00:02
Let's start the problem by recording the information that we know or that is given.
00:06
So we're buying a specific computer, and there's a claim being made by the representative that the average price of a computer is $1 ,249 with a standard deviation of $25.
00:25
They feel that the standard deviation might be larger because they've looked at websites, and found comparison prices.
00:36
So we are trying to test whether standard deviation might be larger.
00:55
And in doing so, they looked at eight different comparison prices.
01:05
And if we put that information into our calculator, so i'm going to put it in right now, if you notice when i hit stat, edit, i already have those prices in.
01:19
So i'm then going to hit stat, calculate, one variable statistics on list one.
01:30
And for those eight, there was an average of $1 ,246 .87 with a sample standard deviation of about 34 .29.
01:48
So because we are running a test centered around a standard deviation, we are going to use the test of a single variance.
02:06
And in order to run that test, we're going to have to use a kai square distribution, and your textbook has provided you with an outline or a framework in the back of the book called your kai square solution sheet.
02:19
So we're going to use that as our framework as well.
02:22
And part a is asking you to write your null hypothesis.
02:27
And with a test of a single variance, your null hypothesis is always a statement of equality.
02:34
And what we're trying to test is a statement of equality.
02:39
So we're going to say that our null hypothesis is sigma equals 25.
02:45
Part b on the framework is asking you to write an alternative hypothesis.
02:49
Well, since we are checking whether this might be larger, our alternative is going to be that it's greater than 25.
02:59
Now, because it is a single test of a variance, we can rewrite both of these in terms of variance and say sigma squared is equal to 25 squared, and sigma squared is greater than 25 squared.
03:18
Part c on the framework is asking you to define the degrees of freedom, and the degrees of freedom for a test of a single variance is found by taking the sample size and reducing it by one.
03:32
Since we had eight comparison prices, our sample size is eight, resulting in a degrees of freedom of seven.
03:43
Part d on the kais square solution sheet asks you to describe.
03:51
The type of distribution that you would use.
03:54
So because this is a test of a single variance, we are going to use the kai square distribution in running this test.
04:20
In part e, you need to find the kai square test statistic.
04:32
And in order to do so, you're applying the formula, the quantity n minus 1 times the sample variance divided by the population variance.
04:43
And since we had eight comparison prices, our sample size is 8, so we'll do 8 minus 1, our standard deviation from our sample was 34 .29.
04:59
So to turn a standard deviation into a variance, we have to square that.
05:05
And our null the whole hypothesis tells us that sigma equals 25, so our denominator will be 25 squared, resulting in a kai square test statistic of 13 .169.
05:24
Parts f and g, i like to tackle simultaneously.
05:32
For part f, you are trying to find the p value, and for part g, you are graphing, a kye square distribution.
05:50
And a kai square distribution is a curve that is skewed to the right, never is negative, so it starts at zero, and the horizontal axis is your kai square values.
06:08
So we now have to go back up to the hypotheses.
06:14
And when you are running a test of a single variance, you could run a right -tailed test, a left -tailed test or a two -tailed test.
06:39
And the nature of the test is determined by your alternative hypothesis.
06:46
And because our alternative hypothesis has a greater -than -symbol, we're performing a right -tailed test.
06:54
If that symbol was a less -than -symbol, it would have been left -tailed.
06:58
And if that symbol was a not equal to sign, it would have been two -tailed.
07:02
So we're running a right -tailed test, which now plays into determining the p -value.
07:10
To find the p -value, you're trying to find the probability that kai -square is greater than 13 .169, which is the test statistic we just found.
07:24
If it was a left -tail test, we would be doing kai -square being less than.
07:28
So because it's a right -tailed test, we are also using a greater -than -sq -square -old.
07:33
Symbol right there.
07:36
So let's look at the graph at the same time.
07:39
So the kye square distribution, the shape of its graph, is dependent on the degrees of freedom.
07:46
And we already stated that our degrees of freedom here was seven.
07:50
And the degrees of freedom is also indicative of the mean of that kai square distribution.
07:57
So we can find the mean slightly to the right of the peak...