Download Review Sheet

Math 101 - Review for Quiz 2
Summary statistics
The distribution and variation of a data set
Range = max value - min value
s
sum (data value − mean)2
Standard Deviation = σ =
#data points − 1
Ex1. The following are the approximate lengths of Beethoven’s nine symphonies and Mahler’s nine symphonies in minutes. Determine the range, the mean, and standard deviation for each data set and
discuss how they are different and what that means.
Beethoven: 28 36 50 33 30 40 38 26 68
Mahler:
52 85 94 50 72 72 80 90 80
Ex2. After recording the pizza delivery times for two different pizza restaurants, you determine that
Restaurant A has a mean delivery time of 45 minutes with standard deviation of 3 minutes, and
Restaurant B has a mean delivery time of 42 minutes with a standard deviation of 20 minutes.
Interpret these numbers and decide which restaurant you would order from assuming you like both
equally.
The Normal Distribution
Shape of the normal distribution
Types of data that normal distributions approximate well (works well - sports statistics, human being
measurements, prices, standardized tests; does not work well - income, easy tests, wait times where the
mode is 0)
68-95-99.7 Rule
standard score/z-score:
z=
data value − mean
standard deviation
nth percentile - the smallest data value such that n% of the data is less than or equal to it.
Ex3. An incoming first-year took her college’s placement exams in French and math. In French she scored
82, and in math she scored 86. The overall results on the French exam had a mean of 72 and a
standard deviation of 8. The mean math score was 68 with a standard deviation of 12. On which
exam did she do better as compared to the other incoming first-years?
Ex4. A individual records the speed of cars driving past his house, where the speed limit is 20 mph. The
mean of 100 readings is 23.84 mph with a standard deviation of 3.56 mph.
a) How many standard deviations from the mean would a car going under the speed limit be?
b) Which is more unusual, a car traveling 34 mph or a car traveling 10 mph?
Statistical Reasoning and Inference
Population vs sample
Steps
1.
2.
3.
4.
5.
of a statistical study:
State Goal
Choose a representative sample form the population
Collect data and summarize
Use sample statistics to infer information about the population
Draw conclusions
Types of statistical studies:
Observational (2 main types - proportions and means)
Experimental (describe with statistical significance - how likely the event is to occur)
Standard error, margin of error, and 95% confidence interval (for proportions):
q
Standard Error = p(1−p)
n , where p = sample proportion and n = sample size.
Margin of Error (M E) = 2 x standard error.
The 95% confidence interval is p ± M E.
“We are 95% confident that the true value lies between p − M E and p + M E”.
Ex5. In a random survey of 226 college students, 20 reported having no siblings. Estimate the proportion
of students nationwide who have no siblings and construct the 95% confidence interval. Interpret
what you find.
Linear Growth
Linear growth occurs when a quantity increases by the same absolute amount in each unit of time, for
example, simple interest (deposit $1000 and earn 3% on the initial amount each month), a school increases
by 50 students per year, etc.
Linear growth can be modeled by A = mt + b, where A is the amount at time t, m is the absolute amount
it increases by in each unit of time, and b is the initial amount (amount at time 0).
Ex6. A town has 2,000 people in 2011 and the town’s population increases by 500 each year. Write an
equation describing the population of the town and determine what the population will be in 2025.
Price Indexes
index =
price
× 100
price in base year
The Consumer Price Index
Rate of Inflation =
CPIyear2 − CPIyear1
CPIyear1
Comparing prices in different years - convert to a base year first:
price in Y dollars = price in Xdollars ×
CP IY
CP IX
where X is the reference/base year and Y is the year you are comparing.
Exponential Growth
Exponential Growth occurs when a quantity increases by the same percentage (relative amount) in each
unit of time, for example compound interest, doubling items, etc.
Exponential growth is modeled by A = P (1+r)t , where A is the amount at time t, P is the initial/starting
amount, and r is the rate of increase/growth rate.
For an account where interest is compounded more frequently that yearly, the equation is A = P 1 +
where n is the number of times it is compounded in a year and Y is the number of years.
r nY
,
n
Exponential growth can also be modeled using the doubling time: A = P × 2t/Tdouble .
One can translate between growth rate and doubling time by using the following formulas:
Tdouble =
log 2
log(1 + r)
r = 21/Tdouble − 1
Ex7. You place $10,000 in a savings account that has a 2% interest rate (APY) compounded monthly.
How much is in your account after 15 years? How long does it take for the money to double?
Populations
Carrying capacity, logistic growth vs “overshoot and collapse”
Demographics and the demographic equation
Birth and death rate, growth rate/natural rate of increase, life expectancy, population momentum
Population density