Is there strong evidence of climate change? Let's consider a small scale example, comparing how temperatures have changed in South Africa from 1968 to 2008, more specifically we wish to analyse by how much temperatures in 2008 differed from 1968. Therefore, we will use the difference between 2008 temperatures and 1968 temperatures (2008 temp - 1968 temp). The daily high-temperature reading on January 1 was collected in 1968 and 2008 for 12 randomly selected locations in South Africa, with the temperatures at the locations shown below. Answer the following questions, and where applicable and unless otherwise stated, Round your answer to 2 (two) decimal places unless otherwise stated. 1 January temperatures Location Year 1968 Year 2008 1 20.7 22.5 2 21.1 20.7 3 20.6 20.8 4 23.6 18.5 5 23.1 22 6 20.6 18.2 7 22.3 18.3 8 22.5 22.9 9 21.9 19.6 10 19.2 18.7 11 22.7 19.7 12 22.9 23.5 b) What is the parameter of interest? Average difference between the 1968 and 2008 temperatures at all locations. Average temperature for 1968 and 2008 at all locations. Average temperature at all locations. Difference between the average temperature in 1968 and 2008. None of the above. c) What is the value of the point estimate, ?¯diff, of interest and what is the standard deviation, ??diff, of the point estimate: (Note: you may use statistical software to compute the standard deviation.) ?¯diff= ____ ??diff= ____ d) Complete the following hypothesis statement to evaluate if these data provide evidence of a change in temperature between 1968 and 2008 at all locations in South Africa. ?0:?diff= _____ ??:?diff≠ _____ e) We wish to calculate the T test statistic for the question. Calculate the Standard Error (SE) and round your answer to FOUR (4) decimal places. f) Using the calculated Standard Error (SE), calculate the T test statistic. g) What is the degrees of freedom associated with this test statistic? h) What is the likelihood of observing the same or a greater average difference as that of the study if the null hypothesis is true? Note: either use statistical software or an online t-distribution calculator to calculate your answer. i) Using a significance level of ?=8%, what is the outcome of the hypothesis test? Note that this part is NOT adaptively marked. Accept the null hypothesis since: ?-value Accept the null hypothesis since: ?-value≥? Do not reject the null hypothesis since: ?-value≥? Do not reject the null hypothesis since: ?-value<? Reject the null hypothesis since: ?-value≥? Reject the null hypothesis since: ?-value<? j) We wish to calculate a 98% confidence interval for the parameter of interest. What is the critical value, ∗t∗, for the confidence interval? Use the ?t distribution tables at the end of the prescribed textbook to determine the answer k) Calculate the Margin of Error (ME) for the confidence interval.
Is there strong evidence of climate change? Let's consider a small scale example, comparing how temperatures have changed in South Africa from 1968 to 2008, more specifically we wish to analyse by how much temperatures in 2008 differed from 1968. Therefore, we will use the difference between 2008 temperatures and 1968 temperatures (2008 temp - 1968 temp).
The daily high-temperature reading on January 1 was collected in 1968 and 2008 for 12 randomly selected locations in South Africa, with the temperatures at the locations shown below.
Answer the following questions, and where applicable and unless otherwise stated, Round your answer to 2 (two) decimal places unless otherwise stated.
| 1 January temperatures | ||
| Location | Year 1968 | Year 2008 |
| 1 | 20.7 | 22.5 |
| 2 | 21.1 | 20.7 |
| 3 | 20.6 | 20.8 |
| 4 | 23.6 | 18.5 |
| 5 | 23.1 | 22 |
| 6 | 20.6 | 18.2 |
| 7 | 22.3 | 18.3 |
| 8 | 22.5 | 22.9 |
| 9 | 21.9 | 19.6 |
| 10 | 19.2 | 18.7 |
| 11 | 22.7 | 19.7 |
| 12 | 22.9 | 23.5 |
b)
What is the parameter of interest?
-
Average difference between the 1968 and 2008 temperatures at all locations.
-
Average temperature for 1968 and 2008 at all locations.
-
Average temperature at all locations.
-
Difference between the average temperature in 1968 and 2008.
-
None of the above.
c)
What is the value of the point estimate, ?¯diff, of interest and what is the standard deviation, ??diff, of the point estimate: (Note: you may use statistical software to compute the standard deviation.)
?¯diff= ____
??diff= ____
d)
Complete the following hypothesis statement to evaluate if these data provide evidence of a change in temperature between 1968 and 2008 at all locations in South Africa.
?0:?diff= _____
??:?diff≠ _____
e)
We wish to calculate the T test statistic for the question.
Calculate the Standard Error (SE) and round your answer to FOUR (4) decimal places.
f)
Using the calculated Standard Error (SE), calculate the T test statistic.
g)
What is the degrees of freedom associated with this test statistic?
h)
What is the likelihood of observing the same or a greater average difference as that of the study if the null hypothesis is true?
Note: either use statistical software or an online t-distribution calculator to calculate your answer.
i)
Using a significance level of ?=8%, what is the outcome of the hypothesis test?
Note that this part is NOT adaptively marked.
-
Accept the null hypothesis since: ?-value<?
-
Accept the null hypothesis since: ?-value≥?
-
Do not reject the null hypothesis since: ?-value≥?
-
Do not reject the null hypothesis since: ?-value<?
-
Reject the null hypothesis since: ?-value≥?
-
Reject the null hypothesis since: ?-value<?
j)
We wish to calculate a 98% confidence interval for the parameter of interest.
What is the critical value, ∗t∗, for the confidence interval? Use the ?t distribution tables at the end of the prescribed textbook to determine the answer
k)
Calculate the Margin of Error (ME) for the confidence interval.
l)
Calculate the 98% confidence interval for the parameter of interest.
CI: ( , )
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