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Understanding Z-scores and Standard Normal Distribution

Learn how to calculate and interpret z-scores, and understand the standard normal distribution. Solve problems related to gas bills, SAT and ACT scores, and data distributions.

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Understanding Z-scores and Standard Normal Distribution

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  1. AP StatisticsHW: p95 22 – 24, 27 (skip part c)Obj: to understand and use a z-score and standard normal distribution Do Now: The mean monthly cost of gas is $125 with a standard deviation of $10. The distribution of the gas bills is approximately normal. • What percentage of homes have a monthly bill of more than $115? • Less than $115? • What bill amount represents the top 16 percent? • What bill amount represents the top 84% • How many standard devations above the mean is a bill of $150? C2 D4

  2. Z-Score (standardized value) • Allows us to identify the position of a data value relative to the μ and σ of its set of data values. z = x – μ σ

  3. Ex: If x = 13.75, μ = 10, and σ = 2.5, then the z-score = 13.75 – 10 2.5 This means that 13.75 is 1.5 standard deviations above the mean of 10

  4. Ex: If x = 100, μ = 120, and σ = 15, calculate the z-score and tell what it means.

  5. If a variable x, which takes on the values x = {x1, x2, …, xn} has a normal distribution N(μ, σ) and we change every data value into its standardized score (z-score), this new variable z takes on the values z = {z1, z2, …, zn} and has the normal distribution N(0, 1) which we call the standard normal distribution

  6. Standard Normal Dist’n

  7. Ex: A student scores 625 on the math section of the SAT and a 28 on the math section of the ACT. She can only report one score to her college. If the SAT summary statistics include μ = 490 and σ = 100 and the ACT summary statistics include μ = 21 and σ = 6, which score should she report?

  8. Ex: For data with a distribution N(0,1) calculate the following percentages: • % of data values between -1 and 1. • % of data values less than 1. • % of data values greater than -1. • % of data values less than 2. • For what data values are 99.85% of the scores lower? • Do p.95 #19, 20

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