Question

A report recently stated that human activities are estimated to have caused approximately 1.0°C of global warming above pre-industrial levels on average, with a 66% confidence interval of 0.8°C to 1.2°C. Assume that the report relied on a sample of 100 global temperatures from 1998 to 2017 to calculate the sample average of 1.0°C (x ̅), and that the standard deviation of the population of global temperatures from 1998 to 2017 is known (σ).

a. What is the approximate distribution of X ̅? What is the standard normal critical value used to create a 66% confidence interval centered at zero?

b. That is, solve for z, where Pr⁡(-z≤Z≤z)=66%. (Note: solve for z to the nearest hundredth; do not interpolate in an effort to get a more precise answer.)

c. Using the answers to a. and b., solve for σ. Having solved for σ, what is the 95% confidence interval for the population mean of global temperatures from 1998 to 2017?

Answer 1

(a)

The approximate distribution of X ̅ will be approximately normal distribution because sample size is large.

(b)

Here we need to find the value of z that has (1-0.66) /2 = 0.17 area at both ends. From z table, z-score -0.95 has approximately 0.17 area to its left. That is area between -0.95 and 0.95 is 0.66.

So,

P(-0.95 <= Z <= 0.95) = 0.66

(c)

From the given 66% confidence interval, the margin of error is

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Now

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The 95% confidence interval is