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7.1 Normal Curves

7.1 Normal Curves. Normal curve Curve is bell shaped with highest point over mean Curve is symmetrical about a vertical line through  Curve approaches horizontal axis but never touches or crosses it.

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7.1 Normal Curves

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  1. 7.1 Normal Curves • Normal curve • Curve is bell shaped with highest point over mean • Curve is symmetrical about a vertical line through  • Curve approaches horizontal axis but never touches or crosses it. • Inflection point (were concave up part of curve meets concave down part) is one  (standard dev.) from the mean ( is measure of spread) • Total area under the curve = 1 • Portion of area under curve between an interval represents the probability that that measurement lies in the at interval • Normal curve can be represented by N(, ) Draw Normal curve and examples of curves that are not normal

  2. 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.

  3. 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.. • 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”

  4. 7.1 Normal Curves • Why are normal distributions important? • Good description (approximation) for some distributions of real data. (SAT scores; heights of people etc.) • Good approximations to results of many kinds of chance observations (coin tossing) • 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)

  5. 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: • HT = 74” b) HT = 65” c) HT= 71.2” Find HT corresponding to (use x = z  + ) a) z = 1.2 b)z = -0.8

  6. 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 z-scores.”  = 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. 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

  8. 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-

  9. 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 tablec. Area in the middle answer is area(z2) – area(z1) For Raw Score use z-score from table and calc x x = z  +  Examples

  10. 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

  11. 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)

  12. 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

  13. 7.4 Sampling Distributions • Statistic description of sample • Parameter description of a population MeasureStatisticParameter Mean x(x bar)  (mu) Variance s2 2 Std. Dev s  (sigma) Probability p (p hat) p

  14. 7.4 Sampling Distributions • Statistics (sample values) are used to make inferences about parameters (population values) • Inferences • Estimation – estimate value of a population parameter • Testing - formulate decision about the value of the population parameter • Regresssion – make predictions or forecasts about the value of the statistical variable.

  15. 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

  16. 76 78 80 82 84 86 76 78 80 82 84 86 76 78 80 82 84 86 7.4 Sampling Distribution Pop heightsSamplen=3x 76 76,78,79 77.7 x only 78 76,78,81 78.3 79 76,78,86 80 81 76,79,81 78.7 P(79  x  81) = 0.50 86 76,79,86 80.3 n = 3 76,81,86 81 78,79,81 79.3 78,79,86 81 P(79  x  81) = 0.80 78,81,86 81.7 n =4 79,81,86 82  = x/n =80

  17. 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

  18. 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)

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