Question

Manufacture of a certain component requires three different machining operations. Machining time for each operation has a normal distribution, and the three times are independent of one another. The mean values are 15, 20, and 30 min, respectively, and the standard deviations are 2, 1, and 1.7 min, respectively. What is the probability that it takes at most 1 hour of machining time to produce a randomly selected component? (Round your answer to four decimal places.)

Answer 1

Concepts and reason

Normal distribution:

Normal distribution is a continuous distribution of data that has the bell-shaped curve. The normally distributed random variable x has meanand variance.

Also, the standard normal distribution represents a normal curve with mean 0 and standard deviation 1. Thus, the parameters involved in a normal distribution are mean and standard deviation.

Standardized z-score:

The standardized z-score represents the number of standard deviations the data point is away from the mean.

• If the z-score takes positive value when it is above the mean.

• If the z-score takes negative value when it is below the mean.

Mean: Mean is obtained by adding the observations in a set of data and then dividing the total value by the total number of observations.

Variance: The variance is the average of the squared deviation of the random variable from its mean. It represents the spread of the data from the mean. It is denoted by or , where X represents the random variable.

Standard deviation: Standard deviation is defined as the square root of the variance. It is denoted by .

Fundamentals

Let , then the standard z-score is found using the formula given below:

where X denotes the individual raw score, denotes the population mean and denotes the population standard deviation.

The properties of mean are:

Let M is denote mean value

The property of variance is:

Let S is denote standard deviation.

Formula for standard deviation is .

Find the total mean time.

From that the given information, the mean values is 15, 20 and 30.

Then,

From that the given information standard deviation values is 2, 1, and 1.7.

Then,

Find the standard deviation.

From the information, it is clear that at most 1 hours is , and

Consider,

From the “standard normal table”, the area to the left of is 0.0375.

Ans:

The probability that it takes at most 1 hour of machining time to produce a randomly selected component is 0.0375.