 Hello there, we are now going to test the claim about a mean using the concept of hypothesis testing So we have to be a little bit careful whenever we use Hypothesis testing for testing a mean because there's two different types of distributions that we would use for this So before I dive into that Please note that the following notation if you see in that sample size X bar sample mean and then the Greek letter Mu is the population mean Now when the population standard deviation sigma is not known We have to use the t distribution To find the p values and to perform the hypothesis test remember the t distribution is kind of like the standard normal distribution The two are practically identical the larger your sample sizes But the t distribution is a little bit more accurate for the sake of doing these sorts of hypothesis tests So you use the t distribution when the population standard deviation is not known When sigma is not known So to use the method we're about to use the sample has to be a simple random sample and either the population is normally Distributed meaning they'll shaped or the sample size is greater than 30 So we're going to practice using the compute tab the t distribution region of Google sheets to find p values So let's find the p value for a right tail test with n equals 12 and a test statistic of 2.573 So let's start with a picture Remember how to find the p value. What do we use to find the p value? We use a test statistic use that test statistic Define the p value I Have a right tail test so on your bell curve You're gonna shade the right region is the test statistic that separates this right hand region from the rest of the bell curve The area of that region is Going to be the p value the area of the tails is going to be the p value in this case. We just have a right tail It's a right tail test So you're going to jump into Google sheets and you'll go to the compute tab to the t distribution region All right, so This is for Google sheets now go to the compute tab Go to t distribution And You'll type the following in You'll type in mu equals zero You'll type in sigma equals one You'll also type in Degrees of freedom member t distribution uses a something called degrees of freedom. It's sample size minus one so 12 minus 1 is 11 and We'll type in your lower bound and your upper bound So you're trying to find area under the curve. So you're using x-axis data values So the lower bound is 2.573. That's where your shading starts and it never stops So six nines is the upper bound So now let's go to Google sheets Google sheets. You're hanging around the compute tab T distribution region zero for mu one for sigma my degrees of freedom in this case is going to be 11 12 minus 1 is 11 the lower bound is going to be 2.573 and upper bound is six nines And look at that you get a p-value of about To four decimal places to nine rounds up to a ten which makes the two go up to a three so point oh one three Point oh one three Point oh one three So ask us what the p-value is the p-value is point oh one three in The event you had a two-tailed test and you had a right tail in the left tail shaded You would double this Just as you can use the test statistic to find the p-value. You can also use alpha to find critical values of Our bell curbs for our test using the t-distribution as well in the compute tab You'll still keep mu as zero and sigma as one You'll still type your degrees of freedom and then you'll input the area to the left We'll do one example of finding the critical value Coming up here in a couple examples We don't spend much time on finding critical values just because the p-value method is so convenient Remember if you get a p-value you compare it to alpha jobs done for the critical value method You got to do some comparison with the test statistic and it's a little bit more work Now to get Google sheets to do everything for you you type in summary statistics Statistics that gives you p-value and then you draw your conclusion from that you'll use the data list tab You'll use the one variable confidence interval p-value t-distribution region and you'll type in x bar s and And then your population mean under consideration You'll also put your sign of your alternative hypothesis. So unlike a proportion we have to type in five pieces of information So Radiation emission measurements corresponding to a sample of 11 cell phones were taken use a point zero five level Significance to test the claim you see that word claim you should wake up Claim that cell phones have a mean radiation less than one So I have a Mean less than one is what they're talking about in my question So what would your hypothesis be? Mune less than one would go where to the null or the alternative. It does not contain equality So it's got to go to the alternative What's the opposite of less than Well, technically it's greater than or equal to but remember for the sake of convenience We like to just write equal to for the null always So we're a claim what they're talking about a question is the alternative hypothesis All right, so in Google sheets you're gonna have to type in a lot of information Well, it's not really that bad. It's better than doing everything by hand So in Google sheets, we're gonna be to the we're gonna go to the daddlest tab to the t-distribution region for confidence intervals and p-values You're gonna type X bar equals 0.938 you're gonna