Refer to Exercise 65. Suppose the strength x of another randomly selected spot weld is measured. a. Find a maximum likelihood estimator of the probability that x is less than 400. That is, find the MLE for P(x 400).

b. Use the result in part (a) with the data from Exercise 65(b) to estimate P(x 400).

Exercise 65

The shear strength x of a random sample of spot welds is measured. Shear strengths (in psi) are assumed to follow a normal distribution.

a. Find the maximum likelihood estimator of the strength that is exceeded by 5% of the population of welds. That is, find a maximum likelihood estimator for the 95th percentile of the normal distribution based on a random sample of size n. Hint: Determine the relationship between the 95th percentile and and , then use the invariance property of MLEs.

b. A random sample of ten spot-weld strengths yields the following data (in psi): 392, 376, 401, 367, 389, 362, 409, 415, 358, 375. Use the result in part (a) to find an estimate of the 95th percentile of the distribution of all weld strengths.