2.2 Standard Normal Calculations Get that Program!! FLIP50

Does It Make Sense Frankie Farmer ‘s wife, Helga, has been bugging him with some serious financial concerns about their farm. She wants to cut back on some crops and animals they raise and she wants Frankie to choose. Here are some of the dilemma’s Helga has hit him with… Vs. Variable POO x Vs. Milk x So How do We Compare the Seemingly Incomparable?

…Using the Standard Normal Distribution  This is a normal distribution with a µ = 0 and σ = 1 If we change our Data into values that fit this curve, then we can compare the incomparable!!! Z- Score x = observation; µ = mean of that distribution σ = std dev of that distribution

What’s A Z Score? Z Score: An observation’s relative position to it’s mean Tells how many STANDARD UNITS an observation is from the mean Average Dairy Cow Daily Poo Production Let‘s look at dairy cow poo production for example… Frankie Farmer’s best cow Bessy produced 147 lbs of poo today. Where is she compared to the rest of the cows? 1.4 Std units ABOVE THE MEAN

This will allow us to help Frankie!! Let’s look at two of Frankie’s star “Poo”er’s. Chicken Little Average’s 1.6lbs of poo per day Overall Chicken Average: 0.9lbs Std Dev:.3lbs Bessy Average’s 152lbs of poo per day Overall Dairy Cow Average: 140lbs Std Dev: 7lbs Which one is the better “Pooer”?

Better or Worse?  A more positive Z score isn’t always better!!  What are some situations where that’s true?

Finding Probabilities using table A Change your value to a Z -Score Draw Std Normal Curve and mark position of z value(s) x Shade “area” looking for (Table A gives this area…) Find area in table A, do appropriate math to find shaded region (Table A is only for areas of less than z)

Table Practice (straight) 1) Find the following probabilities using the table: z < -.43 z < ) Find the following probabilities using the table in a distribution with µ= 64.5 and σ = 2.5: x < 68 x <

Finding Probabilities for “other” Regions  Fancy Math Subtracting different regions  Visualization z > 1.51z <

Finding Probabilities for “other” Regions  Find the area for < z < 2.1 Visualize, find table #’s, do the math  Visualization < z < 2.1z < 2.1z <

Back to the Farm!!! Helga has decided to buy a new goat to help increase goat milk production on the farm. Being the frugal farmer’s wife, she’s decided to calculate the probability of finding a goat that’s better than her best milker, GERTIE. Help her out!! Gertie Produces 4.1 kg of milk per day Natl. Avg = 4.5kg Std Dev. =.25kg =.9452 That means there’s a 94.52% chance that Helga will find a goat that milks better than Gertie! That’s pretty good, jabroney!!!

Going from z -> x  Sometimes you have a Z-Score and need to know what the real (raw) score is: Willy Whitehorse has been rumored around the barn to produce 2.7 std units of poo. Frankie would like to know how much poo he’ll be shoveling out tonight!!! Natl. Avg = 67lbs. Std Dev. = 4.3lbs 78.61lbs of POO!!!

Are They Normal?  Method 1 – Histogram/StemLeaf Construct a histogram or stemplot Look for symmetry about the mean and bell shapedeness Also, mark the 68,95 points on the graph and get a “count” to see if it matches up ○ See if there are the correct percentage of values that fit within the parameters for those percentages (1 or 2σ’s) **Difficult to assess for small data sets

Are They Normal? Method 2 – Normal Probability Plot (Calculator) Put Data into a List 1 Var Stats to compare Mean, Median Boxplot to look for symmetry (or Histogram) Normal Probability Plot is the last graph under “Type” in the stat plot screen Zoom Stat to finish the graph If the points shown from a “line” pattern, the data is said to be normal.

Are They Normal? Flip 50 Program  Run the Flip 50 Program  Fix the window on your histogram (Xscale)  Look at histogram for normality  Stat Plot (see screen capture)  Zoom 9 to see the Normal Probability Plot The Flip50 Program flips a coin 50 times and records the # of heads. It does this experiment 100 times and records the results.

Homework #’s 28-33,36,42-48; Worksheet