00:01
For this question, we are told that certain boards from a lumber company are normally distributed in length, have a mean of 8 feet, and a standard deviation of 0 .003 feet.
00:14
For part a, we are asked for the probability of finding a randomly chosen board that is less than 7 .995 feet.
00:23
So this is the probability that x is smaller than 7 .995.
00:28
So this graph represents the normal distribution for board length for the 8 foot boards, have a mean of 8 exactly in the center, standard deviation of 0 .003, 7 .95 is somewhere over here, and the probability that x is less than 7 .995 is equal to the area under the curve and to the left of 7 .995.
00:59
This probability can be solved using software such as excel.
01:03
In excel, we would start a computation with an equal sign.
01:07
We want to use the normal distribution function, so we select that.
01:10
For the first argument, enter 7 .995, then the mean and the standard deviation.
01:17
And for the cumulative argument, we set it to true because we want the probability that x is anything less than 7 .995.
01:24
We hit enter, and we get a probability of about 0 .0478.
01:37
So less than 7 .995 feet is a board that is too short.
01:45
For question b, we're asked if a customer purchase 120 of such boards, what is the probability that more than five are too short? so the probability of being too short, as calculated in part a is 0 .0478.
02:05
Customer is purchasing 120 boards.
02:10
Let's say the random variable x is the number of boards that are too short.
02:25
So here each of the 120 boards can be thought of as bernoulli trials with two outcomes of interest either too short or not.
02:34
And if we can assume that the boards, lengths are independent of each other, then the number of boards that are too short and a fixed number of independent or newly trials is a binomial random variable.
02:46
So here the number of boards that are too short is a binomial based on probability success 0 .478 and 120 trials.
02:58
Actually in part a we use the random variable x, so let's call this 1y instead.
03:02
So the question is asking, what is the probability that y is greater than 5? this is equal to 1 minus the probability that y is at most 5.
03:22
We can solve this in excel.
03:25
So start with equals 1 minus, and we want to use the binomial distribution function.
03:31
Enter 5 for the number of successes, 120 trials, probability of success, 0 .0478.
03:42
For the cumulative argument it's true because we want the probability of any number of of boards too short from 0 up to 5.
03:50
We hit enter and we get a probability of 0 .5141.
04:02
For part c, we're told that the purchaser is inspecting the boards before purchasing.
04:09
We are asked for the probability that the customer inspects four boards or less before observing the first board that is too short.
04:19
So let's ow the number of boards inspected to find the first board that is too short.
04:50
So again, each inspection of the board can be thought of as a bernoulli trial.
04:54
If each board is independent from the other boards, then the number of independent bernoulli trials until the first success is a geometric random variable.
05:05
So here we can say that w is a geometric with probability success .0478.
05:23
Now the question is asking for the probability that four boards are less are inspected before observing the first board that is too short.
05:33
So that means that w must be five or less, because w is the number inspected to find the first one that is too short.
05:41
So that's including the inspection for the one that is too short...