C HAPTER 2: T HE N ORMAL D ISTRIBUTIONS

D ENSITY C URVES 2 A density curve describes the overall pattern of a distribution. Has an area of exactly 1 underneath it. (Theoretical)

D ENSITY C URVES 3 The median (M) of a density curve is the point that divides the area under the curve in half. The mean ( x ̅ ) of a density curve is the “balance point”. Point that the curve would balance at if made of solid material. Examples:

M ATHEMATICAL M ODEL A density curve is an idealized description of the distribution of data. Values calculated from a density curve are theoretical and use different symbols. Mean Greek letter mu ( x ̅ for data ) Standard deviation Greek letter sigma (s for data) 4

Uniform Density Curves P(X < 2) P(X > 6) P(2 < X < 7) P(x = 2) 5 = 2 (1/8) = 2/8 = 2/8 = 5/8 ≈ 0 Find the proportion of observations within the given interval What would be the median?M = 4

Find the proportion of observations within the given interval P(0 < X < 2) P(.25 < X <.5) P(.25 < X <.75) P(1.25 < X < 1.75) P(.5 < X < 1.5) P(1.75 < X < 2) = 1.0 =.125 =.25 = = What would be the median?

D ENSITY C URVE MODELED BY AN EQUATION A density curve fits the model y =.25x Graph the line. Use the area under this density curve to find the proportion of observations within the given interval P(1 < X < 2) P(.5 < X < 2.5) If the curve starts at x = 0, what value of x does it end at? What value of x is the median? What value of x is the 62.5 th percentile? 7

A NSWERS Graph P(1 < X < 2) P(.5 < X < 2.5) Max Value of X Median 62.5 th percentile =.375 =

D ENSITY C URVE Homework p (not 13a) 9

M EASURING R ELATIVE S TANDING P ERCENTILES : Percentile Percent of observations less than or equal to a particular observation. Example: scores 92, 91, 85, 77, 79, 88, 99, 69, 73, 84 A score of _____ is the ______ percentile? a) 79 b) 88 c) th 70th 100th

N ORMAL DISTRIBUTIONS : Normal curves Curves that are symmetric, single-peaked, and bell-shaped. They are used to describe normal distributions. The mean is at the center of the curve. The standard deviation controls the spread of the curve. The bigger the St Dev, the wider the curve. There are roughly 6 widths of standard deviation in a normal curve, 3 on one side of center and 3 on the other side. 11

N ORMAL DISTRIBUTIONS : Normal curves all have the same overall shape described by mean μand standard deviation σ. 12

N ORMAL C URVE 13

T HE RULE OR E MPIRICAL RULE : 14 68% of the observations fall within 1σof theμ. (approx.!!! Really.6827) 95% of the observations fall within 2σof theμ 99.7% of the observations fall within 3σof theμ

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Example: The distribution of the heights of women is normal with mean of 64.5 and a standard deviation of 2.5. Draw a normal curve. What percent of women are in the following ranges? 1)P(x < 64.5) 2)P(x < 69.5) 3)P(x > 62) 4)P(x > 57) 5)P(57 < x < 67) 6)P(59.5 < x < 67)

Notations: N(μ,σ); example of the women N(64.5, 2.5) FYI Area under a normal curve dx 17

Homework p

S ECTION 2.2: S TANDARD N ORMAL C ALCULATIONS All normal distributions Share many common properties, Are the same if measured in the same units. Use the notation N(mean, standard deviation). The standard normal distribution Has a mean of 0 and standard deviation of 1. N(0,1) Taking any normal distribution and converting it to have a mean of 0 and StDev of 1 is called standardizing. 19

S TANDARDIZING AND Z-S CORES. A standardized value is called a z-score. A z-score tells us how many standard deviations the original observation falls away from the mean, and in which direction. Observations larger than the mean have positive z-scores, while observations smaller than the mean have negative z-scores. To standardize a value, subtract the mean of the distribution and then divide by the standard deviation. 20

Example: Stat class test scores are: 92, 91, 85, 77, 79, 88, 99, 69, 73, 84 If you scored 91, how did you do relative to the class? a)91: b)88: c) 73: Example: A student took a math test and got an 80. He took a Latin test and got a 90. If the math scores had a mean of 70 with a standard deviation of 8 and Latin had a mean of 95 with a standard deviation of 3, in which class did he do relatively better?

C OMPLETE RESPONSE TO A NORMAL DISTRIBUTION QUESTION 1. Check normality – you can only use z-scores if approximately normal!!!! 2. State in terms of x and draw a picture with shading the area. Label with x, μ,σ 3. Standardize x to a z-score. On the picture label the Z-score. 4. use table A or calculator: (2 nd vars) normalcdf (lowerbound, upperbound, μ,σ) 5. Write your conclusion in the context of the problem. Remember you have approximate probability 22

L ET ’ S GO THOUGH AN EXAMPLE OF HOW TO USE THE TABLE. What proportion of all young women are greater than 68 inches tall, given that the distribution of heights for all young women follow N(64.5, 2.5)? 23 Step one – Find P(x > 68) on N(64.5, 2.5)

Step two – standardize x and label picture with z- score Z-scores 24

This value is for area to left of z-score, we need area to right of z-score in this problem. P(x > 1.4) = P(x > 1.4) = Step three – find the probability by using Table A, and the fact that the total area is equal to 1.

Step four – write the conclusion in context of the problem The proportion of young women that are taller than 68 inches is 8.08% 8.08% 26

Using Table A, convert the following z-scores to probability. Draw a picture! 1)Find P(z < 2.3) 1)Find P(z < -1.52) 1)Find P(z > -0.43) 1)Find P(z > 3.1) 1)Find P(-1.52 < z < 2.3) 1)Find P(-3 < z < 3)

Example: For 14 year old boys, cholesterol levels are ≈ N(170, 30) a)What percent of boys have a level of 240 or more? a)What percent of boys are between 170 and 240?

Homework p – 4 p – 8 (not 7d)

F INDING A D ATA VALUE FROM A Z - SCORE To calculate a value in which x% fall above or below the point Use table A to find the z-score. Substitute z, μ and σ into the equation and solve for x. x = μ+ zσ 31

Example: SAT-V scores are ≈ N(505, 110) 1)How high must a student score to be the 30 th percentile? 1)How high must a student score to get in the top 10%? 1)What scores contain the middle 50% of scores?

S ECTION 2.2 Homework: p – 30 p – 36 (not 32b) 34

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36 W EIGHTS OF M OUNDS C ANDY B ARS

37 Computer output of a normal probability plot shows lines as boundaries – if the data falls within the lines, it is approximately normal.

38 In this example, the histogram and the normal probability plot both show that this data is not approximately normal.

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41 Homework: p – 39 (not 39a) p – 61 (not 61c)