M3U9D2 Warm-up:1. At the doctor’s office, there are three children in the waiting room. The children are ages 3, 4, and 5. Another 4 year old child enters the room. a) What will happen to the mean of the

children’s ages when the new child enters?When the new child enters, nothing will

happen to the mean because the child’s age is the mean of the set.

b) What will happen to the standard deviation of the children’s ages? The standard deviation will decrease because the 4th value is exactly equal to the mean, which will give the data less variability than originally 

HW Check:1) systematic2) cluster3) stratified4) cluster5) voluntary response 6) convenience7) stratified8) cluster9) convenience 10) simple random11) stratified 12) systematic 13) simple random 14) cluster

How do you find the mean?Add numbers together then

divide by the number of entries

What is standard deviation? How much variation from

the average exists.

M3U9D2 Normal Distribution and the Empirical Rule

Objective:To fit a data set to the normal

distribution using the mean and standard deviation

ANDTo apply the Empirical Rule to

estimate probabilities for normal distributions

Three main topics in Math III Unit 9:

Normal Distributions

Sampling and Study Design

Estimating Population Parameters

Normal Distributions Video

NORMAL DISTRIBUTIONS:Characteristics of a normal distribution:

1. Continuous random variable2. Symmetric with respect to the mean3. mean = median = mode4. Area under the curve is 1

YOU should know these characteristics!

The Standard Normal Curve…

Z-score: number of standard deviations a value is from the mean on the standard normal curve

µ = 0; σ = 1

What is the meaning of a positive z-score?

What about a negative z-score?

How do you use this?The mean score on the SAT is 1500, with a standard deviation of 240. The ACT, a different college entrance examination, has a mean score of 21 with a standard deviation of 6.

If Bobby scored 1740 on the SAT and Kathy scored 30 on the ACT, who scored higher?

Bobby Kathy

z = 1 z = 1.5

What should a complete answer look like?•Correct mathematical work•Interpretation of that work in context of the problem

For example:Kathy scored higher than Bobby overall – her z-score on the ACT shows that she scored 1.5 standard deviations above the mean while Bobby scored only 1 standard deviation above the mean on the SAT.

The Empirical Rule:

68% of the data falls within ± 1σ

95% of the data falls within ± 2σ

99.7% of the data falls within ± 3σ

When you break it up…

The scores on the Math III midterm were normally distributed. The mean is 82 with a standard deviation of 5. Find the probability that a randomly selected person:

a. scored between 77 and 87 b. scored between 82 and 87c. scored between 72 and 87d. scored higher than 92e. scored less than 77

How do you use this?

82

87

92

97

77

72

67

Draw the curve, add the mean, then add the standard deviations above and below the mean…

a. scored between 77 and 87

b. scored between 82 and 87

c. scored between 72 and 87

d. scored higher than 92

e. scored less than 77

68%34%81.5%2.5%

16%

Sleep Activity

2nd, Vars (DISTR), NormalcdfNormalcdf(min,max,mean,st

dev)Gives %

2nd, Vars (DISTR), InvNormInvNorm(%, mean, st dev)

Gives a boundary

Classwork:U9D2 The Normal Distribution &

distribute calculator directions

Homework:U9D2 Using the

Empirical Rule WS