Question

5.19 Global warming, Part II: We considered the differences between the temperature readings in January 1 of 1968 and 2008 at 51 locations in the continental US in Exercise 5.19. The mean and standard deviation of the reported differences are 1.1 degrees and 4.9 degrees respectively.


(a) Calculate a 90% confidence interval for the average difference between the temperature measurements between 1968 and 2008.
lower bound: _____ degrees(please round to two decimal places)
upper bound: _____ degrees(please round to two decimal places)



(b) Interpret this interval in context.

a. There is a 90% chance that the difference in temperatures in a city from year to year will be between the lower bound and upper bound

b. We are 90% confident that the true mean difference in temperatures is contained between the lower bound and upper bound

c. We are 90% confident that the mean difference in these sample temperatures is contained between the lower bound and upper bound

d. We are 90% confident that 90% of the time the differences in temperatures from year to year will be between the lower bound and upper bound

(c) Does the confidence interval provide convincing evidence that the temperature was higher in 2008 than in 1968 in the continental US? Explain.

a. Yes, because the confidence interval contains negative numbers

b. No, because the confidence interval is not very wide

c. Yes, because the confidence interval contains mostly positive numbers

d. No, because the confidence interval contains 0

 

Answer 1

Concepts and reason

The confidence level of an interval estimate of a parameter is the probability that the interval estimate will contain the parameter, assuming that a large number of samples are selected and the estimation process on the sample parameter is repeated.

A confidence interval is a specific interval estimate of a parameter determined by using data obtained from a sample and by using the specific confidence level of the estimate.

The margin of error is the maximum likely difference between the point estimate of a parameter and the actual value of the parameter.

Fundamentals

The formula for confidence interval for the paired mean is,

Here,

Number of sample values.

Standard deviation of the difference

Sample mean difference

Critical values

(a)

The summary of the statistics are,

The degree of freedom is,

Critical value:

Using the t-distribution tables, the critical value at 10% level of significance with degree of freedom (50) is,

Substitute the values in the formula of confidence interval as follows:

(b)

From part (a), the 90% confidence interval for the average difference between the temperature measurements between 1968 and 2008 is -0.05 and 2.25.

The lower and upper bounds are -0.05 degrees and 2.25 degrees.

Interpretation: We are 90% confident that the true mean difference in temperatures is contained between the lower bound and upper bound.

Hence, the correct answer is option (b).

(c)

From part (a), the 90% confidence interval for the average difference between the temperature measurements between 1968 and 2008 is -0.05 and 2.25.

The lower and upper bounds are -0.05 degrees and 2.25 degrees.

Claim: The temperature was higher in 2008 than in 1968 in the continental US.

No, there is no evidence that the temperature was higher in 2008 than in 1968 in the continental US. Because the confidence interval contains 0

Hence, the correct answer is option (d).

Ans: Part a

The 90% confidence interval for the average difference between the temperature measurements between 1968 and 2008 is -0.05 and 2.25.

The lower and upper bounds are -0.05 degrees and 2.25 degrees.

Part b

We are 90% confident that the true mean difference in temperatures is contained between the lower bound and upper bound.

Part c

No, because the confidence interval contains 0.