otorola discovered that the average defect rate for parts produced on an assembly line varies from run-to-run and is approximately normally distributed with a mean equal to 3 defects per million. Assume that the goal at Motorola is for the average defect rate to vary no more than 1.5 standard deviations above or below the mean of 3. How likely is it that the goal will be met?.
You are looking for the area under a normal curve between -1.5 and 1.5 standard deviations.
Use a calculator to find this area
or Use a z-table.
0.8664
To determine how likely it is that the goal will be met, we need to calculate the probability that the average defect rate falls within the desired range.
First, let's calculate the standard deviation. The problem states that the average defect rate varies no more than 1.5 standard deviations above or below the mean. Since the goal is to have a mean defect rate of 3 defects per million, we need to calculate the standard deviation.
The problem provides that the average defect rate is approximately normally distributed with a mean of 3 defects per million. However, it does not provide the standard deviation directly. We can calculate it by using the fact that the defect rate is given in parts per million.
Since the defect rate is given in parts per million, we know that the variance can be calculated by dividing the defect rate by 1,000,000. Therefore, the variance is 3/1,000,000. To get the standard deviation, we take the square root of the variance:
Standard Deviation (σ) = √(3/1,000,000)
Now that we know the standard deviation, we can calculate the range within which the average defect rate should fall. According to the problem, the goal is for the rate to vary no more than 1.5 standard deviations above or below the mean.
Upper Limit = Mean + (1.5 × Standard Deviation)
Lower Limit = Mean - (1.5 × Standard Deviation)
Next, we need to calculate the probability (p) that the average defect rate falls within the desired range by using a standard normal distribution table or calculator. Since we assume that the defect rate is normally distributed, we can use the Z-score formula:
p = P(x < Upper Limit) - P(x < Lower Limit)
Finally, we can look up the probability value in the Z-table or use a calculator to find the area under the normal curve between the upper and lower limits. This will give us the probability of meeting the goal, expressed as a decimal or percentage.