00:01
Okay, so we've got that the time taken to complete a form is normally distributed with mean 100 minutes and standard deviation 30 minutes, so variance 30 squared.
00:14
Part a says what's the probability that a random person takes less than 85 minutes? we can just standardise that to say that's less, the standard normal variable is less than 85 minus the mean 100 over the standard deviation 30.
00:30
In this case, that's just the probability that z is less than 0 .5 and you can find that in your z tables to be 0 .30, sorry, minus 0 .5, 3085.
00:46
Part b says what's the probability that a random chosen person takes between 70 and 130 minutes? note that that means that x minus mu, mu i'm using for 100, the mean x minus 100 is between minus 30 and plus 30.
01:05
And so that's the same as the probability that the standard normal variable z is between plus or minus 30 over 30, which is plus or minus 1.
01:19
And again, in your z tables, you can find that that is equal to 0 .6826.
01:30
Part c, 5 % of all people take more than how many minutes to complete the form.
01:35
So that's the value x5, where the probability that x is bigger than x5 is 5%, 0 .05.
01:43
We can find in our z tables that the z value above which the probability is 0 .05 is 1 .64.
01:51
And if we standardise this, we find that it's the same as z being bigger than x5 minus 100 over 30.
01:59
If we follow this chain of equalities, then we can see that x5 is 1 .64 times 30 plus 100.
02:09
For part d, it says two people are chosen at random.
02:11
What's the probability that at least one of them takes more than an hour? well, let's first consider just the probability for any random person to take more than an hour.
02:19
Standardising, this is the probability that z is bigger than 60 minus 100 over 30, which is the probability that z is bigger than minus 1 .33.
02:34
And you can find in your z tables that that is equal to 0 .9082.
02:42
So in our situation, if we say, why is the number out of the two people chosen? who take more than an hour? then we're looking for the probability that y is bigger than or equal to 1, which is 1 minus the probability that y is equal to 0.
03:08
And y follows clearly a binomial distribution with two trials and probability 0 .9082 of success of someone taking over 60 minutes.
03:18
So this is just 1 minus 0 .9082 squared.
03:25
And we can find that to be 0 .992 to three decimal places.
03:32
Part e says, what about if four people are chosen and we want the probability that exactly two of them take over an hour? we're going to use the same symbol, but this time y is going to be out of four people chosen.
03:48
So we're going to have the same binomial distribution, but with n equals 4 instead of n equals 2.
03:52
And we want the probability that y is equal to 2, which is just given by the number of trials for choose the number of successes 2 times 0 .982, the probability of success to the number of successes times 1 minus that, which is 0 .0918 to the power of the number of failures, which is 2.
04:15
And we find that to be 0 .0417...