The lifetimes of a particular brand of DVD player are normally distributed with a mean of eight

The lifetimes of a particular brand of DVD player are...

Key Concepts

Probability Calculations

Probability calculations for normally distributed variables involve determining the cumulative area under the probability density function between specified limits. This often requires the use of statistical tables or computational tools to find the probability associated with a given z-score range.

Normal Distribution

A 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 characterized by its mean (?) and standard deviation (?), and is used to model many natural phenomena.

Z-Scores

A z-score is a statistical measurement that describes a value's position relative to the mean of a group of values, measured in terms of standard deviations. This conversion allows for the standardization of any normal distribution, making it possible to utilize standard normal distribution tables or software to compute probabilities.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. By converting values to z-scores, any normal distribution can be related to the standard normal distribution, which simplifies the process of finding probabilities.

Area Under the Curve

In the context of the normal distribution, the probability of an event corresponds to the area under the curve of the distribution over a specific range of values. Calculating the area under the curve between two z-scores provides the probability that a value falls within that interval.

Transcript

00:01 In problem 30, a dvd player lifetime are normally distributed with a mean of 8 years, mean equals 8 years, and the standard deviation of 6 months, standard deviation 6 months, or we can say it as 0 .5 years.

00:23 We want to find the following probabilities where x denote the lifetimes in years.

00:30 For party a, the probability for x, between 7 years and 9 years because these values follows a normal distribution we can use z score and use the values of the standard normal distribution tables we can calculate the z score as it equals x minus mu divided by sigma then we can switch from x to z the probability of z1 between z between z1 and z2 we can calculate z1 using x1 this x1 equals 7 minus mu which is 8 years divided by sigma equals minus 2 and we can get z 2 which equals x2 which equals x2 minus 2 equals 9 minus a divided by half equals 2 by sketching the normal by sketching the probability for the normal distribution we have the bell shape symmetric about x equals 8 years and we have here 7 and we have here 9 and this is the probability we want to calculate we have the normal distribution tables that gives us the value to the left the z score the z score is minus 2 and 2 by entering the tables by minus 2 we have here the value of minus 2 z score of minus 2 and here 0 gives this area 0 0 .2 0 .2 0 .2 0 .2 0275 this is the area to the left of 7 and of course it's the area to the left of 9 because it's symmetric then we can get this probability as 1 which is the total probability minus 2 multiplied by the two tails and these gives 4 .9545 for 4 .545 for part b we want the probability x between 8 .5 and 9 .5 we can switch again it to the probability for z being between z z3 and z4 we can calculate z3 as previous it's 8 .5 minus 8 divided by 0 .5 which is 1.

04:34 And we can get z2 or z4 as 9 .5 minus 8 divided by 0 .5 which is 3 then we want to calculate the probability for being between 8 .5 and here is 9 .5 you want to calculate this area and we have the z score for 8 .5 is 1 and 9 .5 is 1 .5 is 1.

05:08 3 then we can get the area to the left of 1 this area by entering the table by the value of 1 the value of 1 is here then this is the area to the left of 1 and we want to calculate the area to the left of 3 the area to the left of 3 is here 0 .99865 then to the get the shaded area here we can calculate the area to the left of 3 minus the area to the left of 1 gives us this area then the probability for x being between 8 .5 and 9 .5 equals 0 .99865 minus 0 .844 equals 0 .844 equals 0 .19865 equals 0 .844 equals 0 .19165 minus 0 .844 equals 0...

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