2.2 Continued 9.12.2017

Normalcdf in reverse Normalcdf (or Table A) tells us what proportion of the area under a normal distribution is between two values InvNorm tells us the value that has a certain proportion of the area to the left of it (less than it) So if we want to know the score on our chapter 1 test for which 75% of people scored lower 75th percentile, or 3rd quartile invNorm(.75,68,14)=77.44 What if we want to know the score for which 75% scored higher?

Normalcdf in reverse Normalcdf (or Table A) tells us what proportion of the area under a normal distribution is between two values InvNorm tells us the value that has a certain proportion of the area to the left of it (less than it) So if we want to know the score on our chapter 1 test for which 75% of people scored lower 75th percentile, or 3rd quartile invNorm(.75,68,14)=77.44 What if we want to know the score which 75% scored higher? InvNorm(.25,68,14)=58.56

Assessing Normality Many variables can be safely approximated by a Normal distribution Test scores, for example Some other variables often have skewed distributions, so a Normal curve is not a good approximation Income, housing prices, survival times of cancer patients, etc. So we want to be able to assess whether a distribution is (approximately) Normal before we use any calculations like 68-95-99.7 , Table A, normalcdf, or invNorm

Assessing Normality– Option #1 LOOK AT THE DISTRIBUTION Histogram, dotplot, or density curve Does it look Normal? Is it skewed? If yes, not Normal Does it have multiple peaks? Or no peaks? Does it look bell-shaped? If no, not Normal

Assessing Normality—Option #2 Use the 68-95-99.7 Rule Hypothetically, let’s say that you have some data, and 69% are within one standard deviation of the mean, 93% are within two standard deviations of the mean, and 100% are within 3 standard deviations of the mean This is very close to what we would expect from a Normal distribution, so we could say that these data are consistent with a Normal Distribution

Assessing Normality—Option #3 Normal Probability plot Plots Z-scores on the y-axis and values for your variable on the x-axis Some people flip these If the points form a perfectly straight line, then it is perfectly Normal If it is close to being straight, with only small deviations, then we can still say that it is approximately Normal

Approx. Normal Skewed

Chapter 1 Test Scores

How did the Curve Work? Raw scores: Mean: 68.43 St. Dev: 13.52 Max: 96 So the z-score for someone who got a 96 would be (96-68.43)/13.52= 2.04 But I wanted our scores to have a mean of 80, and I wanted to have the highest score be 100 So I created a new Distribution with a mean of 80 But if I kept the same standard deviation, then the person with the 96 would now have above 107% --not what I want So I had to figure out a new standard deviation

The Curve To find the new standard deviation, I am just finding what standard deviation would make that highest score (96) become 100, when the mean is changed to 80 That ends up being 9.81 So now we have changed the mean from 68.43 to 80, and the standard deviation from 13.52 to 9.81

The Curve I calculate everybody’s z-score, and then use it to put them on the new distribution So for someone who got a raw score of 91, their z-score was 1.67 ( (91-68.43)/13.52) Then we can use the z-score formula to put that person onto the new distribution 1.67=((X-80)/9.81) X=96.38 So a 91 raw score was curved to a 96.38

The Curve We can go in the other direction too If your curved score was 76.79 Z=((76.79-80)/9.81) = -.33 -.33=((X-68.43)/13.52) = 64