Comparing Datasets and Comparing a Dataset with a Standard How different is enough?

module 72 Concepts: Independence of each data point Test statistics Central Limit Theorem Standard error of the mean Confidence interval for a mean Significance levels How to apply in Excel

module 73 Independent measurements: Each measurement must be independent (shake up the basket of tickets) Example of non-independent measurements: –Public responses to questions (one result affects the next person’s answer) –Samplers placed too close together so air flows are affected

module 74 Test statistics: Some number that is calculated based on the data In the student’s t test, for example, t If t is >= 1.96, and you have a normally distributed population, you know you are to the right on the curve where 95% of the data is in the inner portion is symmetrically between the right and left (t=1.96 on the right and on the left)

module 75 Test statistics correspond to significance levels “P” stands for percentile P th percentile is where p of the data falls below, and 1-p fall above:

module 76 Two major types of questions: Comparing the mean against a standard –Does the air quality here meet the NAAQS? Comparing two datasets –Is the air quality different in 2006 than 2005? –Or, is the air quality better? –Or, is the air quality worse?

module 77 Comparing mean to a standard: Did the air quality meet the CARB annual stnd of 12 microg/m3? year Ft Smith avg Ft Smith Min Ft Smith Max N_Fort Smith ‘

module 78 Central Limit Theorem (magic!) Even if the underlying population is not normally distributed If we repeatedly take datasets These different datasets will have means that cluster around the true mean And the distribution of these means is normally distributed!

module 79 magic concept #2: Standard error of the mean Represents uncertainty around the mean as sample size N gets bigger, your error gets smaller! The bigger the N, the more tightly you can estimate mean LIKE standard deviation for a population, but this is for YOUR sample

module 710 For a “large” sample (N > 60), or when very close to a normal distribution: A confidence interval for a population mean is: Choice of z determines 90%, 95%, etc.

module 711 For a “small” sample: Replace the Z value with a t value to get: where “t” comes from Student’s t distribution, and depends on the sample size.

module 712 Student’s t distribution versus Normal Z distribution

module 713 compare t and Z values:

module 714 What happens as sample gets larger?

module 715 What happens to CI as sample gets larger? For large samples: Z and t values become almost identical, so CIs are almost identical.

module 716 First, graph and review data: Use box plot add-in Evaluate spread Evaluate how far apart mean and median are (assume the sampling design and the QC are good)

module 717 Excel summary stats:

module 718 N=77 Min0.1 25th7.5 Median th18.1 Max37.9 Mean14.8 SD8.7 1.Use the box-plot add-in 2.Calculate summary stats

module 719 Our question: Can we be 95%, 90% or how confident that this mean of is really greater than the standard of 12? Saw that N = 77, and mean and median not too different Use z (normal) rather than t

module 720 The mean is what? We know the equation for CI is The width of the confidence interval represents how sure we want to be that this CI includes the true mean Now all we need to decide is how confident we want to be

module 721 CI calculation: For 95%, z = 1.96 (often rounded to 2) Stnd error (sigma/N) = (8.66/square root of 77) = 0.98 CI around mean = 2 x 0.98 We can be 95% sure that the mean is included in (mean +- 2), or at the low end, to at the high end This does NOT include 12 !

module 722 Excel can also calculate a confidence interval around the mean: The mean plus and minus 1.93 is a 95% confidence interval that does NOT include 12!

module 723 We know we are more than 95% confident, but how confident can we be that Ft Smith mean > 12? Calculate where on the curve our mean of 14.8 is, in terms of the z (normal) score, Or if N small, use the t score:

module 724 To find where we are on the curve, calc the test statistic: Ft Smith mean = 14.8, sigma =8.66, N =77 Calculate the test statistic, which in this case is the z factor (we decided we can use the z rather than the t distribution) If N was < 60, the test stat is t, but calculated the same way Data’s mean The stnd of 12

module 725 Calculate z easily: our mean 14.8 minus the standard of 12 (treat the real mean  (mu) as the stnd) is the numerator (= 2.8) The stnd error is sigma/square root of N = 0.98 (same as for CI) so z = (2.8)/0.98 = z = 2.84 So where is this z on the curve? Remember at z = 3 we are to the right of ~ 99%

module 726 Where on the curve? Z = 3 Z = 2 So between 95 and 99% probable that the true mean will not include 12

module 727 Can calculate exactly where on the curve, using Excel: Use Normsdist function, with z If z (or t) = 2.84, in Excel : Yields 99.8% probability that the true mean does NOT include 12