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.
The two measurements are independent. The measurements from 1968 do not in any way affect the measurements from 2008, or vice versa. Furthermore, knowing the outcomes of one set of measurements provides no useful information regarding the outcomes from the other set of measurements.
Secondly, the textbook states that sample observations are independent if the observations are from a simple random sample and consist of fewer than 10% of the population. I am going to assume that the population of locations from which the 51 locations were sampled is at least 510 locations.
(b) Write hypotheses for this research in symbols and in words.
H0: \(\mu_{diff} = 0\) There is no difference between average daily high temperatures on Jan 1, 1968 and Jan 1, 2008 (null hypothesis)
HA: \(\mu_{diff} > 0\) Average daily high temperatures on Jan 1, 2008 are higher than Jan 1, 1968 (alternative hypothesis)
I see this as a one-tailed test, because the research question is not concerned with testing for global cooling or warming, just global warming.
(c) Check the conditions required to complete this test
(d) Calculate the test statistic and find the p-value.
# compute standard error (standard deviation / square root of sample size)
se <- 4.9 / sqrt(51)
# compute z-score
z <- 1.1 / se
z## [1] 1.603178# Translate z-score to p-value
1 - pnorm(z)## [1] 0.0544477
(e) What do you conclude? Interpret your conclusion in context.
The exercise does not specify what significance level to use, so I’m going to use \(\alpha = 0.05\) since that seems to be the default from the textbook.
Since the p-value 0.0544 exceeds the \(\alpha\) value, I fail to reject the null hypothesis. The data does not provide strong evidence that the 2008 temperatures are greater than the 1968 temperatures, though it is pretty close.
(f) What type of error might we have made? Explain in context what the error means.
A Type 2 error is when you fail to reject the null hypothesis when the alternative hypothesis is actually true. Since I failed to reject the null hypothesis in e. above, I may be committing a Type 2 error.
(g) Based on the results of this hypothesis test, would you expect the confidence interval for the average difference between the temperature measurements from 1968 and 2008 to include 0? Explain your reasoning.
Yes, I would, because if 0 were outside of the confidence interval, we would have rejected the null hypothesis.
# compute confidence interval using se computed previously
c(1.1 - qnorm(0.95) * se, 1.1 + qnorm(0.95) * se)## [1] -0.02859533 2.22859533