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Assume that a simple random sample has been selected from : normally distributed population and test the given claim. Identify the null and alternative hypotheses; ...

Question

Assume that a simple random sample has been selected from : normally distributed population and test the given claim. Identify the null and alternative hypotheses; test statistic critical value(s) and state the final conclusion that addresses the original claim: 4) A manufacturer makes ball bearings thatare supposed to have a mean weight oE 30 g A retailer suspects that the mean weight is actually less than 30 g The mean weight for a random sample of 16 ball bearings is 28.6 g with standard devi

Assume that a simple random sample has been selected from : normally distributed population and test the given claim. Identify the null and alternative hypotheses; test statistic critical value(s) and state the final conclusion that addresses the original claim: 4) A manufacturer makes ball bearings thatare supposed to have a mean weight oE 30 g A retailer suspects that the mean weight is actually less than 30 g The mean weight for a random sample of 16 ball bearings is 28.6 g with standard deviation 0f 4.4 & At the 0.05 significance level, test the claim that the sample comes from a population with a mean weightless than 30 & Ho = HA = t= Conclusion:



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. Assume that a simple random sample is selected from a normally distributed population. Birth Weights A simple random sample of birth weights of 30 girls has a standard deviation of 829.5 hg. Use a 0.01 significance level to test the claim that birth weights of girls have the same standard deviation as birth weights of boys, which is 660.2 hg (based on Data Set 4 "Births" in Appendix B).

So for this problem we're told the significance of 0.01. The claim is that we pennies are manufactured so that the waves have a standard deviation equal 0.0-30 graphs. And for this problem we want to identify them. No alternative hypothesis testes is six p values, critical values and the make a claim based on what were the data we did based on our process. So, first there's an empire, the no hypothesis. Alright, highlight that this is a protesting for sanity, aviation and that this equals 0.0 230 g from the claim. Yeah. The alternative, we're still testing Sander deviation but there is no greater than or less than So we don't really care as long as it does not equal 0.023 g. So this is a two tailed test. So secondly, let's now find our testes cystic. Yeah. For the test statistics, since we're testing the standard deviation, this is gonna be a chi square distribution. And we can find it to be M -1 squared divided by sigma squared where n is the length of the data. And as squared you can find using this form from the data And if you plug this in, I got a testis cystic of three points 81 for all right. So, once you have that, you can find your p value, you can find the p value is one of two ways either through the table. Um but if you do it through the table method, you won't get an exact value of finding a range that your p value falls under. Or you can use technology like our our stack crunch, which will give you a more accurate p value. So the main function in our final p values where chi square distribution with bp Yeah. Hi, square. And then we're going to have our testes cystic Degrees of freedom and -1 and we're gonna have this lower dot tail equals something At one point on this is a two tailed test. So this p value around to multiply this p value by two and remember this P value is a probability. So if we say the p value is equal to false and we get greater than one then we might we might not be doing the right tail. So let's try P value is equal to true. And we get a p value of zero point two five three and we like this because the p value still less than one. If you were to try this with the lower tell equals of false, you should get a p value greater the one which is not which is no longer a probability. So remember we can make in conclusions. If the p value, it's greater than alpha, we will accept the null hypothesis. Yeah. And if the p value less than alpha, we will reject no of assists. And here we can see that 0.253. Well that's greater than our alpha which is 0.01. So we should accept on our policies process for the sake of completion. Find the critical values and compare it with our testes cystic. Yeah. So you can use the table again. Find the degrees of freedom and alpha. But if you you the degrees of freedom or we're going to be chi squared with one minus off over to degrees of freedom and chi squared Alpha over 2° of freedom. Yeah. And from the table you should get the numbers. You can also use technology like our the functions are gonna be the same will be q chi square and for the most part inputs are the same. You're gonna putting off over to Your N -1. The only thing that's going to change is this lower dot tail. So we're gonna have one that's lower dot l equals true. And this gives him one 34. Yeah. Another one. The whole q chi square alpha over to and -1. Lower dato equals two false. And this gave me 21 point 955. So how do you make conclusions based on the critical values and the testes cystic. So if we have a high score distribution We have these points. The 1.34 And the 21.96. These are giving us two different areas and these are gonna be our alphas over to And both of these are gonna be our rejection region. So if our tested falls within those areas, we reject ever if it falls within the middle, that is our acceptance region. So we won't we will accept our normal processes. So we go back to see what our test statistic was. This is 3.814. So that falls probably somewhere around here, or this was our testes statistic. So again, our testes, it falls within our acceptance region. So we will accept the null hypothesis was which was the same conclusion. We came with the P values in alpha. So This is not statistically significant alpha level of 0.01. So we will have to accept that the standard deviation is equal to 0.023.

Problem. 24 Question eight. It's not that new is smaller than equal toe 3500 and each one equals that mu bigger than 3500. So the the the venue is equal to x bar minus new note over s over square root end, which is 3375135 year review over standard division over square root off end, she is equal to negative 1.571 for question be so. The Big Valley is a number or interpret in the column for Table five with degree of freedom equal to N minus one, which is eight minus one is equal to seven. So accordingly the table that we value is between 4.45 and 4.1 S O. C, which is so the P value is smaller than the significance level than the null hypothesis is rejected. He is bigger than open toe five. So we fail to reject the mile high processes so we can say that there is no sufficient evidence to support the clean

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.

Let us read this question. A light bulb manufacturer guarantees that the mean life off a certain type of light bulb is at least 7 50 hours. So mean life is at least 7 50 years, which means it is greater than or equal toe 7 50 years. So the complement of this is that it is less than 7 50 hours. So let's say that this is a null hypothesis. The first one. The claim forms the null hypothesis and the compliment over here is the alternative hypothesis. A random sample of 25 light bulbs so our end is equal to 25 r n is equal to 25 and they're mean is 7 45. So x bar is equal to 7. 45. Standard deviation off the population is 60. Our sigma is equal to 60. So let's just put in the values. Misers to district will be equal toe 7 45 minus 7 50. This is what I'm checking it against. Upon Sigma is 60 upon root off. What is my end in his 25 which is nothing but five. So if I use a calculator to find this, this is going to be 7 45 minus 7 50 divided by 16 multiplied by five or minus 4.16 Now this is a This is using magnitude, right? Minus 4.16 This Z statistic is used in number. Okay, My Alfa is 0.2 So what is going to be my critical Z value? It is 2.6 My Zed star. Let me say that my said star is 2.6 and this is 2.6 and my value off minus 4.16 falls somewhere around here. So this is in the non rejection region, right? What is my alternative? My, my alternative. Excuse me. My alternative is less than so. This is going to be minus 2.6 My critical value is going to be minus 2.6 So it is not going to be here, but rather it is going to be on the negative side. So this is minus 2.6 My critical value, minus 2.6 on my value falls in the rejection region falls in this region in the rejection region. So I will reject my null hypothesis. What was my null hypothesis? My null hypothesis was the claim so that I can say that I have enough statistical evidence to say that the claim or the guarantee off the light bulb manufacturer is actually false. Okay, He said that it is at least 7. 50. It is at least 7 50 years, but that turns out to be false. That claim is false, and this is going to be my answer.


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