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Ch 6.1 Standard Normal Distribution

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    15900
  • Ch 6.1 Standard Normal distribution

    Normal Density Curve

    A random variable X has a distribution with a graph that is symmetric and bell-shaped, and it can be described by the equation given by

    \( y = \frac{e^{\frac{-1}{2}\cdot {(\frac{x-\mu}{\sigma})}^2}}{\sigma \sqrt{2 \pi}} \) , then it has a “Normal distribution”

    Normal Density Curve:

    Normal Density Curve

    Note: the distribution is determined by μ  and  σ.

     

     

     

    Z-score

    Z-score of x =  \( \frac{x-\mu}{\sigma} \) is the standardized value of x.

    z-score tells the number of standard deviation X is above or below the mean. Positive z implies X is above the mean, negative z implies X is below the mean.

     

    A) Standard Normal

      (also known as z distribution) is a normal distribution with parameters:

    Mean μ = 0 and standard deviation σ = 1.

     The total area under its density curve is equal to 1.

    Standard Normal density curve

    Properties of standard normal (z-normal):

    - area left of z of 0 = 0.5

    - area right of z of 0 = 0.5

    - area left of z =  area right of  - z

    - area right of z = 1 – area left of z

     

    B) Empirical Rule (68-95-99.7)

    If X is normally distributed, 68% are within 1 sd from the mean. 95% are within 2 sd from the mean, 99.7% are within 3 sd from the mean.

    P( Z-score between -1 and 1 ) = 68%

    P( Z-score between -2 and 2 ) = 95%

    P( Z-score between -3 and 3 ) = 99.7%

    Ex1.  Weight of a certain type of dog is normally distributed with mean = 18 lb. and standard deviation of 2 lb.

       write the marking of mean, mean - 1sd, mean - 2sd, mean + sd, mean + 2d on a number line.

    normal curve
    a) What is the z-score of 14lb and 22 lb? What is the probability that a dog weighs between 14 lb and 22lb?

    14 and 22 are 2 sd from the mean, so according to the Empirical rule, the probability is 95%.

    b) What is the z-score of 20lb and 16lb? What is the probability that a dog weighs between 16lb and 20 lb?

    16 and 20 are 1 sd from the mean, so according to the Empirical rule, the probability is 68%.

    c) What is the z-score of 24 lb and 12 lb? What is the probability that a dog weigh between 12 lb and 24lb?

                                                                         24 is 3 sd above the mean, so z-score of 24 lb is 3.

                                                                         12 is 3 sd below the mean, so z-score of 12 lb is -3.

    C) Probability of z-score in standard normal

     Use online Normal distribution calculator

    http://onlinestatbook.com/2/calculators/normal_dist.html

    Specify mean μ =0  standard deviation &sigma&

    -For left area or P(x < a) click below

    -For right area or P(x > a) click above

    -For area between two values a and b P( a < x < b), click between

    -For area outside of a and b, P(x < a or x > b), click outside

    -Click “Recalculate”

     

    Ex1: Find probability that z is between -1.8  and 1.8. Sketch the area.

    Bell curve

     

    Use online Normal calculator Mean = 0, SD = 1

    Click between, enter -1.8, 1.8

    Recalculate: P( -1.8 < z < 1.8 ) = 0.9281

     

    Ex2. Find the probability that z is less than 0.44. Sketch the area.

    Normal bell curve

     

    Use online Normal calculator μ =0 , SD=1

    Click below , enter 0.44

    Recalculate: P( z < 0.44 ) = 0.67

     

     Ex3. Find the probability that z is greater than 1.8.  Sketch the area.

