In Exercises 32-34, assume a normal distribution. Machine...

In Exercises $32-34,$ assume a normal distribution. Machine Accuracy A machine produces screws with a mean length of 2.5 $\mathrm{cm}$ and a standard deviation of 0.2 $\mathrm{cm} .$ Find the probabilities that a screw produced by this machine has lengths as follows. a. Greater than 2.7 $\mathrm{cm}$ b. Within 1.2 standard deviations of the mean

In Exercises $32-34,$ assume a normal distribution.
Machine Accuracy A machine produces screws with a mean length of 2.5
$\mathrm{cm}$ and a standard deviation of 0.2 $\mathrm{cm} .$ Find the probabilities that a screw produced by this machine has lengths as follows.
a. Greater than 2.7 $\mathrm{cm}$
b. Within 1.2 standard deviations of the mean

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetrical about its mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell shape and is defined by two parameters: the mean, which centers the distribution, and the standard deviation, which measures the spread of the data.

Mean

The mean is the central value of a distribution, representing the average of all possible values. In the context of a normal distribution, it is the point of symmetry, and the probability distribution is centered around this value.

Standard Deviation

The standard deviation is a measure of the dispersion or spread of a set of values in a distribution. In a normal distribution, it quantifies the amount of variation or dispersion of the data points from the mean. A smaller standard deviation indicates that the data points are closer to the mean, whereas a larger one indicates more spread out data.

Z-Score Transformation

The z-score transformation is a method for standardizing values from a normal distribution. It converts a value into the number of standard deviations it is away from the mean. This transformation allows one to use standard normal distribution tables for calculating probabilities, regardless of the original scale of the data.

Cumulative Distribution Function (CDF)

The cumulative distribution function in the context of a normal distribution gives the probability that a randomly selected observation falls below a particular value. By using the CDF, one can determine the probability of a value lying within a specific range by subtracting the CDF values at the endpoints of that range.

Transcript

00:01 Okay, so for this question, we've got a random variable x that's normal with mean of 2 .5 and a standard deviation of 0 .2.

00:09 So for the first part, we want to know the probability that x is greater than 2 .7.

00:17 The way that we're going to do this is by normalizing our random variable x.

00:24 And how we normalize or standardize a normal random variable is by subtracting its mean and dividing by its standard deviation.

00:34 And what we do to one side of our inequality we must do to the other.

00:43 And so we're going to get the probability that, and when we do this, when we do this standardization, we get over here the normal random variable z.

00:53 And what we're going to get over here is 0 .2 divided by 0 .2 is exactly 1.

01:00 So from here, we can go and know that this is 1 minus the probability that z is less than or equal to 1.

01:12 And from here, we will check a normal table.

01:18 And i'll actually pull one up here on screen for us to use.

01:26 Let's do this.

01:31 Okay.

01:32 So if we want to know the probability that our normal random variable z is less than or equal to one, we go and find 1 .0 and we get 0 .843.

01:49 3.

01:49 So this is going to be 1 minus 0 .8413, which is equal to 0 .1587.

02:00 So that'll be our first answer.

02:03 And for the next part, we want the probability that x is within 1 .2 standard deviations of the mean.

02:11 So this is going to be the probability that x is between two numbers...

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