MATH104- Ch. 12 Statistics- part 1CNormal Distribution

Normal Distribution

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Excerpt of Normal Chart– see p. 720

Z-score percentile Z-score percentile

-4.0 0.003 0.0 50.00

-3.0 0.13 0.5 69.15

-2.0 2.28 1.0 84.13

-1.0 15.87 2.0 97.72

-0.5 30.85 3.0 99.87

4.0 99.997

Find the given probabilities

Easiest examples• P(z<1)=

• P(z<2)=

• P(z< -2)=

• P(z<1.5)=

Harder examples

• Recall P(z<1)= Try P(z>1)=

• Recall P(z<2)= Try P(z>2)=

• P(z>1.3)=

• P(z> -2.4)=

Find the given probabilities• P( - 1<z<1)=

• P( -2<z<2)=

• P( -1.5<z<1.5)=

• P(1.4 < z < 2.3)=

• P( -1.8<z< 2.3)=

• P(- 2.1<z< -0.7)=

Normal Distribution Problems– Given x, find z, and then find P

• Example #1:• Scores on a standardized test are normal with

the mean = μ= 100 and the pop st dev =σ= 10. Create a normal curve to picture this example.

μ = 100 , σ = 10 Find the probably that scores are:

• Lower than 100

• Lower than 110

• Greater than 110

• Between 90 and 110

• Between 80 and 120…

Continued… Find the probability that scores are:

• Lower than 115

• Greater than 115

• Lower than 108

Calculate z using the formula, and then find probability

• Lower than 93

• Between 93 and 108

• Hint: z =

Example #2-snowfall

• Assume snowfall amounts are normally distributed with mean μ =140, st dev = σ = 20. Find the probability that the amount is:

• Less than 180 inches

• Greater than 162 inches

• Between 134 and 174 inches

Ex 3: Heights- mean = 48 , st dev= 4

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Margin of error (p. 725)—

if a statistic is obtained from a random sample of size n, there is a 95% probability that it lies within of the true populations statistic, where is called the margin of error.

If 1100 people were surveyed about a politician, and 61%

thought favorably of this person, the margin of error would be: So, there is a 95% probability that the true population percentage

is between: