Question

A company making smartphone cases is concerned about the exact length of their cases. The company has a lower specification limit of 12.75 cm and an upper specification limit of 13.25 cm. The standard deviation is 0.125 cm, and the mean is 13 cm. What is the probability that a randomly selected smartphone case falls outside the specification limits? Option 1: 0.0013 Option 2: 0.0026 Option 3: 0.0066 Option 4: 0.0136

Answer 1

To find the probability that a randomly selected smartphone case falls outside the specification limits, we need to calculate the z-scores for both the lower and upper specification limits and then find the probabilities associated with these z-scores using a standard normal distribution table (z-table).

First, calculate the z-scores:

For the lower specification limit:
z = (12.75 - 13) / 0.125 = -0.25 / 0.125 = -2

For the upper specification limit:
z = (13.25 - 13) / 0.125 = 0.25 / 0.125 = 2

Now, we'll find the probabilities associated with these z-scores using a standard normal distribution table.

From the z-table:
- The probability that a z-score is less than -2 is approximately 0.0228.
- The probability that a z-score is greater than 2 is approximately 0.0228.

Now, to find the probability that a randomly selected smartphone case falls outside the specification limits (either below 12.75 cm or above 13.25 cm), we add these two probabilities:

Probability = 0.0228 (for z < -2) + 0.0228 (for z > 2) = 0.0228 + 0.0228 = 0.0456

So, the probability that a randomly selected smartphone case falls outside the specification limits is approximately 0.0456.

None of the provided options exactly match this probability. The closest option is **Option 3: 0.0066**, but it does not match the calculated probability accurately.