To solve this problem, we will use the properties of the normal distribution.

i. To find the probability that the mean life of a random sample of 9 machines falls between 6.4 and 7.2 years, we need to calculate the z-scores for both values and then find the area under the normal curve between these z-scores.

First, we calculate the z-score for 6.4 years:
z1 = (6.4 - 7) / (1 / sqrt(9))
z1 = -0.6 / (1 / 3)
z1 = -1.8

Next, we calculate the z-score for 7.2 years:
z2 = (7.2 - 7) / (1 / sqrt(9))
z2 = 0.2 / (1 / 3)
z2 = 0.6

Now, we need to find the area under the normal curve between these z-scores. We can use a standard normal distribution table or a calculator to find these probabilities.

Using a standard normal distribution table, we find that the area to the left of z = -1.8 is approximately 0.0359, and the area to the left of z = 0.6 is approximately 0.7257.

To find the probability between these two z-scores, we subtract the smaller area from the larger area:
P(6.4

Question

To solve this problem, we will use the properties of the normal distribution.

i. To find the probability that the mean life of a random sample of 9 machines falls between 6.4 and 7.2 years, we need to calculate the z-scores for both values and then find the area under the normal curve between these z-scores.

First, we calculate the z-score for 6.4 years:
z1 = (6.4 - 7) / (1 / sqrt(9))
z1 = -0.6 / (1 / 3)
z1 = -1.8

Next, we calculate the z-score for 7.2 years:
z2 = (7.2 - 7) / (1 / sqrt(9))
z2 = 0.2 / (1 / 3)
z2 = 0.6

Now, we need to find the area under the normal curve between these z-scores. We can use a standard normal distribution table or a calculator to find these probabilities.

Using a standard normal distribution table, we find that the area to the left of z = -1.8 is approximately 0.0359, and the area to the left of z = 0.6 is approximately 0.7257.

To find the probability between these two z-scores, we subtract the smaller area from the larger area:
P(6.4 < x < 7.2) = 0.7257 - 0.0359
P(6.4 < x < 7.2) = 0.6898

Therefore, the probability that the mean life of a random sample of 9 machines falls between 6.4 and 7.2 years is approximately 0.6898.

ii. To find the value of a to the right of which 15% of the means computed from random samples of size 9 would fall, we need to find the z-score that corresponds to the 85th percentile.

Using a standard normal distribution table or a calculator, we find that the z-score corresponding to the 85th percentile is approximately 1.0364.

Now, we can use the formula for z-score to find the value of a:
a = mean + (z * (standard deviation / sqrt(sample size)))
a = 7 + (1.0364 * (1 / sqrt(9)))
a = 7 + (1.0364 * (1 / 3))
a = 7 + 0.3455
a = 7.3455

Therefore, the value of a to the right of which 15% of the means computed from random samples of size 9 would fall is approximately 7.3455 years.

Knowee AI · Accepted Answer