type your sample standard deviation as point four two three You're going to type your sample size As 11 You're going to type your value for mu mu not under consideration the value used in your hypotheses is one And then you have your sign for your alternative hypothesis, which is less than these are the five things you need to input And this is what's going to give us our p-value Let's type these things in the Google sheets now So Google sheets, we're gonna go to the daddlest tab and We're focused on the t-distribution region. We use t because we don't know sigma We don't know the population standard deviation if we did we would go down here and use the z-distribution region So we know that X bar is point nine three eight S is point four two three our sample size in this case is eleven our value from you We're looking at is one and then our alternative hypothesis sign is s then Look at your p-value there. That's a pretty high p-value point three one eight seven point three one eight seven So point three one eight seven Guess what we're gonna do with that We're gonna compare it the alpha You compare that p-value to alpha if they don't give you a level of significance We use point zero five they told us in the question they use point zero five anyway So how does our infamous p-value compare to alpha is definitely greater than? So since we're not under alpha we fail to reject the null hypothesis We fail to reject it All right, so as a result We're stuck on the null. We can't say anything about our all about our alternative, which is our claim So the actual statement is to say there is not Sufficient evidence to support the claim that cell phones have a mean radiation level that is less than one There's not sufficient evidence to support the claim So once again, we failed to reject the null and then our claim did not include equality So we failed to reject the null So we're either in rose three or four and our claim does not include equality That means it's the alternative hypothesis. There is not sufficient evidence to support the claim that da da da da Here's an example we did of confidence intervals, but now we'll do it with Hypothesis testing in a test of the effectiveness of garlic for lowering cholesterol We had 49 subjects that were treated with what raw garlic Cholesterol levels were measured before and after the treatment and the changes in their levels of cholesterol They had a mean of point four and a standard deviation of 21. So that's from our sample Test the claim that with garlic treatment the mean change in LDL cholesterol is greater than zero So we're looking at mu greater than zero And then what does the results suggest about the effectiveness? So we'll state the claim hypothesis test statistic p-value and final conclusion All right. So what I have here Is let's state the hypothesis Mu is greater than zero would go where? It does not contain equality. So it must go with the alternative hypothesis therefore the null is going to be less than or equal to zero or Better put as just equal to zero So greater than zero is what was talked about in the question. So that is my claim Let's look at what we would type in the Google sheets now Google sheets is perfect for calculating and giving you the statistic test statistic in the p-value Remember, this is all coming from the data list tab Because we're just typing in summary statistics It's important you use the t-distribution region because you do not know the population standard deviation So step one what is your x bar? Your sample means 0.4 step 2. What is s your sample standard deviations 21 step 3? What is your sample size? 49 step 4. What is your mean value under consideration? 0 and step 5. What is your alternative hypothesis sign? It's greater than we have what is called a right-tailed test So we're going to type those values into Google sheets now So let's do it remember we're focused on the t-distribution region. So we know that our mean is going to be 0.4 our standard deviation 21 Your sample size n is going to be 49 Zero is the value being taken into consideration and your null hypothesis or your alternative hypothesis sign is going to be greater than So look at your p-value another big p-value point four four seven two Point four four seven two point four four Seven two and we have to compare that to alpha you have to compare that to our significance level Which they said is point zero five So what happens when you compare the p-value to your significance level? Well the p-value is definitely bigger So since we're not under the limbo limbo bar we fail to reject The null hypothesis So null hypothesis Fail to reject Remember we can't say anything now about our alternative about our claim There's not sufficient evidence to support it and we say just that There's not sufficient evidence to support the claim that with the garlic treatment the mean change in LDL cholesterol is greater than zero So remember I failed to reject the null hypothesis And my claim does not include equality meaning it was the alternative hypothesis So we say there is not sufficient evidence supports to support the claim that da da da da All right, let's do a kind of a tricky situation here a Simple random sample of the weights of 19 green M&Ms has a mean of point eight six three five How you like that spent on things talking about the weight of M&Ms And it has a standard deviation of point oh five seven zero