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©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture 1
Lecture 9A: Chapter 7, Section 3 Continuous Random Variables; Normal Distribution o Relevance of Normal Distribution o Continuous Random Variables o 68-95-99.7 Rule for Normal R.V.s o Standardizing/Unstandardizing o Probabilities for Standard/Non-standard Normal R.V.s
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.2
Looking Back: Review
o 4 Stages of Statistics n Data Production (discussed in Lectures 1A-2B) n Displaying and Summarizing (Lectures 3A-6B) n Probability
o Finding Probabilities (discussed in Lectures 7A-7B) o Random Variables (introduced in Lecture 8A)
n Binomial (discussed in Lecture 8B) n Normal
o Sampling Distributions n Statistical Inference
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.3
Role of Normal Distribution in Inference n Goal: Perform inference about unknown
population proportion, based on sample proportion
n Strategy: Determine behavior of sample proportion in random samples with known population proportion
n Key Result: Sample proportion follows normal curve for large enough samples.
Looking Ahead: Similar approach will be taken with means.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.4
Discrete vs. Continuous Distributions n Binomial Count X
o discrete (distinct possible values like numbers 1, 2, 3, …)
n Sample Proportion o also discrete (distinct values like count)
n Normal Approx. to Sample Proportion o continuous (follows normal curve) o Mean p, standard deviation
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.5
Sample Proportions Approx. Normal (Review)
n Proportion of tails in n=16 coinflips (p=0.5) has
n Proportion of lefties (p=0.1) in n=100 people has , shape approx normal
, shape approx normal
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.6
Example: Variable Types o Background: Variables in survey excerpt:
o Question: Identify type (cat, discrete quan, continuous quan) n Age? Breakfast? Comp (daily min. on computer)? Credits?
o Response: n Age: ___________________________ n Breakfast: ______________________ n Comp (daily time in min. on computer): _______________ n Credits: _________________________
Practice: 7.43 p.330
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.7
Probability Histogram for Discrete R.V. Histogram for male shoe size X represents
probability by area of bars n P(X ≤ 9) (on left) n P(X < 9) (on right)
For discrete R.V., strict inequality or not matters.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.8
Definition Density curve: smooth curve showing prob. dist. of
continuous R.V. Area under curve shows prob. that R.V. takes value in given interval.
Looking Ahead: Most commonly used density curve is normal z but to perform inference we also use t, F, and chi-square curves.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.9
Density Curve for Continuous R.V. Density curve for male foot length X represents
probability by area under curve.
Continuous RV: strict inequality or not doesn’t matter.
A Closer Look: Shoe sizes are discrete; foot lengths are continuous.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.10
68-95-99.7 Rule: Normal Random Variable Sample at random from normal population; for sampled value X
(a R.V.), probability is o 68% that X is within 1 standard deviation of mean o 95% that X is within 2 standard deviations of mean o 99.7% that X is within 3 standard deviations of mean
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.11
Example: 68-95-99.7 Rule for Normal R.V.
o Background: IQ for randomly chosen adult is normal R.V. X with .
o Question: What does Rule tell us about distribution of X ? o Response: We can sketch distribution of X:
______________________________________________ Practice: 7.47 p.331
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.12
Example: Finding Probabilities with Rule
o Background: IQ for randomly chosen adult is normal R.V. X with .
o Question: Prob. of IQ between 70 and 130 = ? o Response: _____
Practice: 7.48a p.331
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.13
Example: Finding Probabilities with Rule
o Background: IQ for randomly chosen adult is normal R.V. X with .
o Question: Prob. of IQ less than 70 = ? o Response: ______
Practice: 7.49a p.331
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.14
Example: Finding Probabilities with Rule
o Background: IQ for randomly chosen adult is normal R.V. X with .
o Question: Prob. of IQ less than 100 = ? o Response: _____________________________
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.15
Example: Finding Values of X with Rule
o Background: IQ for randomly chosen adult is normal R.V. X with .
o Question: Prob. is 0.997 that IQ is between…? o Response: _________________
Practice: 7.48c p.331
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.16
Example: Finding Values of X with Rule
o Background: IQ for randomly chosen adult is normal R.V. X with .
