Given that x is a normally distributed random variable with a mean of 28 and a standard deviatio

Given that x is a normally distributed random variable with...

Key Concepts

Z-Score Standardization

Z-score standardization is a process used to convert a normally distributed random variable into a standard normal variable by subtracting the mean and dividing by the standard deviation. This transformation allows one to use the standard normal distribution (with mean 0 and standard deviation 1) to compute probabilities and compare values from different normal distributions, making it a critical tool for probability calculations.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a random variable gives the probability that the variable takes a value less than or equal to a specific value. In the context of the normal distribution, the CDF is used to find probabilities for ranges of values by calculating the difference between CDF values at the endpoints, and it is essential for answering probability questions and working with standardized scores.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve and is defined by two parameters: the mean (which determines the location of the center) and the standard deviation (which measures the spread or dispersion). It is commonly used in statistics because many natural and measurement processes tend to follow this distribution, making it a central concept in probability theory and inferential statistics.

Transcript

00:01 All right, in problem number 48, there are six parts, and all six parts deal with the fact that the average is 28, and the standard deviation will be 7.

00:12 So in part a, you're asked to determine what is the probability that x is less than 28.

00:21 We also knew that the variable was normally distributed.

00:27 28, since it was the average, would be right at the center of the bell, and the probability, that we pick an x value less than 28, which would represent half the bell, which would be 0 .5.

00:42 For part b, we're looking to do the probability that x takes on a value between 28 and 38.

00:52 So again, i would highly recommend drawing that bell -shaped curve.

00:57 Again, 28 is in the center, and 38 would be over here.

01:02 So we're going to utilize z scores in order to solve this problem.

01:07 And the formula for z scores is that z equals x minus mu divided by sigma.

01:17 So the z score associated with 28 would be 28 minus 28 all over 7 or 0, which we should have known that just because the center of the center of the, the bell does represent a z score of zero, and the z score associated with 38 would be found by doing 38 minus 28 divided by 7, and you do get an answer of 1 .43.

01:53 So when we posed the question, what's the probability of being between 28 and 38? it's no different than the question that says what's the probability that z is between 0, and 1 .43.

02:12 You can then calculate the probability that z is less than 1 .43, and from that, subtract the probability that z is less than 0.

02:24 You would then go to your standard normal table in the back of your textbook.

02:29 You would get a value of 0 .9 -236 for the probability that z is to the left of positive 1 .433.

02:39 And the probability that z is less than 0 would be 0 .5 .00 for an answer for part b to be 0 .4, 2, 3, 6.

02:56 In part c, making it a little bit more difficult as we go along, in part c, we're asking what's the probability that x falls between 24 and 40.

03:17 Again i highly recommend drawing that bell -shaped curve, putting 28 in the center because that is our population mean, and we want to be between 24 and 40.

03:30 So we're going to find the z score associated with each, and the z score for 24, we'll do z equals 24 minus 28, divide by the standard deviation of 7.

03:48 And we get negative 0 .57.

03:55 And then the z score associated with 40 would be 40 minus 28 divide by 7, which yields a value of 1 .71.

04:10 So when we say, what's the probability that x is between 24 and 40, it's no different than saying what's the probability that z is between negative 0 .57, and positive 1 .71.

04:29 We would then find the probability that z is less than 1 .71, and subtract from that the probability that z is less than negative 0 .57.

04:43 If you use the table in the back of your book, the standard normal table, you will find the probability that z is less than 1 .71 to be 0 .9564.

04:56 And the probability that z is less than negative .57 you will find to be .284 .3.

05:07 Thus, the answer to part c would be .6721.

05:15 In part d, we are looking to calculate the probability that x falls between 30 and 45.

05:28 So here's our picture.

05:38 We have 28 here.

05:41 So 30 and 45.

05:46 So because both of those values are above 28, both of those z scores should be positive values.

05:52 So we'll find the z score associated with 30 by doing 30 minus 28 divide by 7.

06:04 And when we do that, we will get a value of approximately 0 .2...

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