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(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture
Lecture 17
Continuous Random Variables;
Normal Distribution
Relevance of Normal Distribution Continuous Random Variables 68-95-99.7 Rule for Normal R.V.s Standardizing/Unstandardizing Probabilities for Standard/Non-standard Normal R.V.s (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 2
Looking Back: Review
4 Stages of Statistics Data Production (discussed in Lectures 1-4) Displaying and Summarizing (Lectures 5-12) Probability Finding Probabilities (discussed in Lectures 13-14) Random Variables (introduced in Lecture 15) Binomial (discussed in Lecture 16) Normal Sampling Distributions Statistical Inference (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 3
Role of Normal Distribution in Inference
Goal: Perform inference about unknown population proportion, based on sample proportion Strategy: Determine behavior of sample proportion in random samples with known population proportion Key Result: Sample proportion follows normal curve for large enough samples. Looking Ahead: Similar approach will be taken with means. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 4
Discrete vs. Continuous Distributions
Binomial Count X discrete (distinct possible values like numbers 1, 2, 3, …) Sample Proportion also discrete (distinct values like count) Normal Approx. to Sample Proportion continuous (follows normal curve) Mean p , standard deviation
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 5 Sample Proportions Approx. Normal (Review) Proportion of tails in n =16 coinflips ( p =0.5) has Proportion of lefties ( p =0.1) in n =100 people has , shape approx normal , shape approx normal (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 6
Example: Variable Types
Background : Variables in survey excerpt: Question: Identify type ( cat.,disc.quan., cont.quan.) Age? Breakfast? Comp? (daily time in min. on computer) Credits? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 8
Example: Variable Types
Background : Variables in survey excerpt: Response: Age: Breakfast: Comp (daily time in min. on computer): Credits: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 9
Probability Histogram for Discrete R.V.
Histogram for male shoe size X represents probability by area of bars (on left) (on right) For discrete R.V., strict inequality or not matters.
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 14
68-95-99.7 Rule for Normal R.V.
Looking Back: We use Greek letters to denote population mean and standard deviation. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 15
Example: 68-95-99.7 Rule for Normal R.V.
Background : IQ for randomly chosen adult is normal R.V. X with Question: What does Rule tell us about distribution of X? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 17
Example: 68-95-99.7 Rule for Normal R.V.
Background : IQ for randomly chosen adult is normal R.V. X with Response: We can sketch distribution of X : (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 18
Example: Finding Probabilities with Rule
Background : IQ for randomly chosen adult is normal R.V. X with Question: Prob. of IQ between 70 and 130=?
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 20
Example: Finding Probabilities with Rule
Background : IQ for randomly chosen adult is normal R.V. X with Response: Prob. of IQ bet. 70 and 130=____ (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 21
Example: Finding Probabilities with Rule
Background : IQ for randomly chosen adult is normal R.V. X with Question: Prob. of IQ less than 70 =? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 23
Example: Finding Probabilities with Rule
Background : IQ for randomly chosen adult is normal R.V. X with Response: Prob. of IQ less than 70 =____ (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 24
Example: Finding Probabilities with Rule
Background : IQ for randomly chosen adult is normal R.V. X with Question: Prob. of IQ less than 100 =?
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 32
Example: Finding Values of X with Rule
Background : IQ for randomly chosen adult is normal R.V. X with Response: Prob. is 0.025 that IQ is above ___ (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L14. 33 Example: Using Rule to Evaluate Probabilities Background : Foot length of randomly chosen adult male is normal R.V. X with (in.) Question: How unusual is foot less than 6.5 inches? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L14. 35 Example: Using Rule to Evaluate Probabilities Background : Foot length of randomly chosen adult male is normal R.V. X with (in.) Response: Foot<6.5 _____________________ (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L14. 36 Example: Using Rule to Estimate Probabilities Background : Foot length of randomly chosen adult male is normal R.V. X with (in.) Question: How unusual is foot more than 13 inches?
