In January 2003, the American worker spent an average of 77...

In January $2003,$ the American worker spent an average of 77 hours logged on to the Internet while at work (CNBC, March 15,2003 ). Assume the population mean is 77 hours, the times are normally distributed, and that the standard deviation is 20 hours. a. What is the probability that in January 2003 a randomly selected worker spent fewer than 50 hours logged on to the Internet? b. What percentage of workers spent more than 100 hours in January 2003 logged on to the Internet? c. $A$ person is classified as a heavy user if he or she is in the upper $20 \%$ of usage. In January $2003,$ how many hours did a worker have to be logged on to the Internet to be considered a heavy user?

In January $2003,$ the American worker spent an average of 77 hours logged on to the Internet while at work (CNBC, March 15,2003 ). Assume the population mean is 77 hours, the times are normally distributed, and that the standard deviation is 20 hours.
a. What is the probability that in January 2003 a randomly selected worker spent fewer than 50 hours logged on to the Internet?
b. What percentage of workers spent more than 100 hours in January 2003 logged on to the Internet?
c. $A$ person is classified as a heavy user if he or she is in the upper $20 \%$ of usage. In January $2003,$ how many hours did a worker have to be logged on to the Internet to be considered a heavy user?

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is commonly used to model real-world phenomena and is characterized by its mean and standard deviation, which determine the center and spread of the distribution, respectively.

Standard Deviation

Standard deviation is a measure of how spread out the values in a data set are around the mean. In the context of the normal distribution, it quantifies the amount of variation or dispersion and is critical in accurately determining probabilities associated with different intervals around the mean.

Z-Score Transformation

The z-score transformation is used to normalize a value from a normal distribution, converting it into a value from the standard normal distribution. This transformation simplifies the process of computing probabilities by allowing the use of standard normal tables or software, making it easier to compare outcomes from different normal distributions.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of the normal distribution gives the probability that a random variable will take a value less than or equal to a given point. It is a fundamental tool for calculating the likelihood of an event occurring within a specified range when dealing with normally distributed data.

Percentiles and Quantiles

Percentiles and quantiles are values that divide the data into intervals with equal probabilities. In the context of a normal distribution, they are used to determine thresholds, such as identifying heavy users by finding the value above which a specific percentage of the population falls. This process involves using the inverse of the cumulative distribution function to find the corresponding data value for a given probability.

Transcript

00:01 So this problem follows a normal distribution.

00:07 For the first question, let's find the probability that a worker selected at random spent fewer than 50 hours on the internet.

00:17 We can rewrite this as the probability that x is less than 50.

00:24 So we find the z score that corresponds to an x value of 50 and the formula is right here.

00:32 Therefore we have probability that z is less than the x value of 50.

00:37 Which is 50 minus the mean which is 77 divided by the standard deviation which is 20 we simplify this and we have the probability that z is less than negative 1 .35 now this is equal to the file the normal of negative 1 .35 so we locate negative 1 .35 on the standard z distribution chart we have negative 1 .3 and then under the column .05.

01:42 So this gives us a value of 0 .0885.

01:59 We move on to the next problem b.

02:06 What percentage of workers spent more than 100 hours in january 2003 logged onto the internet? this is equal to finding the probability that x is greater than 100.

02:23 So we find the z score that corresponds to an x value of 100 that is probability that z is greater than the x value 100 minus the mean which is 77 divided by the standard deviation which is 20.

02:47 Therefore we have a probability that z is greater than 1 .15.

03:01 For a normal distribution, the probabilities or the numbers on the chart are valid for the areas to the left of the z value...

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