 Hello there, we're actually now going to use technology to use hypothesis testing to test to claim about a population proportion. So some notation that we're going to be seeing here is n is the sample size, x is the number of successes, p hat is going to be the number of successes divided by the sample size, remember that's the sample proportion, p is the population proportion, and q is always 1 minus p. So to be able to use the method we're about to use, the sample observations must be a simple random sample, the conditions for binomial distribution are satisfied, the number of successes or observations would be 5, and the number of failures or non-successes should be at least 5. So remember p is the assumed proportion, not the sample proportion, p hat is the sample proportion. So we'll be using Google Sheets and we'll be focusing our attention in the data list tab and we'll use the one prop confidence interval p-value region. So we've been to this area before for confidence intervals, but now we're going to kind of tweak the information we type in to give us the p-value we need to run our hypothesis test. So based on a recent survey, 93% of computer owners believe they have antivirus programs installed on their computers. In a random sample of 500 scanned computers, it is found that 450 of them, or 90%, actually have antivirus software programs. Use the sample data from the scanned computers to test the claim, here we go, test the claim that 93% of computers have antivirus software. So we are going to use a level of significance of 0.05. Alright, so let's first write out our hypotheses. 93% of computers have the software, so that means p is equal to 0.93. Where would that go? The null or the alternative hypothesis? The null or the alternative? It contains equal to, so it would have to go with the null. Is the opposite of equal to? Not equal to. Your claim is whatever they mention in your question, so the claim is the null. So now Google Sheets is going to give us a p-value, which we will compare to 0.05, and we'll call it a day. So in Google Sheets, you are going to go to the data list tab, and you are going to go through the one proportion confidence interval p-value region, something like that. And you are actually going to type the following information. First is going to be your number of observations in your sample. So your number of observations is actually going to be your sample size times your sample proportion. So in a sample of 500 scanned computers, it is found that 90% of them, 0.9, had antivirus software. So you do 500 times 0.9, which is 450. They actually gave us that calculation in the question because they are nice, but sometimes they don't always give you that 450. Sometimes they give you sample size, they give you percentage, and they say good luck. All right, next is going to be our sample size, which is 500. Next is going to be what is our claimed value for the proportion? It's 0.93. So we'll put that for p sub 0 and within our Google Sheets spreadsheet. And then last but not least, we are going to actually put our sign for H1. So we need to write our sign for H1, which is going to be not equal to. So we need four pieces of information. We now have those four pieces of information. All right, so in Google Sheets spreadsheet, we will go to the Daedalus tab and to the one proportion confidence interval p value region. Our number of observations was 450 out of 500. Our claimed proportion value, p naught, was 0.93. And our sign for alternative hypothesis is going to be not equal to. That's all you have to do. Just four things. All right, and they give you, they give you your test statistic, which we really don't care about right now, but they also give you the p value. That's what we want to see. The p value rounded the four decimal places is going to be 0.0086, 0.0086. So p value is 0.00 from Google Sheets, 0.0086. All right, so let's compare. Let's compare that p value to alpha. Let's compare 0.0086 to 0.05. Alpha is our level of significance. Well, it looks like the p value is greater, sorry, less than 0.0086 is less than 0.05. Since we are below alpha, since the p value is less than alpha, alas, we can reject the null hypothesis. We can reject the null hypothesis. So basically we can take that null hypothesis and we can say, you're out of here, sorry, so long, see you later, which means all eyes are now pointing at the alternative as being the feasible option. But anyway, remember our conclusion has to be written in terms of the claim. So look at this. Our claim is the null hypothesis, so our claim contains equality and we rejected it. So there is evidence to warrant rejection of our claim. So we say there is sufficient evidence to warrant rejection of the claim that 93% of computers have antivirus software. That is your correct conclusion statement. So once again, we rejected the null and then our claim contained equality. Her claim was the null. Remember the table you can use. We rejected the null hypothesis, so we're either in row one or row two and then the claim includes equality. The claim is the null hypothesis. So we say there's sufficient evidence to warrant rejection of the claim. Let's talk about some M&M, shall we? Data from 100 M&M shows that 8% of them are brown. Use a .05 significance level to test the claim of the Mars candy company that the percentage of brown M&M's is equal