A normal distribution has a mean of 50 and a standard deviation of 4 a. Compute the probability

A normal distribution has a mean of 50 and a standard...

Key Concepts

Normal Distribution

A normal distribution is a probability distribution that is symmetric about its mean, with its shape often described as a bell curve. It is characterized by two parameters: the mean, which determines the center of the distribution, and the standard deviation, which determines the spread or dispersion of the data. This distribution is widely used in statistics due to the Central Limit Theorem, which states that the sum of many independent random variables tends toward a normal distribution, regardless of the original distribution of the variables.

Z-Score Transformation

The Z-score transformation converts a value from a normal distribution into a standardized value, measured in terms of standard deviations from the mean. By subtracting the mean from the value and then dividing by the standard deviation, this transformation helps in comparing scores from different normal distributions and in utilizing the standard normal distribution tables or computational tools to compute probabilities.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a normal distribution gives the probability that a normally distributed variable will take a value less than or equal to a given threshold. When using the CDF in problems involving normal distributions, one typically computes the Z-score first and then finds the corresponding probability from a standard normal table or calculator. The CDF is essential for calculating probabilities for ranges and tails of the distribution.

Transcript

00:01 We're given a normal distribution, and we know that the mean of that normal distribution is 50 with a standard deviation of 4.

00:09 So let's mark a few standard deviations out.

00:14 And let's go back in this direction as well.

00:17 So we'll mark two standard deviations, both above and below the mean.

00:21 And in part a, we want to determine what is the probability of a value being selected between 44 and 55.

00:31 And so 44, we can see us right here.

00:35 And we can go through and use our formula for a z score is equal to the difference minus the mean, which in our case, why don't i just put the mean down? the mean divided by a standard deviation.

00:48 And there's our formula for determining the z score.

00:52 But we can see that that is going to be, this is a negative 1, this is a negative 2, and it's halfway between.

00:56 It's a negative 1 .5.

00:58 Or this difference is negative 6 divided by 4, so negative 1 .5.

01:05 And then our 55 is going to be right here, and that's going to end up being that difference is 5, and so this is going to be 1 .25.

01:15 And to find this probability, we look at the area that is below the upper number, and below 1 .25 corresponds with 0 .8944.

01:30 And then we subtract away the area below this, because this one is too big.

01:35 How much is it too big by? it's too big by this little lower tail.

01:40 And so look up negative 1 .5, and that corresponds with an area of 0 .0668.

01:47 So the 0 .8944, less the 0 .0668, gives us a probability or an area between of 0 .8276.

01:58 Part b, we want to find what is the likelihood that we have a value that is higher than 55.

02:07 And higher than 55 is a being up in this tail.

02:13 Always draw a picture that will help you out to see if your answers make sense...

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