Probability and Statistical Inference

1. Describe the $${{{{z-}}}}$$score as the quotient between the difference of an observed value and the mean by the standard deviation. This is represented by $${{\mathrm{observation-mean}}\over{\mathrm{standard\space deviation}}}$$.
2. Identify the $${{{{z-}}}}$$score from a $${{{{z-}}}}$$score table and convert to a percentile.
3. Describe that the percentile represents the percentage of the population that has scored below the observed value. 
4. Use the normal curve to analyze statistical measures including the mean, standard deviation, and $${{{{z-}}}}$$score. 

• One useful place to find $${{{{{{z-}}}}}}$$score tables is here: http://www.z-table.com/
• Anchor Problem #1 is just a problem to get the initial thinking established and provide an intellectual need for $${{{{{{z-}}}}}}$$score. The majority of the learning will happen through the guiding questions, where the teacher needs to introduce the formula for $${{{{{{z-}}}}}}$$score. 
• EngageNY, Algebra 2, Module 4, Lesson 10 has student answers that show the marking of the mean, the standard deviation, the value of interest, and then noting the $${{{{{{z-}}}}}}$$score. This is a great strategy to ensure that students have a visual representation of the problem they are working on. This is highlighted (out of context) in Anchor Problem #2 but should be represented with context in the Problem Set. 
• Reading a $${{{{{{z-}}}}}}$$score table can be a challenge. Be sure to model to students how to do this rather than have them infer this process.
• If you want more information about the SAT scoring to use with students, this document provides a good amount of information: SAT Understanding Scores 2017.
• Be sure to equate the population percentage calculated through the $${{{{{{z-}}}}}}$$score table with the percentile and clarify what this actually means. 

The weights of cars passing over a bridge have a mean of 3,550 pounds and standard deviation of 870 pounds. Assume that the weights of the cars passing over the bridge are normally distributed. Determine the probability of each instance and explain how you found each answer. 

1. The weight of a randomly selected car is more than 4,000 pounds. 
2. The weight of a randomly selected car is less than 3,000 pounds. 
3. The weight of a randomly selected car is between 2,800 and 4,500 pounds.