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Unit 4 – Normal Distributions

Week 9Ms. Sanchez

+Central Limit Theorem

+Central Limit Theorem

States that if we sample from a population using a sufficiently large sample size, the mean of the samples will approach a normal distribution.

+Normal DistributionThe graph of a normal distribution is

called a normal curve and/or a bell-shaped curve.

+Population Mean and Population Standard Deviation

Mean (μ): average score/number of all your other numbers.

Standard deviation (σ): a quantity calculated to indicate the extent of deviation for a group as a whole.

+Properties of a Normal Curve It helps us find

answers of the data collected, and be able to make predictions as well.

It’s symmetric about a vertical line through the mean.

The highest point is at the mean.

The curve approaches the horizontal axis but never touches or crosses it.

+Sketching a Normal Distribution graph.

1.With mean μ= 15 and standard deviation σ= 3

+Exercise #1Sketch a normal curve for all of the following

1.With μ= 12 and σ= 4

2.With μ= 10 and σ= 2

3.With mean 14 and standard deviation of 5

+The Empirical RuleApproximately 68% of the data

values will lie within one standard deviation on each side of the mean.

Approximately 95% of the data values will lie within two standard deviations on each side of the mean.

Approximately 99.7% of the data values will lie within three standard deviations on each side of the mean.

+Empirical Rule 68-95-99.7

+The Empirical Rule

The Normal Distribution is symmetric about a vertical line through the mean.

+Warm-Up

Using the empirical rule, answer the following.

What percentage of the area under the normal curve lies…1. Below the mean?2. To the right of the mean?3. Between u– 2o and u + 2o4. Above of u+3o

+Example 2.11. 2,000 women at a college campus were

surveyed and their heights are normally distributed with mean μ= 65 in and standard deviation σ=2.5 in

a. Shade the area under the curve that represents shorter than 67.5 in.

b. Find the percentage of area under 67.5 in.

c. How many women have a height above 70in?

+REVIEW. CENTRAL LIMIT THEOREM: APPROACH NORMAL

DISTRIBUTION IF SAMPLE IS LARGE ENOUGH.

+REVIEW NORMAL CURVE:

HELPS FIND ANSWERS AND MAKE PREDICTIONS SYMMETRIC THROUGH THE MEAN. APPROACHES HORIZONTAL BUT NEVER TOUCHES HIGHEST POINT IS AT MEAN.

+REVIEW EMPIRICAL RULE: ALSO KNOWN 68-95-99.7 RULE

+What if… 2,000 freshmen at a State University took a biology

test. The scores were normally distributed with a mean of 70 and a standard deviation of 5. What percentage of students got a 90 or more….

+Understanding z-scoresWhat is a z- score?

It represents how many standard deviations is “X” from the mean.

+What are we finding? We’re finding the area under the normal curve. If we

know that the total percentage under the curve is 100%,therefore its area its in decimal.

+Using a z-score table. Z –score table help us find the area under the curve

easily Find the area to the left of z < 1.18 Find the area to the right of z > 1.18

+ExampleFind the area right of z > 3.00

Find the area right of z > 0.15

Find the area between 1.00 < z < 2.00

+Exercise #3Find the area under the standard normal curve…1. z < - 1.002. z > - 1.003. z > 0.944. - 1.40 < z < 2.705. -0.80 < z < 1.356. z > 3.707. – 3.04 < z < -2.05

+REVIEW EXERCISE1. As z values increase, do the area to the

left of z increase or decrease?

2. If a z value is negative, is the area to the left of z less than or greater than 0.500?

3. If a z value is positive, is the area to the left less than or greater than 0.500?

+Warm-UpFind the area and percentage under the Normal Curve.

1. z < 1.31

2. z > 0.04

3. - 1.80 < z < 3.05

+Using the z-score to find answersX - is your question. Z equation will help you find the answer.

+ExampleThe grades of a statistics class are

normally distributed with an average grade of 70 and a standard deviation of 10.

What percentage of students made a score below 83?

What percentage made above 52?

What percentage made between 75 - 79

+Exercise A student has computed that it takes an

average of 17 minutes with a standard deviation of 3 minutes to drive from home, park the car, and walk to an early morning class. What is the probability that they’ll take more than 25

minutes?

What is the probability that they’ll take less than 16 minutes?

What is the probability that they’ll take less than 10 minutes?

What’s the percentage that it’ll take between 15-20 min

+Warm-Up2000 freshmen at State University took

a biology test. The scores were distributed normally with a mean of 70 and a standard deviation of 5.

1. What percentage of students made above 71?

2. What percentage of students made below 56?

3. What percentage of students made between 86-95?

+Finding x given a percentage. Given an x distribution with mean and standard

deviation, the raw score x corresponding to a z score is

Find x given z. Mean 70 standard deviation of 10

Z > - 3.00

Z < 1.00

+ExampleIf the raw scores of an aptitude test have a mean of 480 and a standard deviation of 70 points.

Karla’s z-score was above 1.74 standard deviations from the mean, what could be her score ?

Adam’s z-score was below 3.00 standard deviations from the mean, what could be his score?

+Exercise #1 The mean weight of an adult American male is

200 pounds with standard deviation of 13 pounds.

If someone’s z-score is more than 1.44, what could be their weight?

If someone’s z-score is less than 0.32, what could be their weight?

What is the probability that someone will weight more than 240 pounds?

+Top/Bottom Percentile If the raw scores of an aptitude test have a

mean of 480 and a standard deviation of 70 points.

What’s the cutoff score to be on the top 10%?

What’s the cutoff score to be on the bottom 15%?

What’s the cutoff score to be in the top 5%?

+Exercise #21. The mean weight of an adult American male is

200 pounds with standard deviation of 13 pounds.

a. What’s the cutoff weight to be on the bottom 20%?

b. What is the cutoff weight to be on the top 5%?

2. The mean GPA for the school is 3.8 with a standard deviation of .04.

a. What is the cutoff GPA to be in the top 10%?

b. What is the cutoff GPA to be in the bottom 10%