Machine Accuracy A machine produces screws with a mean...

Machine Accuracy A machine produces screws with a mean length of 2.5 cm and a standard deviation of 0.2 cm. Find the probabilities that a screw produced by this machine has lengths as follows.
(a) Greater than 2.7 cm
(b) Within 1.2 standard deviations of the mean

Key Concepts

Standardization

Standardization is the process of converting a random variable to a standard normal variable by applying the Z-score formula. This transformation makes it possible to use standard normal distribution tables to compute probabilities and percentiles, as it reduces the problem to a common scale with a mean of 0 and a standard deviation of 1.

Cumulative Distribution Function (CDF)

The cumulative distribution function represents the probability that a random variable will take a value less than or equal to a specific level. For a standard normal distribution, the CDF provides the area under the curve to the left of a given Z-score, which is used to compute the probability of ranges or exceedances.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, where the mean, median, and mode are equal. It is defined by two parameters: the mean (which indicates the central tendency) and the standard deviation (which measures the dispersion or spread of the data). This distribution is commonly used for modeling natural phenomena and measurement errors.

Z-Score

The Z-score is a standardized score that indicates how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the value of interest and then dividing by the standard deviation. This standardization process allows for the comparison of data points from different normal distributions by converting them into a standard normal distribution.

Transcript

00:01 For number 33, we are told to assume normal distribution, so del curve.

00:12 And what they tell us is that this machine produces screws with an average 2 .5 centimeters length and with a standard distribution of 0 .2 centimeters.

00:29 Now, the problem is we need to know what's the probability that the screw is higher than 2 .7 centimeters and what's the probability that it's within 1 .2 standard deviations of the mean.

00:47 That's going to be hard to do if we keep those numbers.

00:55 So what we will have to do is convert them to z scores and then once they're converted into z scores we can use the table at the end of the book.

01:08 Z score for me in green here i'm gonna draw it in the z score representation.

01:24 We know the mean will be 0, and we know that sigma will be 1.

01:31 Because those are the characteristics of a z score curve.

01:36 But to convert 2 .7 centimeters, so a, we have x equals 2 .7.

01:51 And we want to know what's z.

01:54 Z equals 2 .7 minus 2 .5 divided by 0 .2.

02:05 Oh, i'm going to write the formula, actually.

02:08 The formula for z score is x minus mu divided by sigma.

02:12 So that's what i've been doing here.

02:13 2 .7 minus 2 .5 divided by sigma 0 .2.

02:18 And that's 1.

02:22 So what we want to know is the probability that the screw has length greater than 1 in the z score representation.