o Question: Prob. is 0.025 that IQ is above…? o Response: ________
Practice: 7.48d p.331
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.17
Example: Using Rule to Estimate Probabilities
o Background: Foot length of randomly chosen adult male is normal R.V. X with (in.)
o Question: How unusual is foot more than 13 inches? o Response: ______________________________
Practice: 7.49c-d p.331
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.18
Definition (Review) o z-score, or standardized value, tells how
many standard deviations below or above the mean the original value is:
o Notation for Population:
n z>0 for x above mean n z<0 for x below mean
o Unstandardize:
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.19
Standardizing Values of Normal R.V.s
Standardizing to z lets us avoid sketching a different curve for every normal problem: we can always refer to same standard normal (z) curve:
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.20
Example: Standardized Value of Normal R.V.
o Background: Typical nightly hours slept by college students normal;
o Question: How many standard deviations below or above mean is 9 hours?
o Response: Standardize to z = _________________ (9 is ______ standard deviations above mean)
Practice: 7.49e p.331
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.21
Example: Standardizing/Unstandardizing Normal R.V.
o Background: Typical nightly hours slept by college students normal; .
o Questions: n What is standardized value for sleep time 4.5 hours? n If standardized sleep time is +2.5, how many hours is it?
o Responses: n z = ___________________________________________ n ______________________________________________
Practice: 7.49f-g p.331
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.22
Interpreting z-scores (Review)
This table classifies ranges of z-scores informally, in terms of being unusual or not.
Looking Ahead: Inference conclusions will hinge on whether or not a standardized score can be considered “unusual”.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.23
Example: Characterizing Normal Values Based on z-Scores o Background: Typical nightly hours slept by college students
normal; . o Questions: How unusual is a sleep time of
4.5 hours (z = -1.67)? 10.75 hours (z = +2.5)? o Responses:
n Sleep time of 4.5 hours (z = -1.67): __________________ n Sleep time of 10.75 hours (z = +2.5): ________________
Practice: 7.51 p.332
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.24
Normal Probability Problems n Estimate probability given z
o Probability close to 0 or 1 for extreme z n Estimate z given probability n Estimate probability given non-standard x n Estimate non-standard x given probability
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.25
Example: Estimating Probability Given z o Background: Sketch of 68-95-99.7 Rule for Z
o Question: Estimate P(Z<-1.47)? o Response: ______________________
Practice: 7.53 p.332
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.26
Example: Estimating Probability Given z o Background: Sketch of 68-95-99.7 Rule for Z
o Question: Estimate P(Z<+2.8)? o Response: ___________________________________________
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.27
Example: Probabilities for Extreme z o Background: Sketch of 68-95-99.7 Rule for Z
o Question: What are the following (approximately)? a. P(Z<-14.5) b. P(Z<+13) c. P(Z>+23.5) d. P(Z>-12.1) o Response: a. ________ b. _______ c. ________ d. ________
Practice: 7.54 p.332
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.28
Example: Estimating z Given Probability o Background: Sketch of 68-95-99.7 Rule for Z
o Question: Prob. is 0.01 that Z < what value? o Response: ____________
Practice: 7.57 p.332
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.29
Example: Estimating Probability Given x o Background: Hrs. slept X normal; .
o Question: Estimate P(X>9)? o Response: __________________________________
Practice: 7.59 p.333
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.30
Example: Estimating x Given Probability o Background: Hrs. slept X normal; .
o Question: 0.04 is P(X < ? ) o Response: _________________________________ __________________________________________
Practice: 7.61 p.333
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.31
Strategies for Normal Probability Problems n Estimate probability given non-standard x
o Standardize to z o Estimate probability using Rule
n Estimate non-standard x given probability o Estimate z o Unstandardize to x
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture L17.32
Lecture Summary (Normal Random Variables)
o Relevance of normal distribution o Continuous random variables; density curves o 68-95-99.7 Rule for normal R.V.s o Standardizing/unstandardizing o Probability problems
n Find probability given z n Find z given probability n Find probability given x n Find x given probability
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