Lecture 9 Dustin Lueker

2  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 2010 Lecture 9

3  For a normally distributed random variable, find the following ◦ P(Z>.82) = ◦ P(-.2<Z<2.18) = STA 291 Summer 2010 Lecture 9

4  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<z) =.90 ◦ 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 2010 Lecture 9

5  We can also use the table to find z-values for given probabilities  Find the following ◦ P(Z>a) =.7224  a = ◦ P(Z<b) =.2090  b = STA 291 Summer 2010 Lecture 9

6  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

STA 291 Summer 2010 Lecture 97  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

STA 291 Summer 2010 Lecture 98  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 between mean=500 and 700  z=2  Right-tail probability for a score of 700 equals =  2.28% of the SAT scores are above 700 ◦ Now find the probability of having a score below 450

STA 291 Summer 2010 Lecture 99  The z-score is used to compare values from different normal distributions ◦ SAT  μ=500  σ=100 ◦ ACT  μ=18  σ=6 ◦ 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 2010 Lecture 910  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<X<103) ◦ Find the probability that PDI is less than 90  Would you be surprised to observe a value of 90?