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L14. 38 Example: Using Rule to Estimate Probabilities Background : Foot length of randomly chosen adult male is normal R.V. X with (in.) Response: P( X >13) ________________________ 13 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 39
Definition (Review)
z -score , or standardized value , tells how many standard deviations below or above the mean the original value is: Notation for Population: z >0 for x above mean z <0 for x below mean Unstandardize: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 40 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: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 41 Example: Standardized Value of Normal R.V. Background : Typical nightly hours slept by college students normal; Question: How many standard deviations below or above mean is 9 hrs.?
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L14. 48 Example: Characterizing Normal Values Based on z-Scores Background : Typical nightly hours slept by college students normal; Questions: How unusual is a sleep time of 4.5 hours ( z =-1.67)? How unusual is a sleep time of 10.75 hours ( z =+2.5)? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L14. 50 Example: Characterizing Normal Values Based on z-Scores Background : Typical nightly hours slept by college students normal; Responses: Sleep time of 4.5 hours Sleep time of 10.75 hours (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 51
Normal Probability Problems
Estimate probability given z Probability close to 0 or 1 for extreme z Estimate z given probability Estimate probability given non-standard x Estimate non-standard x given probability (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 52
Example: Estimating Probability Given z
Background : Sketch of 68-95-99.7 Rule for Z Question: Estimate P( Z <-1.47)?
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 54
Example: Estimating Probability Given z
Background : Sketch of 68-95-99.7 Rule for Z Response: P( Z <-1.47) (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 55
Example: Estimating Probability Given z
Background : Sketch of 68-95-99.7 Rule for Z Question: Estimate P( Z >+0.75)? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 57
Example: Estimating Probability Given z
Background : Sketch of 68-95-99.7 Rule for Z Response: P( Z >+0.75) (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 58
Example: Estimating Probability Given z
Background : Sketch of 68-95-99.7 Rule for Z Question: Estimate P( Z <+2.8)?
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 65
Example: Probabilities for Extreme z
Background : Sketch of 68-95-99.7 Rule for Z Question: Estimate P( Z >-3.8)? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 67
Example: Probabilities for Extreme z
Background : Sketch of 68-95-99.7 Rule for Z Response: P( Z >-3.8) (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 68
Example: Probabilities for Extreme z
Background : Sketch of 68-95-99.7 Rule for Z Question: Estimate P( Z <+13)? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 70
Example: Probabilities for Extreme z
Background : Sketch of 68-95-99.7 Rule for Z Response: P( Z <+13)
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 71
Example: Probabilities for Extreme z
Background : Sketch of 68-95-99.7 Rule for Z Question: Estimate P( Z >23.5)? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 73
Example: Probabilities for Extreme z
Background : Sketch of 68-95-99.7 Rule for Z Response: P( Z >23.5) (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 74
Normal Probability Problems
Estimate probability given z Probability close to 0 or 1 for extreme z Estimate z given probability Estimate probability given non-standard x Estimate non-standard x given probability (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 75
Example: Estimating z Given Probability
Background : Sketch of 68-95-99.7 Rule for Z Question: Prob. is 0.01 that Z <what value?
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 82 Example: Estimating Probability Given x Background : Hrs. slept X normal; Question: Estimate P( X >9)? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 84 Example: Estimating Probability Given x Background : Hrs. slept X normal; Response: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 85 Example: Estimating Probability Given x Background : Hrs. slept X normal; Question: Estimate P(6< X <8)? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 87 Example: Estimating Probability Given x Background : Hrs. slept X normal; Response: A Closer Look: -0.67 and +0.67 are the quartiles of the z curve.
(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 88
Normal Probability Problems
Estimate probability given z Probability close to 0 or 1 for extreme z Estimate z given probability Estimate probability given non-standard x Estimate non-standard x given probability (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 89 Example: Estimating x Given Probability Background : Hrs. slept X normal; Question: 0.04 is P( X <?) (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 91 Example: Estimating x Given Probability Background : Hrs. slept X normal; Response: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L17. 92 Example: Estimating x Given Probability Background : Hrs. slept X normal; Question: 0.20 is P( X >?)
