Understanding Normal Distribution and Empirical Rule with Height Data Example, Lecture notes of Statistics

Download Understanding Normal Distribution and Empirical Rule with Height Data Example and more Lecture notes Statistics in PDF only on Docsity!

What does a population that is

normally distributed look like?

X 80

 = 80 and  = 10

50 60 70 90 100 110

Empirical Rule

68%

95%

99.7%

68-95-99.7% RULE

Empirical Rule

34% 34% 68%

47.5% 47.5% 95% 49.85% 49.85% 99.7%

Average American adult male

height is 69 inches (5’ 9”) tall with a

standard deviation of 2.5 inches.

Using the Empirical Rule

Let H~N(69, 2.5)

What is the likelihood that a randomly selected adult

male will have a height between 64 and 74 inches?

P(64 < h < 74) =.

Using Empirical Rule-- Let H~N(69, 2.5)

What is the likelihood that a randomly selected adult male would have a height of less than 66.5 inches?

P(h < 66.5) = 1 – (.50 + .34) =.

OR  .50 - .34 =.

Using Empirical Rule--Let H~N(69, 2.5)

What is the probability that a randomly selected

adult male would have a height between 64 and

76.5 inches? (^) P(64 < h < 76.5) = .475 +.

Z-Scores are standardized

standard deviation measurements

of how far from the center (mean) a

data value falls.

Ex: A man who stands 71.5 inches tall is 1 standardized standard deviation from the mean.

Ex: A man who stands 64 inches tall is - standardized standard deviations from the mean.

Empirical Rule & Z-Score

About 68% of data values

in a normally distributed data set have z-scores between – 1 and 1; approximately 95% of the values have z-scores between – 2 and 2; and about 99.7% of the values have z-scores between

• 3 and 3.

Z-Score & Let H ~ N(69, 2.5)

What would be the standardized score for an adult male who stood 71.5 inches?

1

2. 5

3. 5

4. 5

5. 5 69





 

z

z

z

H ~ N(69, 2.5) Z ~ N(0, 1)

Comparing

Z-Scores

Suppose Bubba’s score on exam A was 65,

where Exam A ~ N(50, 10). And, Bubbette score was an 88 on exam B, where Exam B ~ N(74, 12).

Who outscored who? Use Z-score to compare.

Bubba z = (65-50)/10 = 1.

Bubbette z = (88-74)/12 = 1.

Comparing Z-Scores

Heights for traditional college-age students in

the US have means and standard deviations of approximately 70 inches and 3 inches for males and 165.1 cm and 6.35 cm for females. If a male college student were 68 inches tall and a female college student was 160 cm tall, who is relatively shorter in their respected gender groups?

Male z = (68 – 70)/3 = -.

Female z = (160 – 165.1)/6.35 = -.

• z .00 .01 .02 .03.
• -2.0 .0228 .0222 .0217 .0212.
• -1.9 .0287 .0281 .0274 .0268.
• -1.8 .0359 .0351 .0344 .0336.
• -1.7 .0446 .0436 .0427 .0418.
  • P(z < -1.91) = ?.0281 P(z  -1.74) = ?.
  • z .00 .01 .02 .03. Reading a Z Table
  • 1.5 .9332 .9345 .9357 .9370.
  • 1.6 .9452 .9463 .9474 .9484.
  • 1.7 .9554 .9564 .9573 .9582.
  • 1.8 .9641 .9649 .9656 .9664.
• P(z < 1.81) = ?.9649 P(z  1.63) = ?.

Using Z-Table

Find the two tables representing z-scores from – 3.49 to +3.49.

Notice which direction the area under the curve is shaded (Table reads for ONLY shaded area to the left).

The assigned probabilities represents the area shaded under the curve from the z- score and to the far left of the z-score.