use a point zero five significance level to test the claim So we need to wake up and we see the word claim that the mean weight Of all green M&Ms is equal to point eight five three five Which is the mean weight required so that M&Ms have the weight printed on the package label Do green M&Ms appear to have weights consistent with the package label We'll run the whole hypothesis test. So basically we'll do everything. We just did Accept instead of using the p-value approach I'm going to put a spin on things and we'll talk about the critical value approach All right, so I'm going to do a different spin on this So we have a mean weight equal to point eight five three five When I write out my hypothesis, where would that go? That would go as the null hypothesis All right next order of business The opposite of that would be to say mu is not equal to point eight five three five The null hypothesis is my claim. It's what was mentioned in the question All right, so we're going to use google sheets We're going to go to the data list tab This data list time and we're going to type in What is our x bar? Point eight six three five. What is my s? What is my sample size? What is my mean value under consideration? And what is my sign? For the alternative hypothesis And that's what we're going to type in the google sheets So I'm going to type the following information point eight six three five for x bar zero point zero five seven zero for s A sample size of nineteen and a population mean of point eight five three five And our sign is going to be not equal to now you see your test statistic here point seven six Is going to be our test statistic and you see your p-value point four five four three So I'll just take them both to four decimal places and let's write them down All right, so I have a test statistic Which is going to equal point seven six four seven And then I have myself a p-value Which is going to be point four five four three now. There's two ways to run a hypothesis test There's two ways Option one you compared the p-value to add to your significance level and you make your conclusion that way option two Is to use The test statistic Critical value approach It's the less commonly used method, but I do want us to have a little bit of fun with that method So we're going to use that method here I have all my important information that I need to find the critical value and to use the critical value test statistic comparison approach Remember you use alpha To find the critical value So what I need to do now is I need to draw my bell curve So draw your bell curve shape And I'm going to shade two tails because this is a two tail test. Why is it a two tail test? That would be because you have not equal to for the alternative hypothesis Two tail tests have both a negative critical value in a positive one So how do you find it use alpha if alpha in this case is going to be point zero five How much does that leave for each of the tails you split alpha up into the two tails Each tail has an area of point zero two five All right, so now you're going to find a data value along the x-axis using area to the left So you'll go to google sheets. The sheet's the compute tab go to the The t distribution region Mu is zero seven was one degrees of freedom is always sample size minus one So 18 in this case an area to the left is point zero two five This will give us our negative critical value So let's see if we can find our negative critical value go to the compute tab zero One degrees of freedom is going to be 18 in our area to the left as we just discussed is going to be Point zero two five And you'll notice that your critical value is negative two point one or negative 2.10 However, you want to say it So our negative critical value is negative 2.10 because of symmetry About zero the positive critical values 2.10 All right, so we have our critical values. They ask for a positive critical value or a negative one See your negative 2.1 or 2.1 Now, where does this test statistic lie? Where does point seven six four seven lie? Does it lie in the shaded rejection regions or outside of them? Well point seven six four seven Actually lies outside of the rejection region. It is not contained within the rejection or critical regions at all It's in the center part the non shaded part of the bell curve. So We fail To reject the null hypothesis you would get the same thing even if you did the p-value approach You would fail to reject the null hypothesis so fail To reject so we're failing to reject our claim. So there's not sufficient evidence to reject our claim There is not sufficient evidence to warrant rejection of the claim that the mean weight of all green mms is equal to 0.8 535 So what I did was take an alternative spin on things here. I used instead of using the p-value alpha comparison approach I used the critical value test statistic comparison approach Since my test statistic was outside of the critical or rejection regions I Failed to reject the null hypothesis So it's up to you to decide. Do you like the p-value alpha comparison approach the hypothesis testing? Or do you like this critical value test statistic approach? I mean I have my own personal preference, but and I have a feeling I know what yours is too But anyway, I just wanted to give you a different take on how a hypothesis test can be Conducted other than that. I appreciate your time. Thanks for watching