7.1 Normal Curves

7.1 Normal Curves •

Normal curve

1. Curve is bell shaped with highest point over mean 2. Curve is symmetrical about a vertical line through  3. Curve approaches horizontal axis but never touches or crosses it. 4. Inflection point (were concave up part of curve meets concave down part) is one  (standard dev.) from the mean ( is measure of spread) 5. Total area under the curve = 1 6. Portion of area under curve between an interval represents the probability that that measurement lies in the at interval 7. Normal curve can be represented by N(, ) Draw Normal curve and examples of curves that are not normal

7.1 Normal Curves • Empirical Rule – For Normal Distributions – Approximately 68% of the data values lie within one  (standard dev.) on each side of mean – Approximately 95% of the data values lie within two  on each side of the mean – Approximately 99.7% of the data values lie within three  on each side of the mean.

7.1 Normal Curves • •

Ex: If males have a mean height of 69” (= 69”) with a  = 3” What is the probability that a randomly selected male is.. a. b. c. d. e. f. g. h.

Between 66” and 72” Between 66” and 69” Greater than 69” Greater than 72” Between 69” and 75” Between 69” and 78” Between 66” and 75” Between 66” and 78”

7.1 Normal Curves •

Why are normal distributions important?

1. Good description (approximation) for some distributions of real data. (SAT scores; heights of people etc.) 2. Good approximations to results of many kinds of chance observations (coin tossing) 3. Many statistical inference procedures (inference is predicting something about a population by taking a sample) are based on normal distributions * Many sets of real data do not follow distribution (e.g., income)

7.2 Standard Units and Areas Under the Standard Normal Distributions Z - SCORE z – score z = x- 

x – observation of interest  - mean of data for population  - std. dev. of data for population z – number of std. dev. the observation of interest is away from the mean. + z is above mean, -z is below mean N(69”,3”) find z –scores for: a) HT = 74” b) HT = 65” c) HT= 71.2” Find HT corresponding to (use x = z  + ) a) z = 1.2 b)z = -0.8

7.2 Standard Normal distribution • z – score: z = x-  • To find the raw score x = z  +  (point of interest) • Standard Normal Distribution – “normalizes data by using zscores.” =0  = 1 uses the z-score from any normal distribution (draw Std. Normal Distribution)

Areas under Normal Curve Can use empirical rule from –1< z < 1 = 68% from –2 < z < 2 = 95% from –3 < z < 3 = 99.7%

7.2 Standard Normal distribution • Transformation from From  = 10  = 2 Normal Curve

Standard Normal Curve

Since any normal distribution can be transformed into a STANDARD NORMAL DISTRIBUTION, we can utilize the STANDARD NORMAL DISTRIBUTION TABLE for calculating probabilities

7.2 Standard Normal distribution • Standard Normal Distribution Table – Always reports area to left of z-score – To get probabilities below a certain z value, read directly off the table – To get probabilities above a certain value, take value of table and subtract from 1 – Example of z = -1.0 P(z < -1.0) = example of z = 2.18 P(z < 2.19) = example of z = 0.91 P(z >.91) = 1-

7.2 Standard Normal distribution To solve 1. Always sketch Normal curve!!!! 2. Find region of Interest and shade that region 3. Use probability table to find a) area if z-score is known b) z-score if area is known 4. Determine final answer for probability a. Area on left of z-score answer is area directly from table b. Area on right of z-score answer is ( 1 – area) in table c. Area in the middle answer is area(z2) – area(z1) For Raw Score use z-score from table and calc x x=z+ Examples

7.3 More practice • Actual normal curve parameters –  = mean –  = standard deviation – x = raw data point of interest

• Standardized value – z number of standard deviations from mean

• Probabilities of normal curves in – z chart

• Since normal curves always contain the same probabilities (Emperical Rule), z values help us to easily convert to the nearest probability. • Conversion is x value to z to probability • Or probability to z value to x value

7.3 • Practice converting z values in ranges to probabilities • Working with word problems – Always draw normal curve and shade area of interest – Find prob from x values (x to z to %) – Find x values from probability ( % to z to x) Conversion problems – Less than – Greater than – Range

• Warranty problem • Guaranteed time (pizza, oil change)

7.4 Sampling Distributions • Using samples to make estimates • Population – a set of measurements • Sample – a subset of measurements of population. (random is best type) • Parameter – numerical descriptive measure of a population • Statistic – numerical descriptive measure of a sample

7.4 Sampling Distributions • Statistic description of sample • Parameter description of a population Measure Statistic Parameter Mean Variance Std. Dev Probability

x(x bar) s2 s p (p hat)

 (mu) 2  (sigma) p

7.4 Sampling Distributions • •

Statistics (sample values) are used to make inferences about parameters (population values) Inferences 1. Estimation – estimate value of a population parameter 2. Testing - formulate decision about the value of the population parameter 3. Regresssion – make predictions or forecasts about the value of the statistical variable.

7.4 Sampling Distributions • Sampling distribution - probability distribution of a sample statistic based on all possible simple random samples of the same size from the same population.

http://onlinestatbook.com/stat_sim/sampling_dist/index.html

7.4 Sampling Distribution Pop heights 76 78 79 81 86

Sample n=3 76,78,79 76,78,81 76,78,86 76,79,81 76,79,86 76,81,86 78,79,81 78,79,86 78,81,86 79,81,86

x 77.7 78.3 80 78.7 80.3 81 79.3 81 81.7 82

x only 76

80

82

84

86

P(79  x  81) = 0.50 n=3 76

78

80

82

84

86

P(79  x  81) = 0.80 n =4 76

 = x/n =80

78

78

80

82

84

86

Frequency

Cholesterol levels of 38-41 year old men 14 12 10 8 6 4 2 0 0

100

200 Cholesterol Level

300

400

7.5 Central Limit Theorem • x distribution, given x distribution is Normal • Theorem – for a normal Probability distribution – The x distribution is a normal distribution – The mean of x distribution is  – The standard deviation of the x distribution is / n x =  x= / n

z = x - x = x -  x / n ex  = 10.2,  = 1.4, n = 5 Standard error is the standard deviation of a sampling distribution

7.5 Central Limit Theorem • Central Limit Theorem – Given any distribution of the random variable x, as the sample size increases, the x bar distribution approaches a normal distribution ( n  30) • (draw example)