to 13%. So we're dealing with a percentage, so that's a proportion P, and we're testing the claim of P equaling .13. We did something similar with confidence intervals, and usually a hypothesis test and confidence interval will agree with each other most of the time. There's a little bit of difference because of the one-tailedness sometimes of a hypothesis test because confidence intervals are built symmetrically with two tails in mind. All right, so let's talk about those hypotheses. Let's talk about H1, let's talk about H0. P equals .13, is that the null or the alternative? It has equal to, so that is totally the null. So the alternative would have to be not equal to. So the claim is the null hypothesis. So in Google Sheets, let's practice. What's the number of observations? N times P hat. In our sample, how many M&M's were brown? Well, out of 100, there were 8%, so you do 100 times .08. There were 8 M&M's that were brown. What's my sample size? 100. What's my claim proportion value? .13. And what is my H1 sign? What is my alternative hypothesis sign? Not equal to. I've seen four things, and that's what's going to spit out, your P value. So this is the nice way to do hypothesis testing. All right, so let's type in those four things in the Google Sheets. So I had 8 out of 100 M&M's that were brown. My claim proportion is going to be .13. And then my sign is still not equal to for my alternative hypothesis. So what's your P value? Well, your P value is actually going to be .137. The 0 goes up to a 1, because there's a 5 or higher in the following place value. .1371. .1371. .371. So let's do our comparison. Let's compare .1371 to our significance level. They didn't give us 1, so you just assume it's .05. That's always the default significance level. So I'm comparing the P value to the significance level alpha, and clearly the P value is totally greater than alpha. Because I'm not under the limbo bar, because I'm greater than .05, I fail to reject the null. I fail to reject H0, so fail to reject the null hypothesis. That means all eyes are now looking at the null hypothesis as being a reasonable conclusion here. We always write our conclusion in terms of the claim. So my claim is the null hypothesis, because it contains equal to, and I'm failing to reject it. As a result, we say that there's not sufficient evidence to warrant rejection of the claim that the percentage of brown M&Ms is equal to 13%. So remember the key here is to use the table. In a research poll, 1,002 adults were asked if they felt vulnerable to identity theft. 531 of the 1,002 said yes. Use a .05 significance level to test the claim that the majority of adults feel vulnerable to identity theft. So what does that mean when we say majority? That means more than half, right? Greater than .5? So think about what your hypothesis would look like, H0 and H1. So P is greater than .5, the proportion of adults that feel vulnerable to identity theft, would that be the null or alternative? Well, it doesn't contain equal to, so it goes in the alternative. So for the null, we can just flat output equal to .5. That's always the default equal to .5. You could have also done less than or equal to .5. These are both suitable answers for the null hypothesis. Now that that's said, let's figure out what the P value of this exciting test is going to be. So in Google Sheets, how many observations did we have? How many people said they felt vulnerable? Well, it tells you 531. Out of how many, what's the sample size? 1,002. What's our claimed proportion or the value that's used in our hypothesis? .5. And what's the sign of H1? H1 sign is actually going to equal or be greater than. So Google Sheets spreadsheet, let's do some magic for us please. So in the spreadsheet, I know that I had 531 people out of 1,002 that said they felt vulnerable or claimed proportion value used in the hypothesis would be 0.5. And then our sign is going to be greater than. So look at that nice P value right there, .029, .029 is our P value, .029. Let's compare that P value to alpha. Let's compare .029 to .05 and see what happens. In this case, my P value of .029 is actually going to be less than my significance level of .05, my alpha. So as a result, I'm under that limbo bar so I can reject the null hypothesis. Rejected. It'd be more fun if we had a stamp that said rejected on it and you can just go bam, rejected. Alright, so that means the null hypothesis is out of here and all eyes are pointing to the claim, all eyes are pointing to the alternative hypothesis. So we rejected the null hypothesis and our claim is the alternative meaning it does not contain equality. So our statement is as follows. There is sufficient evidence to support the claim that the majority of adults feel vulnerable to identity theft. There is sufficient evidence to support the claim that the majority of adults feel vulnerable to identity theft. Remember how they get the structure of that sentence from our table. We rejected the null hypothesis, so we're either in row one or two, and the claim does not include equality, meaning the claim belongs to H1. So that puts us in row one. That's why we say there is sufficient evidence to support the claim that blah blah blah blah. So anyway, that's how to use Google Sheets to do most of the work for you when testing a claim about a proportion. I hope you enjoyed. Thanks for watching.