The mean duration of television commercials on a given...

Name: Problem 32, Chapter 2: Introduction to Probability and Statistics
Uploaded: 2021-04-01T15:24:49-08:00
Duration: 4 min 9 s
Channel: Sheryl Ezze
Description: The mean duration of television commercials on a given network is 75 seconds, with a standard deviation of 20 seconds. Assume that durations are approximately normally distributed. a. What is the approximate probability that a commercial will last less than 35 seconds? b. What is the approximate probability that a commercial will last longer than 55 seconds?

The mean duration of television commercials on a given network is 75 seconds, with a standard deviation of 20 seconds. Assume that durations are approximately normally distributed. a. What is the approximate probability that a commercial will last less than 35 seconds? b. What is the approximate probability that a commercial will last longer than 55 seconds?

The mean duration of television commercials on a given network is 75 seconds, with a standard deviation of 20 seconds. Assume that durations are approximately normally distributed.
a. What is the approximate probability that a commercial will last less than 35 seconds?
b. What is the approximate probability that a commercial will last longer than 55 seconds?

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its bell-shaped curve. It is defined by two parameters, the mean and the standard deviation, and is widely used to model natural phenomena where values cluster around a central mean with symmetrical tails on either side.

Mean

The mean is the average value of a dataset and serves as the central point of a normal distribution. It helps determine the location of the distribution's peak and acts as a reference point for calculating deviations.

Standard Deviation

Standard deviation is a measure of the variability or spread of data points around the mean. In a normal distribution, a smaller standard deviation indicates that the data points are tightly clustered around the mean, while a larger standard deviation indicates a wider spread.

Z-Score

A z-score represents the number of standard deviations a particular value is from the mean of the distribution. It is computed by subtracting the mean from the value and then dividing by the standard deviation, allowing for the standardization of values and the use of z-tables to calculate probabilities.

Cumulative Distribution Function (CDF)

The cumulative distribution function for a normal distribution gives the probability that a random variable takes on a value less than or equal to a specific value. It is essential for determining the probability associated with intervals in a normal distribution.

Transcript

00:01 So we know we have an approximate normal distribution, so i like to draw my bell shape curve.

00:06 And we know that the mean amount of time that a commercial is supposed to last is 75 seconds.

00:12 And we know that the standard deviation, they're telling us, is 20 seconds.

00:17 And so if i just place where that is right here, just so i get an idea of what the distribution will look like, this will be at 95 seconds.

00:25 Here's two standard deviations higher, and that's going to be at 115 seconds.

00:29 Wow, that's a long commercial.

00:32 And subtract 20 seconds and we're down to 55 seconds.

00:36 And subtract 20 more seconds and we're down to 35 seconds.

00:40 So we want to know on part a, what's the likelihood that a commercial will last less than 35 seconds? now, we can actually figure out an approximation if we use that idea that this is the 68, 95, and then if we went out three standard deviations, which i didn't mark on here, that 99 .7%.

01:07 And we could say up 95 % in between here.

01:11 Looks like this is two standard deviations below.

01:14 And so this is going to be about 2 .5%.

01:17 So we could just say that.

01:18 But let's look at the process that you would go through if this wasn't just as easy.

01:22 We would need to convert that to a z value, which i can see already.

01:25 It's going to end up being negative two.

01:27 But the formula would be to take what we got, minus the mean.

01:31 And divided by the standard deviation.

01:34 And we can see that that is going to come out to be negative 40 divided by 20 is going to be negative 2.

01:41 And then we could look that up on our table to find out what that probability is...