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(xant {cpIndicate the correct decision ("reject" or 'do not reject" the null hypothesis), the reason for it, and write the appropriate conclusio...

Question

(xant {cpIndicate the correct decision ("reject" or 'do not reject" the null hypothesis), the reason for it, and write the appropriate conclusion @) Alpha (Enter an exact number as an integer, fraction , decimal )(ii) Decision: ~reject the null hypothesis do not reject the null hypothesis(i) Reason for decision: Since & < p-value we reject the null hypothesis Since a > p-value, we reject the null hypothesis Since a > p-value we do not reject the null hypothesis_

(xant {cp Indicate the correct decision ("reject" or 'do not reject" the null hypothesis), the reason for it, and write the appropriate conclusion @) Alpha (Enter an exact number as an integer, fraction , decimal ) (ii) Decision: ~reject the null hypothesis do not reject the null hypothesis (i) Reason for decision: Since & < p-value we reject the null hypothesis Since a > p-value, we reject the null hypothesis Since a > p-value we do not reject the null hypothesis_ Since & < p-value_ we do not reject the null hypothesis Conclusion: There is sufficient evidence to conclude that the machine needs to be recalibrated: 0 There is not sufficient evidence to conclude that the machine needs to be recalibrated.



Answers

A machine is considered to be operating in an acceptable manner if it produces $0.5 \%$ or fewer defective parts. It is not performing in an acceptable manner if more than $0.5 \%$ of its production is defective. The hypothesis $H_{o}: p=0.005$ is tested against the hypothesis $H_{a}: p>0.005$ by taking a random sample of 50 parts produced by the machine. The null hypothesis is rejected if two or more defective parts are found in the sample. Find the probability of the type I error.

Okay, so we know there are no hypothesis. H not. Is that P is equal to 0.1 on our alternative hypothesis. H A is that p is less than Europeans, your one which was given to us in the question. First, we can calculate the proportion just x over and which is 16 over to 1234. Sorry, which gives us European zero 13 Now you can calculate dizzy the statistic, which is proportion. This peaches are hypothesis off the square off Q p over, which will give us 1.6 if you plug in all of these values. Cute is right. Yeah, one minus piece. All right, so now you can find the P value. The P C is less than 1.6 on from the table that is equal to 0.855 for on. Since the P values getting a significant level, which is your 0.5 there is not sufficient evidence to claim support the claim that but scanners less than 1% of sales are overcharged

For this problem. We are given the following claim machines suspense amounts with standard deviation greater than 0.15 oz and we have the following information off to the side. So the first for this hypothesis tests the state or no hypothesis. So each not. We're talking about standard deviation. So sigma And it's gonna be our 1.5 oz are alternative since the claim is stating greater than it'll be sigma, Somebody greater than 0.15. So once we have our go on alternative hypothesis, let's find our testes cystic. Since we're talking about sanity, aviation, we're gonna be using the chi square distribution and the chi square testes cystic can be found by and before end minus one times as squared divided by sigma squared. Yeah, So this is going to be 27 -1. Yeah. Time 0.17 Squared Divide by 0.15 Squared. And the chi square tests should be approximately equal to 33 points 396. So I was the testes cystic, you can do the p value. There's two ways to do the people, you can either do since the sample size 27 we have a degrees of freedom. 26 But in the end -1. Since this degrees of freedom is relatively small, we can find it within the table or we can find the inequality arranged within the table. And since this is a right to test and the high score table, I'm using finds area right to the tip. Other critical values what we wanna do. We don't want to go down the degrees of freedom. Do we get to 26? We want to find the range where 33.396 is between the two numbers. So I found that is between 17 point 292. And it's gonna be less than 35 point 563. So now if we go all the way up we'll find that probably are the area to the right of the curve. So the 17.292 equivalent to 0.935.563 equivalent to 0.1. And now we have our P value. So RP values between 0.9 or 0.1 and 0.9. A more accurate where you can get your P value though is using software is like our you use the Arco p chi square which does it take the input of your chi square testes, cystic degrees of freedom and then lower dot L. Equals true or false. So the lord dot tails telling us whether it's a left tail test or right tail test. Since the right tail test, you want that to be false. If you plug in this information we should get to this p value is about 0.1 509. Which again agrees within the table method table inequality. So if you have the p value can make a conclusion by comparing your p value. What's your alpha. But you can also find your critical values. To find your critical value of critical values. Your again going to your chi square table is gonna be Since we're doing air to the right we just want our alpha. And so we are degrees of freedom. Remember that our alpha is 0.05. Our degrees of Freedom 26. So if we go to our high score table we can find were those two intersect and are critical values. 38 .885. All right. So, you can make conclusion with the critical values and your testes cystic. If you have a chi square distribution looking something like that, you're critical value. The states here Just 38.885. Okay, so you're critical value. We will reject any or don't have process. If it falls within this blue region, this is also called our rejection region. And if it falls anywhere within the non stated region, we will accept it. Which is also called our acceptance region. We see our test statistic as 33.396. So if you try to play set somewhere here that maybe belong around over there. Yeah. So that's archive square test. Yeah. And it falls within our acceptance region. So in this case for comparing test statistics and critical values we fail to reject If you want to compare with the P value and alpha. So a p value is greater than alpha. We failed to reject. H. Not. Yeah. and if P value is less than alpha, we reject H. Not again, we can look really quickly. We have AP value of 0.1509. And after a value of 0.05, our p value is greater than our alpha, so we failed to reject the null hypothesis. So in both cases we came to the same conclusion, which was we fail to reject H not.

