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0540 while the sample of 27 has mean of 41 In a hypothesis test; the claim normally distribuled dala set In (his and sample slandard deviation ol 5.9 from wouild 7 ...

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0540 while the sample of 27 has mean of 41 In a hypothesis test; the claim normally distribuled dala set In (his and sample slandard deviation ol 5.9 from wouild 7 test slatislic be used or A t test statislic and why? hypolhesis tesl

0540 while the sample of 27 has mean of 41 In a hypothesis test; the claim normally distribuled dala set In (his and sample slandard deviation ol 5.9 from wouild 7 test slatislic be used or A t test statislic and why? hypolhesis tesl



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For Exercises 5 through $20,$ assume that the variables are normally or approximately normally distributed. Use the traditional method of hypothesis testing unless otherwise specified. Age of Psychologists Test the claim that the standard deviation of the ages of psychologists in Pennsylvania is 8.6 at $\alpha=0.05 .$ A random sample of 12 psychologists had a standard deviation of $9.3 .$

We have a sample with sample size and equals 81 X bar sample mean equals 20 and sample standard deviation S equals three. We want to construct a confidence interval about this sample to determine the population, meaning you. But first we want to answer a In a student's T distribution appropriate to use for this interval. The answer is immediately yes, because we have and greater than equal to 30 unnecessary criterion to use the students the distribution next let's construct a 95% confidence interval and interpret our findings. So, first to find this interval, we have to identify the T score. To do so we can use a tea table which maps for for a specific confidence level and for specific degree of freedom, the exact T score corresponding we have degree of freedom, a confidence level alpha equals 0.95 So from this we get T C equals 1.990 next to identify the margin of error given by the formula here on the left, we see that we can calculate E as the T. Value times asked about it by route and in this case that gives E equals 1.99 times three. Or route 81 or equal 0.66 So we can use to construct the confidence interval expo minus is less than new. Is that expire plus E? Which gives us 19.34 is less than new, is less than 20.66 Finally, we can interpret this to mean that we can say with 95% confidence that the meetings between 19.34 and 20.66

What's upstart cots In this video, we are given an example, and we are asked to perform either a chef or a two key test. So this is the example where provided. So considering we have a neat, uneven sample, sizes were gonna want to perform a chef test. So this is the formula to calculate our test statistic. So let's go ahead to the Excel spreadsheet where I have the raw data and heard. So I have entered the data and performed an Innova because through the example, were given, we are told that, um, the differences are there's already a significant difference. So I just wanna had ended theano of a test. So I think the first thing we should do is calculate our critical value. So our critical value is found by taking the critical value from our F test and multiplying it by the degrees of freedom between groups. So that is our critical value. And now we're going to calculate the test statistic. So what we're gonna do is I'm gonna go ahead and pair these categories up, so cars I'm just gonna use one SUV will be too, and trucks will be three. So I'm gonna pair cars and SUVs, SUVs and trucks and then trucks and cars. And the first thing I'm gonna do to calculate the test statistic is I'm gonna do the differences in means squared. So I'm gonna do equals parentheses. And I'm going to go to the mean of cars cause that's one and subtract the mean of SUV's gonna close the parentheses, and I mean square. So I'm going to do the same thing for the rest of these. All right, So that's the differences in our means squared. If we go back to our formula now, we have to do the variance within and multiply it by one divided by our sample sizes. Okay, so let's go back to our excel spread she So I'm gonna do okay first. Going to calculate that one over n nonsense. So let's do over here. It's too. He fools one over and then and will be the sample size for car. So 1/5 waas one over the sample size for SUVs. Well, I have us over a lot going on. Okay, So let me just type for than against, Okay, so I'm gonna do that for the rest of them. So one over. Sample size for SUV plus one over sample size for Chuck's man. Last one. Okay, that's just gonna make it easier for us when we do our numerator divided by our denominator. Okay, so next thing we're gonna do is we're gonna take our means square with n group, and we're gonna multiply it by that value we just calculated. And then now we can calculate our f our test statistic really easily. So we just take our numerator divided by our denominator, and I can just pull out formula down or not. Okay, so we've calculated our critical values and our test statistics, so let's go back to our whiteboard and fill them in. So are critical value. Waas e 0.52 on. Yeah, Okay. Just making sure 8.52 and our test statistic for cars and SUVs. Waas 2.1. And for SUVs and trucks, it was 17.64 And for cars and trucks, it was 27.9 to Okay, So in order to know which of these is significant, we want to ask which of these is larger than our critical value. So it was 2.1 greater than 8.52 No. Is 17 greater than a yes? And is 27 greater than a yes? So these so SUVs and trucks. So the differences between those means or different and the difference between the means of cars and trucks are different. So to summarize this, we're gonna say there is sufficient evidence to conclude a difference. And I mean cost the drive. These this between hybrid cars and trucks and between SUV's and checks. Alright, guys, that's it for this video. I hope you learned a lot else. You next time.

