Topcis

1. Describing Location in a Distribution
   • Percentiles and z-scores can be used to measure an individual's location
     • The percentile is the percent of the distribution that is at or below the value of that observation
     • The z-score is how many standard deviations x is from the mean
   • A cumulative relative frequency graph shows the accumulating percent of observations as you move through the classes.
   • Data can be transformed by adding/subtracting and multiplying/dividing
     • (+) and (-) by b
       • Center (mean and median) and location (quartiles and percentiles) changes by b
       • Measures of spread and shape do not change
     • (x) or (/) by b
       • Center, location, and spread change by a factor of b
       • Shape does not change
2. Density Curves and Normal Distributions
   • density curves can describe the overall pattern of a distribution.
     • Always is on or above the horizontal axis
     • Always has an area of one
     • The $\mu$ of a density curve is the balancing point
     • The median divides the area under the curve in half
     • The $\sigma$ is at the POI
   • A normal distribution is described by a symmetric density curve called the normal curve
     • It is described by N($\mu$,$\sigma$)
     • A normal distribution obey's the 68-95-99.7 rule
   • A standard normal distribution has a $\mu$ of 0 and a $\sigma$ of 1
   • A normal probability plot can tell you if a distribution is normal

Formulas

Calculator Functions

normCdf(lower bound, upper bound, $\mu$, $\sigma$) tells you the area under a normal curve (in other words the probability from [a,b])

invNorm(area to the left, $\mu$, $\sigma$) tells you the value for a given area

normPdf(x value, $\mu$, $\sigma$) tells you the probability at a given point $\text{normCdf(a,b)}=\int_{a}^{b} \text{normPdf(x) } dx$

Terms

Review

1. What rule does a normal distribution obey?
   • A) 69-94-98.9
   • B) 65-96-99.6
   • C) 68-95-99.7
   • D) 67-97-99.5
   • A B C D
2. If $\sigma_x=5$ is transformed by the formula $y=2\sigma_x+4$, what is the new $\sigma$?
   • A) 5
   • B) 14
   • C) 9
   • D) 10
   • A B C D
3. A dataset containing the number of times someone dies within an hour after an unspecefied leader tweets followes a normal distribution with a $\mu$ of 50,000 and a $\sigma$ of 10,000. What is the probability that less than 40,000 die?
   • A) .274
   • B) .159
   • C) .841
   • D) .726
   • A B C D
4. An approximately normal distribution has a mean of 23.8 and a standard deviation of 4.2. What is the probability that the value is within [22, 24]? (round to the nearest thousandth)
5. A group of students are having a competition to see who can get the highest score in the game, "Send Me to Heaven" in which you toss your phone as high up in the air as possible. The scores are approximately normal with a mean of 17ft and a standard deviation of 12. What percentile is a score of 23ft? (enter as a decimal rounded to the nearest hundreth)