00:01
For this question, we consider a random variable x that follows a normal distribution with a mean of 60 and a variance of 64.
00:10
Let's first calculate the standard deviation.
00:14
It's the square root of the variance.
00:16
Squared of 64 is 8.
00:20
And for part a, we were asked for the probability that x is greater than 70.
00:25
So mathematically, we read it like this.
00:31
So if this graph represents the normal distribution of x, the mean of 60 is exactly in the center, standard deviation is 8, 70 is somewhere around here, and the probability that x is greater than 70 is equal to the area under the curve and to the right of 70.
00:55
Since the total area under any density curve is 1, that means that the area under the curve to the right of 70 is equal to 1 minus the area under the curve to the left of 70.
01:07
That means that we can express this probability as 1 minus, the probability that x is at most 70.
01:16
And we can solve this probability using standard normal tables.
01:20
Do so, we first have to standardize the random variable according to this formula.
01:28
And if we do that, we have 1 minus the probability that z is less than or equal to 1 .25.
01:41
And so if we look up z equals 1 .25 in the standard normal table, that course corresponds to a cumulative probability of 0 .8944, probability comes out to 0 .1056.
02:08
And for part b, we want the probability that x is greater than 45 and less than 72.
02:26
So suppose 72 is here, 45 somewhere around here.
02:33
The probability that x is between 45 and 72 is equal to the area under the curve between these two valleys.
02:42
It corresponds to the area of this blue -shaded region.
02:46
So this can be expressed as the probability that x is less than 72 minus, i'll write it on the second line, minus the probability that it's at most 45.
03:08
To solve this we can once again use the standard normal table just as we did in part a.
03:13
But let's use excel instead just for a change.
03:16
So we have two terms here.
03:18
We have the probability that x is less than 72, subtract the probability that x is less are equal to 45.
03:24
So if we go to excel, we start a computation with an equal sign.
03:28
We want to use the normal distribution function that's highlighted here in blue, so we select that.
03:33
For the first argument we enter 72, the first argument for the first term.
03:39
The mean is 60, standard deviation is 8.
03:41
For the cumulative argument we enter true because we want the probability that x is anything up to 72.
03:47
Then we subtract the second term.
03:48
Same function, except the first argument, is 45, and the other arguments are the same.
03:58
It gives us a probability of 0 .9028.
04:11
And for c we were asked, probability that x is less than 65 is what? all of this using excel as well.
04:26
So equals normal distribution function, they enter 65, then the mean standard deviation.
04:33
Simulative argument is true once again because we want the probability.
04:36
That x is anything less than 65, we hit enter, and we get a cumulative probability of .7340.
04:54
For part d, the probability is 0 .1 that x is greater than what value.
05:01
So the probability of the random variable being greater than some value is .1...