Question

Although errors are likely when taking measurements from photographic images, these errors are often very small. For sharp images with negligible distortion, errors in measuring distances are often no larger than .0004 inch. Assume that the probability of a serious measurement error is .05. A series of 150 independent measurements are made. Let X denote the number of serious errors made.

a. In finding the probability of making at least one serious error, is the normal approximation appropriate?

Answer 1

The normal approximation is appropriate for determining the probability of making at least one error out of 150 measurements because both np and n(1-p) are greater than 5. The calculation shows that this criterion is met, hence the approximation can be used.

Step-by-step explanation:

The question asks whether the normal approximation is appropriate when finding the probability of making at least one serious error out of a series of 150 independent measurements, given that the probability of a serious error is 0.05. The normal approximation to the binomial distribution can be applied when both the np and n(1-p) values are greater than or equal to 5, where 'n' is the number of trials, and 'p' is the probability of success (in this case, making an error).

You would calculate np = 150 × 0.05 = 7.5 and n(1-p) = 150 × (1 - 0.05) = 142.5. Both these values are greater than 5, which means the normal approximation is indeed appropriate for this situation. To find the probability of at least one error, we could use the normal approximation to calculate the complementary probability of making no errors and subtract it from 1.