1 global warming part i is there strong evidence of global...

1. Global warming, Part I. Is there strong evidence of global warming? Let's consider a small scale example, comparing how temperatures have changed in the US from 1968 to 2008. The daily high temperature reading on January 1 was collected in 1968 and 2008 for 51 randomly selected locations in the continental US. Then the difference between the two readings (temperature in 2008 - temperature in 1968) was calculated for each of the 51 different locations. The average of these 51 values was 1.1 degrees with a standard deviation of 4.9 degrees. We are interested in determining whether these data provide strong evidence of temperature warming in the continental US. (a) Is there a relationship between the observations collected in 1968 and 2008? Or are the observations in the two groups independent? Explain. (b) Write hypotheses for this research in symbols and in words. (c) Check the conditions required to complete this test. (d) Calculate the test statistic and find the p-value. (e) What do you conclude? Interpret your conclusion in context. (f) What type of error might we have made? Explain in context what the error means. (g) Based on the results of this hypothesis test, would you expect a confidence interval for the average difference between the temperature measurements from 1968 and 2008 to include 0? Explain your reasoning.

Transcript

00:01 Okay, so we have a few different parts to this question.

00:03 So we have a bunch of data from 1968 temperatures versus 2008 temperatures, and they took 51 cities in from 1968 and the same 51 cities from 2008 and looked at their january 1st temperature for each year.

00:23 Okay? and then they took all of those, and they subtracted them.

00:30 Okay? and when they did that, the average of the 51 differences was, for the differences, was 1 .1 degrees with a standard deviation of those differences of 4 .9 degrees.

00:46 So we want to see if there is a relationship between the observations collected in 1968 in 2008 or whether they're independent.

01:00 So there should be a relationship, right? because each of the 51 cities in 1968 were again tested, well, not tested necessarily, right, but we looked at again for the 2008 data, right? so each of those cities has data for 2008 as well.

01:40 So there is a relationship, right? it's not like they're completely different random values each time.

01:47 For b, the hypothesis is going to be, well, we're trying to see if the temperature is increasing, right? so the null hypothesis is going to be that it's not increasing.

02:02 So that would be that the mean difference is not equal or is equal to zero, right? or sometimes depending on the class, they like you to have it as like the mean difference is less than or equal to zero.

02:15 So you'll have to see which one that is.

02:18 For the alternative hypothesis, it's going to be that the mean difference is greater than zero.

02:24 Because if the mean difference is greater than zero, that means that the temperature is increasing.

02:31 So that's in symbols, right? this would be that the mean temperature for 2008 is not greater than the mean for 1968.

02:52 And then in words for this one, it would be that the mean temperature for each of the cities for 2008 is greater than the mean difference for 1968.

03:17 Right? because if the, yeah, because if we're looking at the difference, the mean difference, right? another way that you could say is, you know, the main difference between the two years is significant or not.

03:36 So for c, to check the conditions required to complete this test, the first is always that we have to have randomness.

03:45 And it's told that we do have randomness because the 51 locations were chosen randomly.

04:02 Okay? another thing that we need is we need to, well, so this is also one that depends on the class.

04:10 Sometimes they say that it's good enough to just have a large sample size, and other times they say that you need to have approximately normally distributed data.

04:21 But we don't have any indication here that we can even check for normality of the data.

04:25 So we're going to check if there's a large enough n, and so we're going to say yes.

04:30 And usually if you say like greater than 30, that's oftentimes what you're looking for.

04:35 And then for this one, they don't have to be independent.

04:41 Well, each, like, each of the 51 locations have to be independent...

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