as a random variable having a normal distribution with mean 20 seconds. find its standard deviation if the probability is 0.25 that it will take a value greater than 31.5 seconds.

Answer 1

To find the standard deviation for the given probability, convert the probability to a z-score using the standard normal distribution table. Then, use the formula z = (x - mean) / standard deviation to solve for the standard deviation.

Step-by-step explanation:

To find the standard deviation, we need to find the z-score for the given probability and use it to solve for the standard deviation using the formula z = (x - mean) / standard deviation. Here's how:

Step 1: Convert the probability to a z-score using the standard normal distribution table. Since the probability is 0.25 and we want to find the z-score for the value greater than 31.5 seconds, we need to find the z-score for the complement of the probability (1 - 0.25 = 0.75).

Step 2: Find the z-score using the standard normal distribution table. The closest z-score to 0.75 is 0.674. This means that 0.75 of the area under the normal distribution curve is less than the value corresponding to a z-score of 0.674.

Step 3: Use the formula z = (x - mean) / standard deviation to solve for the standard deviation. Rearranging the formula, we get standard deviation = (x - mean) / z. Substituting the values, we get standard deviation = (31.5 - 20) / 0.674 ≈ 17.21 seconds.