chapter 7 section 2

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 1 of 32

Chapter 7Section 2

The StandardNormal Distribution

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 2 of 32

Chapter 7 – Section 2

● Learning objectives Find the area under the standard normal curve Find Z-scores for a given area Interpret the area under the standard normal curve

as a probability

1

2

3

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 3 of 32

Chapter 7 – Section 2

● Learning objectives Find the area under the standard normal curve Find Z-scores for a given area Interpret the area under the standard normal curve

as a probability

1

2

3

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 4 of 32

Chapter 7 – Section 2

● The standard normal curve is the one with mean μ = 0 and standard deviation σ = 1

● We have related the general normal random variable to the standard normal random variable through the Z-score

● In this section, we discuss how to compute with the standard normal random variable

X

Z

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 5 of 32

Chapter 7 – Section 2

● There are several ways to calculate the area under the standard normal curve What does not work – some kind of a simple formula We can use a table (such as Table IV on the inside

back cover) We can use technology (a calculator or software)

● Using technology is preferred

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 6 of 32

Chapter 7 – Section 2

● Three different area calculations Find the area to the left of Find the area to the right of Find the area between

● Three different area calculations Find the area to the left of Find the area to the right of Find the area between

● Three different methods shown here From a table Using Excel Using StatCrunch

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 7 of 32

Chapter 7 – Section 2

● "To the left of" – using a table● Calculate the area to the left of Z = 1.68

● "To the left of" – using a table● Calculate the area to the left of Z = 1.68

Break up 1.68 as 1.6 + .08

● "To the left of" – using a table● Calculate the area to the left of Z = 1.68

Break up 1.68 as 1.6 + .08 Find the row 1.6

Enter

● "To the left of" – using a table● Calculate the area to the left of Z = 1.68

Break up 1.68 as 1.6 + .08 Find the row 1.6 Find the column .08

Enter

Enter

● "To the left of" – using a table● Calculate the area to the left of Z = 1.68

Break up 1.68 as 1.6 + .08 Find the row 1.6 Find the column .08

● The probability is 0.9535 Read

Enter

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 8 of 32

Chapter 7 – Section 2

● "To the left of" – using Excel● The function in Excel for the standard normal is

=NORMSDIST(Z-score) NORM (normal) S (standard) DIST (distribution)

Enter

Read

Read

Enter

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 9 of 32

Chapter 7 – Section 2

● To the left of 1.68 – using StatCrunch● The function is Stat – Calculators – Normal

EnterRead

Enter

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 10 of 32

Chapter 7 – Section 2

● "To the right of" – using a table● The area to the left of Z = 1.68 is 0.9535

Read

Enter

Enter

● "To the right of" – using a table● The area to the left of Z = 1.68 is 0.9535

● The right of … that’s the remaining amount● The two add up to 1, so the right of is

1 – 0.9535 = 0.0465

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 11 of 32

Chapter 7 – Section 2

● "To the right of" – using ExcelRead

Read

Enter

● "To the right of" – using Excel

● The right of … that’s the remaining amount of to the left of, subtract from 1

1 – 0.9535 = 0.0465

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 12 of 32

Chapter 7 – Section 2

● "To the right of" – using StatCrunch

ReadEnter

● "To the right of" – using StatCrunch● Change the <= to >=

● "To the right of" – using StatCrunch● Change the <= to >= (the picture and number

change)

ReadEnter

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 13 of 32

Chapter 7 – Section 2

● “Between”● Between Z = – 0.51 and Z = 1.87● This is not a one step calculation

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 14 of 32

Chapter 7 – Section 2

● The left hand picture … to the left of 1.87 … includes too much

● It is too much by the right hand picture … to the left of -0.51

Includedtoo much

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 15 of 32

Chapter 7 – Section 2

● Between Z = – 0.51 and Z = 1.87

We want

We start out with,but it’s too much

We correct by

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 16 of 32

Chapter 7 – Section 2

● We can use any of the three methods to compute the normal probabilities to get

● The area between -0.51 and 1.87

● We can use any of the three methods to compute the normal probabilities to get

● The area between -0.51 and 1.87 The area to the left of 1.87, or 0.9693 … minus

● We can use any of the three methods to compute the normal probabilities to get

● The area between -0.51 and 1.87 The area to the left of 1.87, or 0.9693 … minus The area to the left of -0.51, or 0.3050 … which

equals

● We can use any of the three methods to compute the normal probabilities to get

● The area between -0.51 and 1.87 The area to the left of 1.87, or 0.9693 … minus The area to the left of -0.51, or 0.3050 … which

equals The difference of 0.6643

● We can use any of the three methods to compute the normal probabilities to get

● The area between -0.51 and 1.87 The area to the left of 1.87, or 0.9693 … minus The area to the left of -0.51, or 0.3050 … which

equals The difference of 0.6643

● Thus the area under the standard normal curve between -0.51 and 1.87 is 0.6643

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 17 of 32

Chapter 7 – Section 2

● A different way for “between”

We want

We delete theextra on the left

We delete theextra on the right

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 18 of 32

Chapter 7 – Section 2

● Again, we can use any of the three methods to compute the normal probabilities to get

● Again, we can use any of the three methods to compute the normal probabilities to get

● The area between -0.51 and 1.87 The area to the left of -0.51, or 0.3050 … plus The area to the right of 1.87, or .0307 … which equals The total area to get rid of which equals 0.3357

● Again, we can use any of the three methods to compute the normal probabilities to get

