The average commute to work (one way) is 25 minutes...

Name: Problem 10, Chapter 6: Elementary Statistics a Step by Step Approach
Uploaded: 2020-06-05T19:09:41-08:00
Duration: 4 min 51 s
Channel: Kylie Patel
Description: The average commute to work (one way) is 25 minutes according to the 2005 American Community Survey. If we assume that commuting times are normally distributed and that the standard deviation is 6.1 minutes, what is the probability that a randomly selected commuter spends more than 30 minutes commuting one way? Less than 18 minutes?

The average commute to work (one way) is 25 minutes according to the 2005 American Community Survey. If we assume that commuting times are normally distributed and that the standard deviation is 6.1 minutes, what is the probability that a randomly selected commuter spends more than 30 minutes commuting one way? Less than 18 minutes?

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, displaying a characteristic bell-shaped curve. It is determined by two parameters, the mean (which indicates the center of the distribution) and the standard deviation (which indicates the spread). This distribution model is frequently used in statistics to represent real-valued random variables whose distributions are not known but that tend to cluster around the mean.

Z-Score Standardization

Z-score standardization is a technique used to transform a normal random variable into a standard normal variable with a mean of zero and a standard deviation of one. This process involves subtracting the mean and dividing by the standard deviation, allowing for the use of standard normal distribution tables to compute probabilities.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) for a continuous distribution, such as the normal distribution, gives the probability that the random variable is less than or equal to a specific value. In practice, the CDF is used to determine the area under the curve to the left of a given point, thereby indicating the likelihood of observing a value within that range.

Complement Rule for Probabilities

The complement rule is a fundamental principle in probability that states the probability of an event occurring is equal to one minus the probability of it not occurring. This is especially useful when calculating the probability of a value falling in the tail of a distribution, where one computes the probability of being below a threshold (or above) and then subtracts that from one to obtain the desired probability.

Transcript

00:02 For our problem, we're given that the average commute is 25 minutes and the standard deviation is 6 .1 minutes.

00:09 We first need to find the probability that a commute is greater than 30 minutes.

00:14 So we are going to start by changing our value of 30 minutes to its z score or z value using our formula, z equals the value minus the mean over or divided by the standard deviation.

00:30 So 30 minus 25 divided by 6 .1.

00:39 And this comes out to be approximately 0 .82.

00:45 We're going to round to two decimal places here because our standard normal distribution table goes to two decimal places for our z scores.

00:54 So we are finding the probability that the z score or z value is greater than 0 .82.

01:08 Now if we're to draw a sketch of what this would look like, we would draw our standard normal distribution.

01:17 The mean would be 0, and then we'd have one standard deviation away, two, standard deviations away, three, and you can also go in the other direction.

01:31 Our z value is going to be, is 0 .82, so we're going to sketch the approximate.

01:38 Here and we're looking for the probability that it's greater than that so the area to the right will give us the probability that it's greater than 0 .82.

01:50 Now our standard normal distribution table gives us the area under the curve to the left of a z score so since we're trying to find the area to the right we need to do one minus the value we're given in the table.

02:05 So if we go to our table and we look up the z score 0 .82, we're going to see the value .7939.

02:19 And we do one minus that value to give us the probability or area to the right.

02:24 So that's going to be 0 .2061.

02:30 Or we can write it as a percent...

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