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IB Math Studies Unit 3 Statistics: Normal Distribution
The Normal Density Curve:
The Empirical Rule: 68% of the data fall within 1 standard deviation of the mean
95% of the data fall within 2 standard deviations of the mean
99.7% of the data fall within 3 standard deviations of the mean
Example 1: Suppose IQ scores of students at IM Smart College follow a normal distribution with µ = 116 and 𝜎 = 12.
a.) Draw and label the normal curve:
b.) What % of students scored between 103 and 127?
c.) What % scored between 115 and 127?
d.) What % scored higher than 139?
EXPECTED VALUE = the probability times the total number in the sample
Example 2: The heights of 250 twenty-year-old women are normal distributed with a mean of 1.68 m and a standard deviation of 0.06 m.
a.) Draw and label the normal curve:
b.) Find the probability that a woman has a height between 1.56 and 1.74 m.
c.) Find the expected number of women with a height greater than 1.8 m.
WHAT IF YOU WANT TO KNOW A DATA VALUE THAT DOES NOT FALL ON AN EXACT STANDARD DEVIATION?
z-score : tells us exactly how many standard deviations it is away from the mean. You find this by subtracting the value and the mean, then divide by the
standard deviation. You would use a table to find the percentage. In this class, we will not use z-scores/table – we will use the calculator. However, if you
choose this for your IA, you will investigate using z-scores/table.
In calculator: To find the percentage/probability given boundaries
Distribution menu (2nd VARS)  choose option 2:normalcdf(  normalcdf(lower boundary, upper boundary, mean, standard deviation)
In calculator: To find the boundaries given a percentage/probability
Distribution menu (2nd VARS )  choose option 3: invNorm(  invNorm(percentage to the left of the boundary, mean, standard deviation)
Example 3: In the video we watched in class, it said that the Caucasian baby birth weight distribution has a mean of 7 lbs. 5 oz. and a standard deviation of 1.2
pouns. The African American baby birth weight distribution has a mean of 6 lbs. 9 oz and a standard deviation of 1.4 pounds. The Hispanic baby birth weight
distribution has 7 lbs. 3 oz. and a standard deviation of 1.2 pounds.
Answer the following for each population:
a.)
b.)
c.)
d.)
Sketch and label a normal curve.
Calculate the probability of a baby weighing less than 5 lbs.
Calculate the probability of a baby weighing between 6 and 8 lbs.
Calculate the probability of a baby weighing more than 10 lbs.
Caucasian:
African American:
Hispanic:
e.) Your cousin had a Caucasian baby and the doctor said it was in the 80th percentile for weight. Find the baby’s weight.
f.) 30% of African American babies weigh more than p weight. Find p.
Extension: If there are 500 babies born in a year at WakeMed Hopsital, 40% of them are Caucasian, 35% of them are African American, and 20% of them are
Hispanic, how many of each subpopulation would we expect to weigh between 6 and 8 lbs?