● The area between -0.51 and 1.87 The area to the left of -0.51, or 0.3050 … plus The area to the right of 1.87, or .0307 … which equals The total area to get rid of which equals 0.3357

● Thus the area under the standard normal curve between -0.51 and 1.87 is 1 – 0.3357 = 0.6643

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 19 of 32

Chapter 7 – Section 2

● Learning objectives Find the area under the standard normal curve Find Z-scores for a given area Interpret the area under the standard normal curve

as a probability

1

2

3

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 20 of 32

Chapter 7 – Section 2

● We did the problem:

Z-Score Area● Now we will do the reverse of that

Area Z-Score

● We did the problem:

Z-Score Area● Now we will do the reverse of that

Area Z-Score● This is finding the Z-score (value) that

corresponds to a specified area (percentile)● And … no surprise … we can do this with a

table, with Excel, with StatCrunch, with …

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 21 of 32

Chapter 7 – Section 2

● “To the left of” – using a table● Find the Z-score for which the area to the left of

it is 0.32

● “To the left of” – using a table● Find the Z-score for which the area to the left of

it is 0.32 Look in the middle of the table … find 0.32

Find

Read

Read

● “To the left of” – using a table● Find the Z-score for which the area to the left of

it is 0.32 Look in the middle of the table … find 0.32

The nearest to 0.32 is 0.3192 … a Z-Score of -.47

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 22 of 32

Chapter 7 – Section 2

● "To the left of" – using Excel● The function in Excel for the standard normal is

=NORMSINV(probability) NORM (normal) S (standard) INV (inverse)

Read

Read

Enter

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 23 of 32

Chapter 7 – Section 2

● "To the left of" – using StatCrunch● The function is Stat – Calculators – Normal

Read Enter

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 24 of 32

Chapter 7 – Section 2

● "To the right of" – using a table● Find the Z-score for which the area to the right of

it is 0.4332● Right of it is .4332 … left of it would be .5668● A value of .17

Enter

Read

Read

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 25 of 32

Chapter 7 – Section 2

● "To the right of" – using Excel● To the right is .4332 … to the left would be .5668

(the same as for the table)

Read

Read

Enter

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 26 of 32

Chapter 7 – Section 2

● "To the right of" – using StatCrunch● Change the <= to >= (watch the picture change)

Read Enter

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 27 of 32

Chapter 7 – Section 2

● We will often want to find a middle range, to find the middle 90% or the middle 95% or the middle 99%, of the standard normal

● The middle 90% would be

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 28 of 32

Chapter 7 – Section 2

● 90% in the middle is 10% outside the middle, i.e. 5% off each end

● These problems can be solved in either of two equivalent ways

● We could find The number for which 5% is to the left, or The number for which 5% is to the right

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 29 of 32

Chapter 7 – Section 2

● The two possible ways The number for which 5% is to the left, or The number for which 5% is to the right

5% is to the left 5% is to the right

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 30 of 32

Chapter 7 – Section 2

● The number zα is the Z-score such that the area to the right of zα is α

● The number zα is the Z-score such that the area to the right of zα is α

● Some useful values are z.10 = 1.28, the area between -1.28 and 1.28 is 0.80

z.05 = 1.64, the area between -1.64 and 1.64 is 0.90

z.025 = 1.96, the area between -1.96 and 1.96 is 0.95

z.01 = 2.33, the area between -2.33 and 2.33 is 0.98

z.005 = 2.58, the area between -2.58 and 2.58 is 0.99

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 31 of 32

Chapter 7 – Section 2

● Learning objectives Find the area under the standard normal curve Find Z-scores for a given area Interpret the area under the standard normal curve

as a probability

1

2

3

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 32 of 32

Chapter 7 – Section 2

● The area under a normal curve can be interpreted as a probability

● The standard normal curve can be interpreted as a probability density function

● The area under a normal curve can be interpreted as a probability

● The standard normal curve can be interpreted as a probability density function

● We will use Z to represent a standard normal random variable, so it has probabilities such as P(a < Z < b) P(Z < a) P(Z > a)

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 33 of 32

Summary: Chapter 7 – Section 2

● Calculations for the standard normal curve can be done using tables or using technology

● One can calculate the area under the standard normal curve, to the left of or to the right of each Z-score

● One can calculate the Z-score so that the area to the left of it or to the right of it is a certain value

● Areas and probabilities are two different representations of the same concept

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 34 of 32

Example: Chapter 7 – Section 2

● Determine the area under the standard normal curve that lies

a. to the left of Z = –2.31. (0.0104)

b. to the right of Z = –1.47. (0.9292)

c. between Z = –2.31 and Z = 0. (0.4896)

d. between Z = –2.31 and Z = –1.47. (0.0603)

e. between Z = 1.47 and Z = 2.31. (0.0603)

f. between Z = –2.31 and Z = 1.47. (0.9188)

g. to the left of Z = –2.31 or to the right of Z = 1.47. (0.0812)

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 7 Section 2 – Slide 35 of 32

Example: Chapter 7 – Section 2

● The Graduate Record Examination (GRE) is a test required for admission to many U.S. graduate schools. The Department of Molecular Genetics at Ohio State University requires a GRE score no less than the 60th percentile. (Source: www.biosci.ohio-state.edu/~molgen/html/admission_criteria.html.)

a. Find the Z-score corresponding to the 60th percentile. In other words, find the Z-score such that the area under the standard normal curve to the left is 0.60. (0.25)

b. How many standard deviations above the mean is the 60th percentile? (0.25)