Okay, So the objective off the study is the test and determine whether polygraph results are correct. Less than 80%. So less than 80%. Yes, the time. Okay. Let p the note the proportion of correct results. Let no hypothesis be defined as the polygraph results are correct. 80% time on that is given to be 0.80 Alternative parts is age. Not if each sub a sorry represents the probability that the polygraph results are correct less than 80% of the time. So it's just a probability less than 0.8. Okay, now we can find a sample proportion, which is just X over, and Andi were given those values 74 subsidize off 98 to give us a sample portion of 0.755 Okay, so now, to calculate the test statistics, Z portion of P over B Q over on que is equal to or more people. So now substituted values your from 75. That's your to screw. There's your 0.8 times sure to over 98 which will give us give 1.41 pictures are Z statistic value. Okay, enough last step an alternative hypothesis is left. Tales like calculation P value done like this piece she Dustin negative mhm, which is equal to 0133 five from the standard table. And since the level of significance Alfa is equal to 0.5 it could be observed. P value is greater, then the level of significance and therefore we can fail to reject. I don't know that's is so. It could be concluded that there is not sufficient evidence to support the claim that the polygraph results are correct less than 80% of time.

So the author purchased a slot machine and tested it by playing it 1197 times. He recorded his results in 10 10 different categories of outcomes, and he wants to test a claim that the actual outcomes agree with the expected frequencies. So we're going to run a hypothesis test to test this claim. So in order to do that, we will have to generate a hypothesis and we always generate are no hypothesis based on the observed data fitting or agreeing with the expected data. So therefore, our claim will be our null hypothesis, and our alternative hypothesis will be that the actual outcomes do not agree with the expected frequencies. And any time we want to compare or test whether observed data fits what is expected, we run a hi square goodness of fit test, which then we'd have to find a Chi Square test statistic. And there is a formula for this, and the author has already calculated it, and he or she has calculated out to be 8.185 so we then need to calculate a P value, and the P value is the probability that your chi square is greater than that test statistic. And in order to calculate that, I always recommend taking a look at what the graph would look like. So we are running a chi square goodness of fit test. So we're going to draw a chi square graph, which is a skewed right graph, and the shape of the graph is dependent on the degrees of freedom. And we find degrees of freedom by calculating K minus one, and K represents the number of categories that you're going to separate your data into. So if we go back up here, we were already told that the author separated his data into 10 categories. So therefore, R K value is 10, making our degrees of freedom to be nine. Now. Not only do the degrees of freedom give us the shape of the graph. It also gives us information about the average or the mean of the chi square distribution. So the mean is also going to be nine. Now, on our Chi square distribution graph, the mean can always be found slightly to the right of the peak. So we now know there's a nine right here on our chi square axis and for us to find the P value we're trying to determine what's the probability that Chi Square is greater than 8.185 So we're trying to find this shaded region. In order to find that shaded region, we will use our chi square cumulative density function from our calculator. And the cumulative density function requires you to input information about the lower boundary of the shaded area, the upper boundary of the shaded area as well as the degrees of freedom. So our lower boundary of this shaded region is the 8.185 and the upper boundary. Keep in mind that that curve continues infinitely to the right, and as we keep going, the tail is getting smaller and smaller. So envision a very large number out at the end of that tail. So we're going to use 10 to the 99th Power, and our degrees of freedom is nine. So I'm going to bring in the graphing calculator and show you where you can access the Chi Square cumulative density function, and we'll access it by hitting the second button and the bears button. And it's number eight on this menu. so are low. Boundary is 8.815 are upper boundary is 10 to the 99th Power and are degrees of freedom was nine. So our P value for this hypothesis test is going to be Let's let's change that. I noticed I have a typo, so we want to do 8.185 not 8.815 and then tend to the 99th in our degrees of freedom or nine. So we end up this time with the correct value of 515 six. So that is referring to this area that is shaded. So that's our P value. Now, when you run a hypothesis test, you can also find a critical value, and our critical chi square value can be found by looking in the table in the back of your textbook so you would go to the Chi Square distribution table. And it does records some information about degrees of freedom down the left side and your level of significance across the top. And we're running this hypothesis test at a level of significance or an Alfa of 0.5 So in your table across the top you're looking for 0.5 and down the side you're looking for nine and where those to meet up would be your critical chi square value and you get 16.919 So now it's time to make a decision based on the components of this test And to make your decision, we could do one of two things We can either use RPI value or we can use our chi squared critical value. If you decide you're going to use your p value, then you want to ask yourself if Alfa the level of significance is greater than the P value and if it is, then your decision will be to reject the null hypothesis. So our Alfa is 0.5 and our P value was 0.5156 So, as you can see, our Alfa is not greater than our P value. So our decision then will be fail to reject the null hypothesis. Now, the other way we can make this decision is by use using that critical chi square value. And again I like to draw a picture and what I I would like to do is to place that critical value that 16.919 on the curve. And by putting that on the curve, you have now separated your curve into two regions. The tail would be classified as your reject H O region, and the other side to the left of that critical value is going to be your fail to reject the null hypothesis region. And then you would go back up and look at your test statistic, and our test statistic was already pre calculated to be 8.185 So you'll come back here and on this chi square access. You'll also plot that value, and since 16 is here, then the test statistic would be approximately here at eight point 185 and that value is falling in the fail to reject the null hypothesis region. So our decision, no matter which method we use, is to fail to reject the null hypothesis. So therefore, let's go back to our hypotheses. If we fail to reject the null hypothesis, that means this is a viable option. It's not definitive. Our data is not conclusive that it definitely happens, but we can't throw it out either, So the data is inconclusive for us to rule that out, and if we don't rule that out, then we can't rule out this claim either. So therefore, our conclusion is that there is insufficient evidence to reject the no hypothesis. Thus, there is not enough evidence to reject the claim that the observed outcomes agree with the accepted frequencies. So since we're not rejecting the claim, we could then say slot machine appears to be functioning as expected and that concludes your hypothesis test.


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