A major television manufacturer has determined that its 19...

A major television manufacturer has determined that its 19-inch color TV picture tubes have a mean service life that can be modeled by a normal distribution with a mean of six years and a standard deviation of one-half year. a.) what probability can you assign to service lives of at least (1) Five years? (2) Six years? (3) Seven and one-half years? b.) If the manufacturer offers service contracts of four years on these picture tubes, what percentage can be expected to fail from wear-out during the service period?

Transcript

00:01 For this question we are told that the service life of tv tubes is normally distributed with a mean of 6 years and a standard deviation of 0 .5 years.

00:14 And for part a we are asked for the probability of service lives of at least 5 years, 6 years, and 7 .5 years.

00:24 So for 5 years we want to find the probability that x is greater than or equal to 5.

00:30 So this graph represents the normal distribution for the tv picture tubes.

00:35 Lives we have a mean of 6 in the center, standard deviation of 5, and 5 would be approximately here.

00:46 Probability that x is greater than 5 is equal to the area under the curve to the right of 5.

00:52 So that's the area of this blue shaded region.

00:57 Because the total area under a density curve is 1, the area under the curve to the right of 5 is equal to 1 minus the area under the curve to the left of 5.

01:04 So this can be expressed as 1 minus the probability that x is at most 5.

01:11 We can use the standard normal table to solve this probability, but we first must standardize the random variable according to this formula.

01:23 And if we do so we have 1 minus the probability that z is less than or equal to minus 2.

01:33 Now we can look up z equals minus 2 in the standard normal table.

01:38 And we see that that corresponds to a cumulative probability of 0 .0228.

01:49 This probability comes out to 0 .9772.

01:55 So that means the probability of a service life of at least 5 years is 0 .9772.

02:04 Do the same thing for 6 years.

02:19 For 6 years it's a pretty intuitive answer because the normal distribution is symmetrical about its mean.

02:28 And the mean is 6, so exactly half of the distribution lies to the right of 6 years.

02:33 So we can state right away that this probability is 1 half...

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