Length of pregnancies The length of human pregnancies from conception to birth varies according

Length of pregnancies The length of human pregnancies from...

Key Concepts

Normal Distribution

A continuous probability distribution that is symmetric about the mean, with its shape defined by the mean and standard deviation. It is widely used in statistics for representing real-valued random variables whose distributions are not known, and many natural phenomena can be approximated with it.

Z-Score

A measure that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. It is calculated by subtracting the mean from the value and then dividing by the standard deviation, facilitating comparison across different normal distributions.

Standardization

The process of converting a normally distributed random variable to a standard normal variable (with mean 0 and standard deviation 1) using the Z-score formula. This allows for the use of standard normal distribution tables regardless of the original distribution's parameters.

Percentiles

Statistical measures indicating the value below which a given percentage of observations in a group falls. In the context of a normal distribution, percentiles are used to understand the relative standing of a particular observation within the distribution.

Cumulative Distribution Function (CDF)

A function that shows the probability that a random variable takes a value less than or equal to a specific value. For a normal distribution, it is used to determine the percentile rank of a given observation.

Inverse Cumulative Distribution Function (Quantile Function)

A function that yields the value of the random variable corresponding to a given cumulative probability. It is used to find threshold values such as the cutoff for the longest 20% of observations in a normal distribution.

Transcript

00:02 The length of human pregnancies is normally distributed with a mean of 266 and a standard deviation of 16.

00:09 So i've already filled out my bell curve here.

00:12 And the first question we want to answer is, what percent of the pregnancies last less than 240 days? so here's right around 240 in my number line.

00:22 And i want to figure out essentially the area underneath the bell curve to the left of that.

00:28 So we need to calculate the z score, first of all, for 240.

00:31 We're going to do the observed value minus the mean divided by our standard deviation, and that's going to give us a z score of negative 1 .625.

00:51 And now we're going to reference the standard normal probability table in the back of your book, and we're going to look up a z score of negative 1 .63.

01:01 So negative 1 .63 gives us a probability of 0 .05.

01:08 So the chance of a human pregnancy lasting less than 240 days is about 5%.

01:23 Now our next question, we want to know what percent of pregnancies last between 240 and 270 days.

01:31 So i'm going to estimate 270 right here.

01:35 And we want to figure out this probability from 240 up to 270.

01:43 So we're going to need our z score at 270.

01:50 Now this is going to come out positive because 270 is higher than the mean.

01:54 And that comes out to be 0 .25.

01:59 So now we'll go to our standard normal probability table and look up 0 .25, which is 5987.

02:11 Now, keep in mind that .5987 is all the way to the left from 270.

02:21 So to find just this space, this area between 240 and 270, we're going to have to take our 0 .5987 and subtract the 0 .0516...

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