1 MET 136 Statistical Climatology - Lecture 11 Confidence Intervals Dr. Marty Leach San Jose State University Reading: Gonick Chapter 7

2 Sampling  We previously studied how samples of large populations were distributed.  Now, we’ll look at one sample, and study what we can determine from this alone.

3 Confidence Intervals   Used in election polls (watch it!)   The average global air temperature near the Earth's surface increased 0.74  0.18ºC (1.33  0.32 º F) during the 100 years ending in (IPCC 2007)C

4 Confidence Intervals  Are used extensively in science  Used in election polls (watch it!)  Example:  The average global air temperature near the Earth's surface increased 0.74  0.18ºC (1.33  0.32 º F) during the 100 years ending in (IPCC 2007)C

5 Example 1  Election Numbers  x?g=252060cf-f1d3-49bc-80ed- 24d0c9122b49&d=0 x?g=252060cf-f1d3-49bc-80ed- 24d0c9122b49&d=0  Let’s look at the numbers

6 Poll  Surveyed 661 likely to vote people  N=661  Randomly selected  Result:

7 Poll  Surveyed 661 likely to vote people  N=661  Randomly selected  Result:

8 Standard deviation of normal  To determine the accuracy of this probability, we need to calculate the standard deviation:  Only problem…we don’t know true probability, p.

9 Standard deviation of normal  To determine the accuracy of this probability, we need to calculate the standard deviation:  Only problem…we don’t know true probability, p.

10 Standard Error  Only thing we can do is use the standard error (which uses the sampled probability (p- hat) 

11 Standard Error  Only thing we can do is use the standard error (which uses the sampled probability (p- hat)  This is called the standard error

12 Standard Error  So now we can estimate the confidence interval at the 95% level 

13 Standard Error  So now we can estimate the confidence interval at the 95% level  This says that 95% of the time, the true probability p will fall within these two values.

14 Calculate confidence interval  Let’s calculate the 95% confidence interval for the presidental poll in CA.  N=661   

15 Calculate confidence interval  Let’s calculate the 95% confidence interval for the presidental poll in CA.  N=661  So that now, p is within the range:  0.53±1.96*0.019  p=0.53 ± 0.038

16 Interpretation  So what does this mean?  p=0.53 ±   Slight oversight… 

17 Interpretation  So what does this mean?  p=0.53 ±  ≤ p ≤  Slight oversight…  Obama: 53 McCain: 43 Undecided/other: 4

18 20 samples with n=1000; assume true value p=0.5. Shown are 95% confidence interval. On average 1 in 20 will not cover 0.5

19 Improve the results  Suppose we want a more confidence, say 99%, what can we do?  

20 Improve the results  Suppose we want more confidence, say 99%, what can we do?  Widen the confidence interval  Increase the sample size

21 Example  Redo the confidence interval at the 99% level  Result:     But now our margin of error is larger… (e.g. I’m 100% confident the probability will be between 0 and 1!

22 Example  Redo the confidence interval at the 99% level  Result:  0.53±2.58*0.019  p=0.53 ±  ≤ p ≤  But now our margin of error is larger… (e.g. I’m 100% confident the probability will be between 0 and 1!

23 Sample Size  But what if we are not happy that our error has gone up. The other way to keep the error down and the confidence high is to increase the sample size. 

24 Sample Size  But what if we are not happy that our error has gone up. The other way to keep the error down and the confidence high is to increase the sample size.  Where Z is from the normal table (pg 84), p* is the estimate of the probability and E is the margin of error.

25 Example  So now calculate the sample size required to produce a margin of error of 0.01 and a 99% confidence level.  Result   Limits to polling…

26 Example  So now calculate the sample size required to produce a margin of error of 0.01 and a 99% confidence level.  Result  More then 16,000 respondents!  Limits to polling…

27 Confidence intervals for the mean  Now, we’ll look at confidence intervals for the mean, not the probability.

28 Confidence intervals for the mean  Now, we’ll look at confidence intervals for the mean, not the probability.

29 Standard Error  The standard error of the mean is defined as: 

30 Standard Error  The standard error of the mean is defined as:  Where s is the sample standard deviation

31 Example  Suppose that you calculate the average winter low temperature in Silicon Valley during the last 25 years to be 41.5F and the standard deviation is 3.2F.  Compute the 95% confidence interval for the mean temperature.  If temperatures below 40F are required for fruit to start growing in the valley, would you expect this to happen in a typical winter?

32 Student’s t  We’ve discussed that as the sample size increases, the distribution approaches a normal distribution.  We can quantify this using the degrees of freedom.  If you have x 1, x 2, …x n data points, then you have n-1 degrees of freedom.  So, we can choose a t-distribution for n-1 degrees of freedom.

33 t-distribution

34 Mean using a t-distribution 

35 Mean using a t-distribution  So, using a t-distribution, the mean and the confidence interval is given by:

36 Notation: 

37 t-distribution table