Lecture 9 Dustin Lueker

  Perfectly symmetric and bell-shaped Characterized by two parameters ◦ Mean = μ ◦ Standard Deviation = σ  Standard Normal ◦ μ = 0 ◦ σ = 1  Solid Line STA 291 Summer 2008 Lecture 9 2

 ◦ For a normally distributed random variable, find the following P(Z>.82) = ◦ P(-.2

 For a normal distribution, how many standard deviations from the mean is the 90 th percentile?

◦ What is the value of z such that 0.90 probability is less than z?

◦   P(Z

If 0.9 probability is less than z, then there is 0.4 probability between 0 and z  Because there is 0.5 probability less than 0  This is because the entire curve has an area under it of 1, thus the area under half the curve is 0.5

z=1.28

 The 90 th percentile of a normal distribution is 1.28 standard deviations above the mean STA 291 Summer 2008 Lecture 9 4

  We can also use the table to find z-values for given probabilities ◦ Find the following P(Z>a) = .7224

 a = ◦  P(Z

b = STA 291 Summer 2008 Lecture 9 5

  When values from an arbitrary normal distribution are converted to z-scores, then they have a standard normal distribution The conversion is done by subtracting the mean μ, and then dividing by the standard deviation σ

z



x

   STA 291 Summer 2008 Lecture 9 6

   The z-score for a value x of a random variable is the number of standard deviations that x is above μ ◦ If x is below μ, then the z-score is negative The z-score is used to compare values from different normal distributions ◦ Calculating Need to know    x μ σ

z



x

   STA 291 Summer 2008 Lecture 9 7

 SAT Scores ◦ μ=500 ◦     σ=100 SAT score 700 has a z-score of z=2 Probability that a score is above 700 is the tail probability of z=2 Table 3 provides a probability of 0.4772 between mean=500 and 700  z=2 Right-tail probability for a score of 700 equals 0.5-0.4772=0.0228

 2.28% of the SAT scores are above 700 ◦ Now find the probability of having a score below 450 STA 291 Summer 2008 Lecture 9 8

 The z-score is used to compare values from different normal distributions ◦ SAT ◦    μ=500  σ=100 ACT μ=18 σ=6

z z SAT ACT

 

x x

        100 6   1.5

1.17

◦  What is better, 650 on the SAT or 25 on the ACT?

Corresponding tail probabilities?

  How many percent have worse SAT or ACT scores?

In other words, 650 and 25 correspond to what percentiles?

STA 291 Summer 2008 Lecture 9 9

 The scores on the Psychomotor Development Index (PDI) are approximately normally distributed with mean 100 and standard deviation 15. An infant is selected at random.

◦ Find the probability that the infant’s PDI score is at least 100 ◦ ◦   P(X>100) Find the probability that PDI is between 97 and 103  P(97

STA 291 Summer 2008 Lecture 9 10