In exercise four, we're considering a random sample that is drawn from a normally distributed population and this time the standard deviation for the population is unknown. We're going to use the given information to construct a 98% confidence interval for the population mean. So what are we given were given? The sample size is the sample means? And also the sample standard deviation. Now the sample mean and standard deviation, He's 58.3 and 4.1 respectively, for the two cases before the first case we have a small sample of eight, and the second one is also a small sample uh of 27. So let's begin. Um First of all look at the formula that we're going to use to get the 98% confidence interval. So we're going to use the formula in dread X bar plus critical value of T for the given level of significance, multiplied by the sample standard deviation developed by the square root of the sample size end. So let's begin by determining the critical value for these tests. And uh for for us to get the critical values for the test, we have to first get the level of significance. Uh That is common for both cases and that's going to be alpha. So the alpha equals 1 -0.98 because 0.98 is simply 98% and that gives us 0.02. And next we want to have that because the two tailed kind of test, so divided by two, it will be 0.01. Next you want to get the degrees of freedom uh for the first case. So the degrees of freedom given by N -1 Which equals 8 -1 and that equal seven. So the critical value of T. That corresponds to these two tails uh test here would be equal to two point nine 98. And now we have everything we need to substitute um into the main formula. So we have 58.3 which is x bar plus or minus the critical value of T. 2.998 times the value of s the sample standard deviation 4.1, invited by the square root of eight which is a sample size. And when you work that out you're going to get 500, sorry 58.3 plus or minus 4.35. And so the margin of error for this one is 4.35. Now let's proceed to the second case and fast get the decrease of freedom. So the degrees of freedom will be N -1, which in this case is 27 -1 and that gives 26. So the critical value for the 0.01 level of significance equals 2.4 79. Next we're going to substitute everything into the main formula to it's going to be five, Plus or -2.479 times 4.1 Divided by the Square Root of 27. And when you walk that out, going to become 58.3 plus or minus 1.96. So the margin of error in this case is 1.96. When we quickly compare the two margins of error margins of error, we noticed that when the sample science is bigger and we have a smaller margin of era, meaning that the interval will also be much shorter. Mhm.

What's up? Stock cats? In this video, we are asked to perform either a chef or two key test, and those are post talk tests for a one way and over. So this is the example were given, and these are. These are the costs of what costs oven oven her watts of the oven. So because the sample sizes are uneven, we are going to perform a chef test. So this is the test statistic. This is the formula for the test statistic. This is our CV says are critical value. And we are told that the Alfa level for the one way Unova tests was performed at a 10.1 Alfa level. So let's go to the Excel she that I have our raw data on. So I've already performed the one way and Nova because that's how they have the example set up. So now we are going to pair up our oven watts because a chef test is a pair wise test. So we're gonna pair up 1000 watts and 900 watts, 900 watts and 100 watts and 100 watts and 1000 watts. So to calculate the numerator for our test statistic. We're gonna want to take the difference of the means squared. So let's go ahead and do that. So we're gonna take the mean for the 1000 watt of in. We're going to subtract the mean of the 900 watt of in, and we're going to square that value, and we're gonna do that for the rest of the pair. Wise pair wise pairs. No cheese. Hey, off to a rocky start. But we can recover. So the this is our numerator, So are enumerators done. And now we want to calculate our denominator. So the first thing I'm gonna do is I'm actually gonna calculate, um, this portion here and then because we already have our difference within groups. We already have our means squared within groups already calculated. So we need to calculate the other part, which I'm gonna do right here. So one over sample size of 1000 want plus one over sample size of 900 watt. All right, so now too calculate our denominator. We are going to take our means square within groups, and we're going Teoh, multiply it by the other value we just calculated I'm gonna go ahead and pulls. Actually, I can't pull it down. Okay? Finally, to calculate our test statistic, we're going to take our numerator over our denominator, and I can pull the this value, these values down. So for 1000 and 900 watt oven, we have two point 91 for a test statistic. And for the 908 100 we have eight point 40 And finally, for the 800 the 1000 we have 19 0.28 Okay, so last thing we have to do is calculate are critical value, because that's how we're going to decide which pairs air. Significant. So we're just going to take our critical value from the Unova test and multiply it by our degrees of freedom between groups. So are critical. Value is 5.21 Okay, so let's go ahead and go back to our white boards so you can copy these numbers down, make it easy for us. Teoh reject or failed to reject, so to decide whether we have a significant para not We're gonna take the critical. I'm sorry. We're going to take the test statistic of that pair, and if it is larger than our critical value. Then we can reject our no hypothesis. So that means the pair will be significant. So is 2.91 larger than 5.21 No. Is 8.40? Yes. Is 19.28? Yes. So we have to significant pairs at the 0.1 council level. So let's go and summarize our results. So there is fish in evidence. Uh, the 0.1 for love will to conclude there is a difference in me and cost between between 908 100 watt of ends and 801,000 walk ovens. All right, that's it for this video. I hope you guys learned a lot Chelsea next time.


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