00:01
For this problem, we have a study of time spent shopping in a supermarket, and the time spent shopping showed an approximately uniform distribution between 10 and 20 minutes.
00:18
So if we imagine our x -axis here with 10 and 20 labeled on it, then the probability density function just looks like a straight line, and you can cut it off like this, if you like, such that the area of this rectangle is 1, because it's a probability density function, so the total area under it has to add up to 1.
00:50
And now, using this distribution, we want to figure out a few things.
00:55
First, we want to find what the probability that a probability a, that the shopping time will be between 13 and 17 minutes.
01:04
So let x be a random variable shopping time for randomly a selected shopper.
01:14
Then we're interested for part a in calculating the probability that x is between 13 and 17 minutes.
01:24
So we're interested in this probability.
01:30
And the way we can calculate this, since it's a uniform distribution, any probability of an interval like this between 13 and 17 is just the ratio of the interval in question, the length of the interval in question, over the total length of the interval.
01:56
So what i mean is that we need to take on the numerator 17 minus 13 because this is the length of the length of the interval.
02:05
The interval here in the picture that goes from 13 to 17, say it looks like this.
02:13
So on the numerator, i've got this length of this blue line, 17 minus 13.
02:20
And on the denominator, well, that's going to be the length of the whole interval, which is going to be 20 minus 10.
02:28
So 17 minus 13 gives me 4 and 20 minus 10 gives me 10.
02:36
So this is 410, i don't know if we want it as a decimal or a fraction, but we can reduce this down at least to two -fifths, which is just equal to 0 .4 exactly.
02:49
So whichever way you like your answer, there it is.
02:53
Now for part b, what is the probability that the shopping time will be less than 16 minutes? well, this time we are interested if, say, 16 is here.
03:05
The interval of interest will be this guy, which i've drawn lower so it doesn't clutter up the picture.
03:13
And the probability we want to calculate is the probability that x is less than or equal to 16.
03:24
So again, we just use the same rule as before.
03:27
It's the ratio of lengths of intervals.
03:30
So the interval in question is from 10 to 16.
03:34
So it has length 16 minus 10.
03:37
And again, we take the length of the total interval in the denominator which is 20 minus 10 and that is 6 over 10 which is 3 5ths or 0 .6 as a decimal so there's our answer for part b make a little space here for part c now for part c this time we want to calculate the mean and standard deviation of the shopping time...