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.
Is there a relationship between the observations collected in 1968 and 2008? Or are the observations in the two groups independent? Explain.
According to the details of the question, each observation from the 51 sampling locations for 1968 were later collected again in 2008. This means there are two sets of 51 special correspondence data points for the same geographic location, makes the observation paired, and not independent.
Write hypotheses for this research in symbols and in words.
\(H_0 : \mu_2 - \mu_1 = 0\). Null Hypothesis is that there is no difference between the mean temperature of Jan 1, 1968 and Jan 1, 2008.
\(H_1 : \mu_2 - \mu_1 > 0\). Alternative Hypothesis is that the difference between the mean temperature of Jan 1, 1968 and Jan 1, 2008 is greater than 0, suggesting an increase in temperature, hence evidence of global warming.
Check the conditions required to complete this test.
Calculate the test statistic and find the p-value.
# The standard error [sd / sqrt(n)]
SE <- 4.9 / sqrt(51)
# The z-score (mean / se)
Z <- 1.1 / SE
# Z-score to p-value
p <- 1 - pnorm(Z)
c(Z, p)## [1] 1.6031778 0.0544477What do you conclude? Interpret your conclusion in context.
At a 95% confidence level, with a z-score < 1.96, the critical level and p-value > 0.05, we fail to reject the null. This suggest that there is no evidence that the difference between the mean temperature of Jan 1, 1968 and Jan 1, 2008 is greater than 0, hence there is no evidence of global warming given the sample data.
What type of error might we have made? Explain in context what the error means.
A Type 2 error is when there is a failure to reject the null hypothesis when the alternative hypothesis may actually be true. There may be an increase, but with our sample, we were unable to prove it.
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.
Yes, in this case, the confidence interval will include 0 because we couldn’t reject the null that test for no difference in the temperature means.