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In Exercises $21-24,$ use these results from the "I-Panel-THC" test for marijuana use, which is provided by the company Drug Test Success: Among 143 subjects with positive test results, there are 24 false positive results; among 157 negative results, there are 3 false negative results. (Hint: Construct a table similar to Table 4-1, which is included with the Chapter Problem.) Testing for Marijuana Use If one of the test subjects is randomly selected, find the probability that the subject did not use marijuana. Do you think that the result reflects the general population rate of subjects who do not use marijuana?

Yeah. All right guys, the first thing we need to do here is we do identify on are null and alternative hypothesis are no hypothesis. Excuse me being that the variations are going to be the same. And the turn of prosthesis is that they're going to have different variations as you can see right here. Okay, so the next thing I did to solve the problem was I opened a google she and I am putting a lot of the data. You'll notice all the data is and put it right there. Okay. And then, so what I'm gonna do from there, what I'm gonna do from there is this I'm gonna go ahead and calculate the variances for each. The first variance I'm gonna calculate using google sheets. So I'm gonna type equals var open parentheses and then just highlight everything I want. I know have an extra box in there. That's not a big deal. And there is my first variant. Second variants will be found the exact same way it looks like it already knows what I want, which is great. And there is our second variance. Alright, so from there I need to find my f statistic. Okay, so my F statistic is just going to be the greater variance divided by the smaller variance and there we go, there's my f statistic. Once I found my F statistic, I'm ready to find my P value. Actually. Sorry, scratch that. Once I find my F statistic, I'm almost ready to find my P value. But there's one more step in order to find a P value. Using google sheets with an F statistic. First need to calculate degrees of freedom. Okay, very easy to do in google sheets. So degrees of freedom is just all the data, the amount of data you have minus one. So how do you find that? Google sheets is equals count. Just highlight everything. I have noticed how I do have an extra box there. That's not a big deal to subtract one. And that I have 20 in this case 24 degrees of freedom. And in our second case again equal count Open parentheses, highlight everything and I do need to subtract one. And the other case have 15 15 degrees of freedom. Okay, so now I'm ready to find my P. Value. So my P value uh through here is going to be something called an F. Distribution or F. Dist. And then open parentheses. I need to put three values. Number one, my F statistic. The second value is going to be the number of entries from S one squared. I'm sorry the degrees of freedom from s one square in this case is 24. And my last and final value is the degrees of freedom from my S two squared. So that's right here. I close all that up and there is my P value. However, I'm not quite finished yet. Okay, that is a P value for a one tailed distribution. What do I mean by that? That's a good P value. If that was a greater than or less than sign, this is a not equal to sign. Okay. That's not equal to sign. Which means um uh that which means that I have actually had the wrong P value right now. Since this is for a one tailed distribution and I need a two tailed distribution because they're not equal sign, I need to take this value multiplied by two. So equals I click on the value multiply symbol in google sheets. Asterisk hit A two and I'm done. That is my P value. Notice that P value is just a touch above 0.05, meaning I'm going to fail to reject the no hypothesis. I'm gonna fail to reject. And all hypothesis

Try to look at this question. It is given that among 143 subjects with positive test results, So let over here, let this be positive people who are actually positive like this with people who are actually negative. And let this be the test side. So who have tested positive? And these are the people who have tested negative. All right, So it is given that among 143 subjects with positive test results with positive test results, which means that the some of this column is 1 43. Yeah, right. In this case, Okay. 24 are false. Positive results. 24 are false. Positive results. Which means this is 24. They're actually negative by the tested positive. So what is the remaining one? 43 minus 24. That it's 119 after that, Among 157 negative results. This is 1 57 among 1 57 negative results. There are three false negative results. There are three false negatives, false negatives. What is the meaning of false negatives? That means that they are actually positive, but they have tested negative. So this is three. This is 1 53. Okay. How many subjects are included in the study? How many? Some things are included in the study 300. How do I find this? The addition of these two 300 subjects are included in the study. What is the next question? The next one is how many subjects had two negative result. How many subjects to negative result 1 53 1 53. After that. What is the probability that a randomly selected subject had a true negative result? The probability that the randomly selected subject had a true negative result. It will be 1 53 upon 300 1 53 divided by 300. Now, what is going to be the answer? In this case, we need to use a calculator. And so just give me a moment. This is turning out to be 0.513 By the way, what we're doing wrong is 1 57 minus three is 1 54. So this is 1 54. So this answer is 1 54. This answer is 1 54 and hence this is also 1 54. Now, let's try to calculate this 1. 54.300 0.513 This answer is zero point 513 These are my ancestors. Thats all I needed, right? Yeah, these are my answers.

What we've got to find is what our true positives and negatives, they're simply going to subtract the totals From the false readings. So 143 -24 is 119 and 1 57 -3 is gonna be 154. This means that we had 300 people that were sampled for this. If we just add our totals for the positives and negatives and then for part B were asked of not using marijuana would be uh the results that turned out to be truly negative and then our false positive. So the number that didn't use marijuana would be 154 Plus 24 and we will get 178 Now for the probability that a randomly selected subject did not use marijuana, That would be our previously found number. 178. Yeah, We have 178th out of our 300 MLB 0.593 or 59.3% chance.

The probability is that the person actually used marijuana is going to be our true positive and then we're going to have our false negatives for the ones that are using marijuana and then we'll take the ones that the test selected as well. So this is gonna be um so what we're gonna be looking at is 119 Plus The three from the false negative. And then we're just gonna divide that, bar a total of 300 And this is going to be 0.407. So about 40.7%. And uh it's probably unlikely that the whole population, 40% of the population smokes pot just saying mm