    Normal bell curve

     

    Use online Normal calculator μ =0 , SD=1

    Click above , enter 1.8

    Recalculate: P( z >1.8) = 0.0359

     

    Ex 4. Find the probability that z is less than – 1.2. Sketch the area.

    clipboard_effa1b5f56d3aa1fc703e68fe8a610853.png

    Use online Normal calculator μ =0 , SD=1

    Click below , enter -1.2

    Recalculate: P( z < -1.2) = 0.1151

    D) Find percentile of a z-score

    kth percentile bell curvek Percentile corresponds to a value that is higher thank% of all values. Or k% of data are less than the k percentile value. This corresponds to left area of k%.

     

     

    Ex5. What percentile is the z-score 2.2?

    clipboard_eb50606309f43868fe67f6004ae0064a3.png

    Use online Normal calculator Mean =0 , SD=1

    Percentile is referring to 2.2 or less. Click below , enter 2.2

    Recalculate: P( z < 2.2) = 0.9861 = 98.6%

    Round to whole percent = 99th percentile

     

    Ex 6. What percentile is the z-score -1.35

    clipboard_eb50606309f43868fe67f6004ae0064a3.png

    Use online Normal calculator μ =0 , σ=1

    Click below , enter -1.35

    Recalculate: P( z < -1.35) = 0.0885 = 8.9%

    Round to whole percent. 9th percentile

     

    E) Find z-score given area or percentile.

     Online Inverse Normal calculator is use to find the z-score that is the cut-off for the left area, right area or percentile.

    http://onlinestatbook.com/2/calculators/inverse_normal_dist.html

    Specify area, mean= 0 and SD= 1

    Select if area is below, above, between or outside.

    Click “Recalculate”

     

    Ex1. Find the z-score that corresponds to bottom 10% of all values.

    Use Inverse Normal Calculator. Convert 10% to  0.1.

    Specify area = 0.1, Mean = 0, sd =1,

    Click below.

    Recalculate.   P( z < __-1.28_____ ) = 0.1,   cut-off z = -1.28

     

    Ex2.  Find the cutoff for top 20%.

    clipboard_eb50606309f43868fe67f6004ae0064a3.pngUse Inverse Normal Calculator. Convert 20% to  0.2.

    Specify area = 0.2

    Mean = 0, sd =1,

    Click above  ( for top percent). Recalculate.

    P( z >  __0.842______) = 0.2  z-cutoff = 0.84

     

    Ex3. Find P91 , 91th percentile of all z.

    clipboard_eb50606309f43868fe67f6004ae0064a3.png

    Use Inverse Normal calculator.

    Specify area = 0.91

    Mean = 0, sd =1,

    Click below. Recalculate.

    P( z <  __1.341______) = 0.91  91th percentile of all z = 1.34.

     

    Ex4. Find P15, 15th percentile of all z.

    Convert 15% = 0.15. Use Inverse Normal calculator.

    clipboard_eb50606309f43868fe67f6004ae0064a3.png

    Specify area = 0.15, Mean = 0, sd =1,

    Click below. Recalculate

     P( z <  _-1.036_______) = 0.15  15th percentile = -1.04

     

    F) Find Critical value Zα

    α = significant level. The probability of unlikely, default is 0.05 if not specify.

    Critical value zα : the positive z-value that separates significantly high values of  z with non-significant z.

    Significant level bell curveNote: Significantly low critical value = - zα 

     

     

     

    To find zα:  Use Inverse Normal calculator

    Specify area = α, Mean = 0, sd =1,

    Click above. Recalculate.

     

    Ex1. Given α = 0.02, Find the critical value Z0.02. 

    clipboard_eb50606309f43868fe67f6004ae0064a3.png

    Use Inverse Normal calculator

    Specify area = 0.02, Mean = 0, sd =1,

    Click above. Recalculate.

    Z0.02 = 2.054

     

    Ex2. Given α = 0.06, find the critical value Z0.06

    clipboard_eb50606309f43868fe67f6004ae0064a3.png

    Use Inverse Normal calculator

    Specify area = 0.06, Mean = 0, sd =1,

    Click above. Recalculate.

    Z0.06 = 1.555

     

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