5.19 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.

It is a paired observation, since for each observation in the data set, there is one corresponding observation in another for the same location.

Paired data: Two sets of observations are paired if each observation in one set has a special correspondence or connection with exactly one observation in the other data set.

  1. Write hypotheses for this research in symbols and in words.

Null hypotheses: There was no change in average temperature between 1968 to 2008. Alternative Hypotheses: There is significant difference between average temperatures in 1968 and 2008.

  1. Check the conditions required to complete this test.

The sample size is at least 30. Location is said to be random and it represents less than 10% of all locations. We can’t see the actual data, so we don’t know how skewed it is.

  1. Calculate the test statistic and find the p-value.

    N=51
M=1.1
SD =4.9
SE <- SD/sqrt(N)
SE

Test Statistic for Null Hypotheses.

T = (M-0)/SE
T

qt(.05,df=50)*-1

P-Value

P <- pt(T, df=N-1, lower.tail =FALSE)
P
  1. What do you conclude? Interpret your conclusion in context.

Since p value is greater than 5%, fail to reject null hypotheses.

  1. What type of error might we have made? Explain in context what the error means.

It should be noted that our P value is very close to our significance level of 5%, so it is possible we may have made a Type II Error. Failing to reject null hypotheses when it is false.

  1. Based on the results of this hypotheses 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, we failed to reject the null hypotheses that the average difference in temperature from 1968 and 2008 is 0