2.2 Standard Normal Calculations

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Presentation transcript:

2.2 Standard Normal Calculations AP Statistics 2.2 Standard Normal Calculations

Learning Objectives: Calculate and interpret a z-score Standardize data to a standard normal distribution Calculate the proportion of observations under an interval on a normal curve

All normal distributions are the same if we measure in units of size σ about the mean μ as the center. Changing to these units is called: standardizing

Standardizing and z-scores If x is an observation from a distribution that has a mean μ and standard deviation σ, the standardized value of x is: ***A standardized value is called a z-score***

Ex: The heights of young women are approximately normal with μ=64 Ex: The heights of young women are approximately normal with μ=64.5 inches and σ=2.5 inches. The standardized height is: A women’s standardized height is the The # of standard deviation by which the height differs from the mean height of all young women.

What is the standardized height of a woman who is 68 inches tall? What does this mean? 1.4 standard deviation above the mean height

What about a woman who is 5 feet tall? What does this mean? 1.8 standard deviation below the mean height

What does z-scores do? Make all normal distributions into a single distribution. (with mean =0 and a standard deviation of 1)

Standard Normal Distribution The standard normal distribution is the normal distribution N(0,1) with μ= 0 and σ= 1.

Ex: SAT versus ACT- Who did better? Compare: Amy scored 680 on the math section of the SAT. The mean score is 500 with a standard deviation of 100. Tom scores 27 on the ACT. The test is normally distributed with a mean of 18 and a standard deviation of 6. Find the standardized scores for both students. Amy: Tom:

Show both scores on a standard normal curve and determine who did better!

To get on calc: 2nd/VARS/2/Normalcdf Normalcdf(1.6,1000,0,1) Ex: Find the proportion of observations from a standard normal distribution that are greater than 1.6? Solution: P(z>1.6)= 0.0548 To get on calc: 2nd/VARS/2/Normalcdf Normalcdf(1.6,1000,0,1)

Ex: Find the proportion of observation from a standard normal distribution that are less than -2.35? Solution: P(z<-2.35)= 0.0094 To get on calc: 2nd/VARS/2/Normalcdf Normalcdf(-1000,-2.35,0,1)

Step to Finding Normal Proportions Step 1: Draw a curve and shade your area of interest Step 2: Write in terms of x, then convert to a z-score. P(x>___ ) = P(z>___ ) Step 3: Find the probability Step 4: Write a conclusion in CONTEXT OF THE PROBLEM!!!!!

About 1% of 14 yr. old boys have cholesterol that is above 240 mg/dl. Ex: Blood cholesterol of 14 year old boys have a mean of 170 mg/dl with a standard deviation of 30 mg/dl. Any levels above 240 mg/dl require medical attention. What percent of 14 year old boys require medical attention? P(x>240)=P(z>2.3333)=0.0098 About 1% of 14 yr. old boys have cholesterol that is above 240 mg/dl.

Assessing Normality Method 1: Given data, make a graph (histogram ,stem plot, etc….) Method 2: Given statistics (no actual data), assume normality.