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.

Answer: Each observation in the data set for 2008 has a connection with exactly 1 observation in the data set 1968. Hence these are paired sets.

(b) Write hypotheses for this research in symbols and in words.

Answer:

H0 : The average temperature in 2008 is same as average temperature in 1998

  mu-temp-2008   -    mu-temp-1998  =  0

HA : The average temperature in 2008 is greater than the average temperature in 1998

  mu-temp-2008   -    mu-temp-1998  >  0

We have assumed one-sided Alternate hypothesis here to suggest that the temperature has increased over the years due to global warming.

(c) Check the conditions required to complete this test.

Answer:

  1. The number of observations in the sample is 51, which is less than 10% of the overall number of locations on the earth or even the U.S. Hence the observations can be said to be independent.

  2. The sample size is quite large as n = 51 which means n > 30 in this case.

  3. There is no data given here for the skewness of the data. But generally this can be ignored unless there is a very strong skewness.

Hence the conditions for the inference are satisfied and this sample can be used to check the hypothesis.

(d) Calculate the test statistic and find the p-value.

Answer:

mu_point_value_5.19 <- 1.1
n_5.19 <- 51
sd_5.19 <- 4.9
null_5.19 <- 0

SE_5.19 <- sd_5.19 / sqrt(n_5.19)
T_value_5.19 <- (mu_point_value_5.19 - null_5.19) / SE_5.19

T_value_5.19
## [1] 1.603178
P_VALUE_5.19 <- pt(T_value_5.19, df = 50, lower.tail = FALSE)
P_VALUE_5.19
## [1] 0.05759731

(e) What do you conclude? Interpret your conclusion in context.

Answer: p-value = 0.0576, by default significance value = 0.05. In this case, p-value > significance value, hence Null Hypothesis cannot be rejected. That means there is not sufficient evidence to confirm that the average temperature increased from 1968 to 2008 or in other words, there is no evidence of global warming in the U.S.

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

Answer: In this case, the p-value came out to be very close to 0.05. As it was slighly larger than significance value of 0.05, we failed to reject H0. Hence in this case, it seems quite possible that the global warming might have actually been happening over years, but we failed to suggest the same. Hence we might have made Type-2 error.

Type-2 error - when HA is true, but the results show that H0 cannot be rejected.

(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.

Answer: Yes, as we failed to reject H0, that means there is a possibility of difference of average tempetratures to be equal to 0. That means a confidence interval should have 0 (Null-value) within the range.