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The sales representative for a manufacturer of a new machinepart claims that the new machine part will decrease cost permachine to at least 38$ per unit. A line man...

Question

The sales representative for a manufacturer of a new machinepart claims that the new machine part will decrease cost permachine to at least 38$ per unit. A line manager installs theproduct on 30 of the machines and finds that the average cost wasonly 42$ with a standard deviation of 10.2$. Test the claim ofsales representative when the probability of rejecting a true nullhypothesis is equal to .10? Interpret your findings. Show the stepsof hypothesis testing precisely. Use proper notation.

The sales representative for a manufacturer of a new machine part claims that the new machine part will decrease cost per machine to at least 38$ per unit. A line manager installs the product on 30 of the machines and finds that the average cost was only 42$ with a standard deviation of 10.2$. Test the claim of sales representative when the probability of rejecting a true null hypothesis is equal to .10? Interpret your findings. Show the steps of hypothesis testing precisely. Use proper notation.



Answers

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section. In a study of store checkout-scanners, 1234 items were checked for pricing accuracy; 20 checked items were found to be overcharges, and 1214 checked items were not overcharges (based on data from "UPC Scanner Pricing Systems: Are They Accurate?" by Goodstein, Journal of Marketing, Vol. 58). Use a 0.05 significance level to test the claim that with scanners, $1 \%$ of sales are overcharges. (Before scanners were used, the overcharge rate was estimated to be about $1 \% .$ Based on these results, do scanners appear to help consumers avoid overcharges?

Hi. This question. We need to use many tab. So, um, you have to go toe stat. Basic statistics, one proportion. So you're gonna find that just click the other choice, which is summarized data number off events to as described in the in the question 50. And check this, please. And with the value off point or five and then just go ahead and click options on. Try to find the hypothesis is quicker than then press. OK, you're gonna find the exact value of P value. It is point or 26 so we can conclude that reject, you know?

Yeah. Okay. So way knows that. Yeah. No hypothesis. H not. Is that p is equal to 0.5 on that alternative hypothesis. H A. The P is not equal to 0.5. Okay, Now we can calculate our proportion, which is excellent. And her ex is 123 on on is 280. Uh huh. She'll give us your credit for For came. Now we can calculate the test statistic value using a calculator which will give us negative. Too unsure. Three. Um, And now you can use that to find the p value using the table. So give us a P value. She is equal to 0.0 for two. Three. Finally, since we know Alfa is your opinions your one under, if you use a two tailed critical value be find the table value for Z Alfa two, which is 2.5 eight. So now, since it's two tailed critical values, Yes, arranged to 58 Okay. And since 0.423 is greater than our the significance level of 0.1 we can fail to reject the null hypothesis. Um, and therefore it could be concluded that there was not sufficient evidence supporting the rejection of the claim that touch therapists use method equivalent around guesses failed to reject.

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.

So we're given that an is equal to 300. Alfa is equal to 0.5 Mm yeah, which is our level. Significance on X is equal to 27. Therefore we can kill. Okay, X over n which is equal to 27. Okay, 300 literally. Call. Does your point. Zero? No. So, Donald, Hypothesis H nine is a P is equal to 01 andare alternative hypothesis is a P is less thens you're going so we can now use our formula to find those eat statistic. Yes. Yeah. Square P time zone minus p over. And which is he called? 009 minus zero. Mhm. Over the square was your one Times one nice. You should you. And over 300. She'll give us natives Europe 58 Now we can find g critical value, which is equal to negative 1.645 She was in our value. If she opens your Fife. Andi since r p value a z less than two negatives Europe on the side eight, which is equal to 0.280 man. Okay, it's there for the p value is greater than yeah, pretty cool significance level and therefore you can fail to reject the age, not there is not enough evidence to conclude that the proportion of test, which are wrong, is less than 10%.


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