00:01
So we have an expensive watch that is expected to last five years.
00:06
So the expected lifetime is the same as its mean lifetime.
00:16
And we are going to suppose the life of the battery has a standard deviation of 0 .7.
00:28
And we're going to suppose that it is normally distributed.
00:33
So that means we could draw the normal curve.
00:37
Or the bell shape curve.
00:41
We want to, in part a, determine the probability that a watches battery will last longer than 5 .5 years.
00:54
So translating that, we're looking for the probability that x is greater than 5 .5.
01:01
So if i were to redraw that bell -shaped curve and place the mean in the center, and 5 .5 would be to the right, so the probability we're looking for is going to be equated to the area of the bell shape curve to the right of 5 .5.
01:25
So in order to calculate this probability, we're first going to need to find the z score associated with 5 .5.
01:34
Keeping in mind that the z score associated with the mean is always zero, so we know that this z score is definitely going to be greater.
01:44
Than zero.
01:45
It's going to be positive.
01:46
And to calculate a z score, we could use the formula x minus mu over sigma.
01:55
So that would be 5 .5 minus the mean of five divided by the standard deviation of 0 .7, which is going to yield a z score of 0 .714 -285 -7143.
02:13
And because when we use our z score normal distribution tables, they are usually out to two decimal places, we're going to say it's .71, or actually we're going to say it's 0 .71.
02:31
So the next thing we want to do is we want to go to your standard normal table, and that is usually found in the back of your textbook.
02:40
And what you'll do is you'll look for the ones or units place and the tenths place down the side of the table and across the top you'll be looking for the hundredth place and where those two line up give you an area of 0 .76115.
02:59
Now your table in the back of your textbook might only go to three decimal places.
03:03
It might go to four.
03:05
The one that i happen to use is going to five decimal places.
03:09
So that value describes the area that extends into the left tail of the curve.
03:18
So the area over here is 0 .76115.
03:27
So for us to determine the area on the other side, the area of our interest here, we have to keep in mind that the two parts of this bell -shaped curve have to add up to one.
03:39
So if 0 .76115 plus the shaded area equals 1, then the shaded area is the same as 1 minus 0 .76115.
03:54
And when we perform that subtraction, we are going to get an area or a probability of 0 .2385.
04:05
And your direction said round to four decimal places.
04:10
So we're going to say that that's 0 .2389.
04:15
Now part b works the same way.
04:19
This time we want to determine what is the probability that the battery will last more than 4 .15 years.
04:28
So that would be probability that x is greater than 4 .15.
04:33
So again, we're going to draw that bell -shaped curve.
04:39
We'll place our mean in the center.
04:42
And 4 .15 is going to be to the left of center this time.
04:49
And again, we are interested in being greater, so we're interested in this area right here...