Practice Using the Z-Table

Tiger on the Range On the driving range, Tiger Woods practices his swing with a particular club by hitting many, many balls. When Tiger hits his driver, the distance the ball travels follows a Normal distribution with mean 304 yards and standard deviation 8 yards. What percent of Tiger’s drives travel at least 290 yards? Let x = the distance that Tiger’s ball travels. The variable x has a Normal distribution with μ = 304 and σ = 8. We want the proportion of Tiger’s drives with x ≥ 290 Hint: Find the z-score first Then use the z-table (Table A)

Hint: DRAW A PICTURE!

Tiger on the Range (Continued) What percent of Tiger’s drives travel between 305 and 325 yards? As in the previous example, let x = the distance that Tiger’s golf ball travels. We want the proportion of Tiger’s drives with 305 ≤ x ≤ 325

Hint: Find the z-scores! Then find the corresponding percentages and subtract accordingly …

More Practice Page 131 41-43 47-50 54 70-72

Example Suppose SAT math scores are nearly normally distributed with a mean of 550 and a standard deviation of 30

Example You are finding the area to the left of 600 Suppose you score a 600, how many people did you do better than? You are finding the area to the left of 600

Example To calculate this, we will need to calculate a z-score and use the “z-table”

Example =550,  = 30, y = 600  z=(600-550)/30 = 1.67 Area to left of 1.67 is 0.9525 thus you are in the 95th percentile

Example You can also go the other way… meaning if you want to know what the 75th percentile is, you : Look up where 0.7500 is on the z-table, then find the z-score by finding the column and row header.. Between z=0.67 and z=0.68 so lets call it z=0.675

So… z =(y- )/  0.675 = (y-550)/30 <--- solve for y Therefore the 75th percentile is a score of 570.25. NOTE: The area under the standard normal curve, below a certain z-value is commonly written as: p(z<0.675) = 0.75

Using Ti-83s for Areas WITHOUT Z-Scores Press 2nd VARS [DISTR]. Scroll down to 2:normalcdf( Press ENTER The proper syntax (arguments) are: normalcdf(lower bound, upper bound, µ, σ)

Using Ti-83s for Areas WITHOUT Z-Scores For our purposes, if our smaller bound is the beginning of the normal curve, then we will put “1E -99”. This means 1x10-99. The E is done by hitting EE, or 2nd -> “comma”. This implies that the lower bound is a tiny number. If our upper bound is the end of the normal curve, then we will put “1E99”. This implies that the upper bound is a huge number.

Practice So going back to our Tiger Woods example … We wanted to find the probability here that he would hit a golf ball further than 290. Our calculator arguments would therefore look like: normalCDF(290, 1E99, 304, 8) Because our arguments are: normalCDF(lower bound, upper bound, mean, SD)

Practice How would we enter the other Tiger Woods problem into the calculator?

Using Ti-83s for Areas WITH Z-Scores Press 2nd VARS [DISTR]. Scroll down to 2:normalcdf( Press ENTER The proper syntax (arguments) are: normalcdf(lower bound z-score, upper bound z-score)

Using Ti-83s for Areas WITH Z-Scores For example, if I wanted to find the area between z-scores of 1 and 2, I would enter: And that tells me the area here: 0.136, or 13.6% of the population

Using Ti-83s for Areas WITH Z-Scores Suppose now, I wanted to find the area from 1 and up. What would I enter?

Assessing Normality How do we know if something is normal so that we can actually use the standard normal model??? What we can do is make a NORMAL PROBABILITY PLOT

Normal Probability Plot First, you have to have your data in one of the lists (ideally list L1). Then, you have to plot a NORMAL PROBABILITY PLOT. Go to Stat Plot and for the Type, choose the last one.

Normal Probability Plot Let’s check whether data for unemployment in the USA is normal. The following information is the unemployment level in every state. 4.1 4.5 5.0 6.3 6.3 6.4 6.4 6.6 6.7 6.7 6.7 6.9 7.0 7.0 7.2 7.4 7.4 7.4 7.8 8.0 8.0 8.2 8.2 8.4 8.5 8.5 8.6 8.7 8.8 8.9 9.1 9.2 9.5 9.6 9.6 9.7 10.2 10.3 10.5 10.6 10.6 10.8 10.9 11.1 11.5 12.3 12.3 12.3 12.7 14.7 First, use the calculator to create a histogram in List L1. Does it appear normal? Second, let’s use the calculator to create a normal probability plot. Remember to press Zoom -> 9 for ZOOMSTAT to fit the graph into the window

Finished Normal Probability Plot Your finished plot should look something like this:

What Does This Tell Us? If the points on a Normal probability plot lie close to a straight line, the plot indicates that the data are Normal. Systematic deviations from a straight line indicate a non-Normal distribution. Outliers appear as points that are far away from the overall patter of the plot. Based on our Normal Probability Plot, we can say that “There is a strong linear pattern, which suggests that the distribution of unemployment rates is close to Normal.” We will always be using language like this now, NEVER say that the data IS NORMAL because it’s rarely ever perfectly normal. It only is “CLOSE TO NORMAL,” or even better yet, “APPEARS NORMAL.”

What a non-Normal Probability Plot Looks Like Guinea Pig Survival Rates NON-LINEAR!