Publish on 11th November 2019
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SP 225Lecture 9

The Normal Curve

The Normal Curve

‘Bell’ ShapedUnimodal in centerTails extend to infinity in either direction

The Ideal Normal Curve

Naturally Normal

IQ testsStandardized testsErgometric MeasuresManufacturing Processes

Normal Curve as an Approximation

Approximation

No data set is perfectly normalMany are very closeAllows us to ‘sum up’ distributions with two pieces of information: mean and standard deviationAllows us to draw conclusions based on calculations vs. data inspection

Empirical Rule for Data with a Bell-Shaped Distribution

Empirical Rule for Data with a Bell-Shaped Distribution (2)

Empirical Rule for Data with a Bell-Shaped Distribution (3)

Number of Standard Deviations

Use the z-scoreZ-score represents the number of standard deviations from the mean of a single data pointUse it no matter what the mean or standard deviation is.

z=

x-x

s

Properties of Z-score

+ if above the mean- if below the meanUnits are number of standard deviations

Example

Einstein had an IQ of 160 points. Amongst the population, IQ has a mean value of 110 with a standard deviation of 16.What is the difference between Einstein and the average person?What is the difference in number of standard deviations?What is the z-score for Einstein’s IQ?

Areas under the Normal Curve

Areas correspond to proportions of data.Areas are looked up in the z-score table

Special Properties of the Normal Distribution

Total area under the curve is 1Partial areas represent proportionsSymmetric

Areas Example

Suppose your weight fluctuates each week with a mean of 0 and a standard deviation of 1 and is normally distributed.What is the probability you lose 1.5 lbs or more?What is the probability you gain less than 1.5 lbs?What is the probability you gain 1.5 lbs or more?What is the probability your weight fluctuates by less than 1.5 lbs?

Areas Example (2)

Find a value, c, where:You gain c or less pounds 75% or the timeYou lose c or more lbs 20% of the timeYour weight fluctuates by c or less with 80% probability

Z-score Example

GPAs of SUNY Oswego freshman biology majors have approximately the normal distribution with mean 2.87 and standard deviation .34.In what range do the middle 90% of all freshman biology majors’ GPAs lie?Students are thrown out of school if their GPA falls below 2.00. What proportion of all freshman biology majors are thrown out?What proportion of freshman biology majors have GPA above 3.50?

http://www.oswego.edu/~srp/stats/normal_wk_4.htm

Z-score Example (2)

The length of elephant pregnancies from conception to birth varies according to a distribution that is approximately normal with mean 525 days and standard deviation 32 days.What percent of pregnancies last more than 600 days (that’s about 20 months)?What percent of pregnancies last between 510 and 540 days (that’s between 17 and 18 months)?How short do the shortest 10% of all pregnancies last?

http://www.oswego.edu/~srp/stats/normal_wk_4.htm

Probabilities

Areas under the curve represent probabilities as well as proportionsProportions make statements about existing dataProbabilities make statements about future actions

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Lecture9PSStats