Use a computer or calculator to find the probability that one randomly selected value of x from

step 1 In problem number 70, we're going to use a computer or a calculator. I'm going to show you how t

Use a computer or calculator to find the probability that...

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its symmetrical, bell-shaped curve, where most values cluster around the mean and probabilities for values further from the mean taper off. It’s defined by two parameters: the mean (which locates the center of the distribution) and the standard deviation (which measures the spread of data around the mean).

Standardization (Z-score Transformation)

Standardization is a process of converting a normal random variable into a standard normal variable by subtracting the mean and dividing by the standard deviation. This process results in a variable with a mean of 0 and a standard deviation of 1, simplifying the computation of probabilities using standardized tables or computer functions.

Cumulative Distribution Function (CDF)

The CDF of a distribution provides the probability that a random variable is less than or equal to a given value. In the context of the normal distribution, the CDF is used to determine the probability associated with specific values or intervals by calculating the area under the curve up to those values.

Probability Calculation for Intervals

Calculating probabilities for ranges in a normal distribution involves finding the difference between the CDF values at the endpoints of the interval. This difference represents the area under the curve between those endpoints, which corresponds to the probability that the variable falls within that interval.

Using Tables vs. Computational Tools

Both standard normal tables and computational methods (e.g., calculators, computer software) can be used to find probabilities for a normal distribution. However, slight differences may arise due to rounding errors, interpolation methods in the tables, or numerical precision differences in software outputs. These differences should generally be minor and are worth examining if discovered.

Transcript

00:01 In problem number 70, we're going to use a computer or a calculator.

00:06 I'm going to show you how to use the ti graphing calculator, the texas instruments, and we need to find the probability that one randomly selected value of x from a normal distribution with a mean of 584 .2 and a standard deviation of 37 .3 will have a value that is less than 525 for part a.

00:51 So this is part a.

00:53 So when it comes time to use your calculator, on the graphing calculator, and i'll pull my graphing calculator in, you're going to hit the second key.

01:12 So we're going to hit the second key.

01:17 And then you're going to hit the vers key.

01:24 So i'm going to hit the vers key.

01:28 Try again.

01:29 Second key, then the variables key.

01:31 And then we're going to use number two.

01:38 And that's your normal cdf.

01:42 So it's a cumulative density function.

01:44 So when we use normal cdf, notice below it, it talks about lower bound, upper bound.

01:57 So we're going to bring it to our home screen.

02:01 So we're going to have our lower boundary.

02:05 We're going to have our upper boundary.

02:09 We're then going to have our average and our standard deviation.

02:16 So in this particular problem, our lower bound is infinitely into the left tail.

02:24 So if i think about what the bell would look like, here's 584 .2.

02:33 We want to be less than 525.

02:39 So we're going all the way into this tail.

02:43 So in order to do that, our left bound is going to have to be a really big negative number.

02:50 So what we're going to type in is we're going to type in negative 1 times 10 to the 9.

02:57 99th, which is a super, super, super negative number.

03:02 So we'll have negative 1 times 10 raised to the 99th power.

03:11 Then we're going to do a comma, and we're going to do our upper boundary.

03:15 So what we have just put in is a number all the way back here.

03:20 So our upper boundary is going to be the 525.

03:30 So we're going to type in 525.

03:34 Follow it up with a comma.

03:36 Then we have to go with our average, and our average was the 584 .2.

03:45 So we'll type in the 584 .2, and we'll follow it up with our standard deviation, and our standard deviation was the 37 .3.

04:00 So we are getting an answer for our problem, the probability, that x is less than 500, 125 is 0 .0562.

04:22 So i'm going to tuck that away.

04:24 I'm going to finish up what was hidden there.

04:27 Remember that last argument for the cumulative density function is going to be that standard deviation.

04:36 Okay, so for part a, what's the probability that a randomly selected x value is less than 525 would be 0 .056.

04:47 2.

04:48 And we used our graphing calculator.

04:51 And again, i used the t .i.

04:53 Graffing calculator.

04:55 The 84 or the 83 would work.

05:04 All right.

05:05 So let's go to part b.

05:13 In part b, you are asked to find the probability that x lies between 525 and 590.

05:20 So the probability that x is between 525 and 590.

05:29 And again, we're going to use our normal probability, normal cdf, which means cumulative density function.

05:41 And remember, it goes lower bound, upper bound, average, standard deviation.

05:52 So we'll use normal cdf.

05:55 In this case, our lower bound is the 525.

06:03 Our upper bound is the 590.

06:06 The average was the 584 .2 and the standard deviation was 37 .3.

06:18 So i'm going to slide my calculator in and i'm going to do the second vairs.

06:25 Select number 2.

06:27 And this time lower is 525.

06:30 Separate them by the 1 .5.

06:31 Comma, 590 separated by the comma, 584 .2, separate by the comma, and 37 .3.

06:45 So for the answer here, we get .5055.

06:58 And i'm going to have to move my calculator.

07:01 So let's go to part c.

07:04 Part c is a probability of at least 590.

07:13 Probability well at least 590 means it's going to be greater than 590 so if you think about what the bell would look like then we're talking about going into this tail so our upper boundary is going to have to be a super large number so we're going to set it up by saying normal cdf again it's lower boundary upper boundary average and standard deviation...

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