Name:________________________________
Statistics: Easter Break Assignment
Date:_______ Period:______
Algebra II & Trig
Statistics
Statistics is the study and interpretation of numerical data. If only one variable is being
analyzed, it is called univariate statistics. There are a number of ways that data can be gathered
for statistical studies.
Data Gathering Methods:
 Surveys: used to collect information from a group of people. A survey is an efficient way
of obtaining a wide range of information from a large number of people. Validity of
surveys depends on the honesty of its participants.
 Census: type of survey in which an attempt is made to reach every member of an entire
population. For example, the United States has a census every 10 years to count how
many people are in the country.
 Sample: a subset of the entire population. Many times it is not cost-effective or
productive to gather information from an entire population. When this is the case, a
sample is used. It is always important to make sure that the survey sample is
representative of the entire population.
 Simulation: Scientists and engineers use this to determine information about an event.
For example, to determine the amount of weight a bridge can hold, engineers first
simulate the effects statistically on a computer. Then, they use that data to draw
conclusions about how the bridge should be constructed to bear the weight that it will
be holding.
 Controlled Experiment: a study that is used to gather information about the effect of
some kind of intervention. For example, medicine or an exercise program. The results
obtained in from an experimental sample are compared to the results from a control
sample. A control group is identical to the experimental group, except without one
aspect whose effect is being tested.
 Observational Study: individuals are observed or certain outcomes are measured. Data
is gathered without any intervention. There is no attempt made to affect the outcome.
Bias and Random Sample:
In statistics, it is important to determine if the information that is presented is somewhat
biased, data used in the sample have come from sources that have a particular interest in the
impact of the statistics. When this is the case, the sample is not representative of the general
population, and it is said to be biased or skewed.
In order that the sample reflects the properties of the entire group, the following three
conditions must exist:
1. The sample must be representative of the group being studied.
2. The sample must be large enough to be effective.
3. The selection should be random or determined in such a way as to eliminate any bias.
Model Problem 1:
Car manufactures want to explore the desirability of pre-installed infant seats in their new
car designs. Which of the following groups would be most likely to provide an unbiased
sample for a survey?
(1) adult shoppers at a supermarket with young children in shopping carts
(2) couples signing up to win a honeymoon package at a bridal expo
(3) adults visiting a car show
(4) a group of teenagers playing ball at the park
Answer: (3) is the least biased because (1) contains people who already need car seats; (2)
involves couples who might be planning on having kids; (4) involves teenagers for whom
parenthood is not relevant.
Model Problem 2:
A new medicine intended for use by adults is being tested on five men whose ages are 22,
24, 25, 27, and 30. Does the sample provide a valid test?
Answer: No. The sample is too small, the sample includes only men, and the sample does
not include adults over the age of 30.
Problems:
1. Which would be the most unbiased group of people to ask about methods of Social Security
reform?
(1) adults at a senior citizen center
(2) college students in a sociology class
(3) members of a children's choir
(4) shoppers at a mall
2. Reporters on a news show want to survey adults about their exercise habits. Where should
they go to find an unbiased sample?
(1) the boardwalk on the beach
(2) the post office
(3) an exercise gym
(4) a rock-climbing expo
3. The city board of directors is considering raising the cost of parking tickets. Which group of
citizens would be least biased on this issue?
(1) fifty citizens who have never received a parking ticket
(2) fifty citizens who have had at least three parking tickets
(3) the first fifty people encountered on a city street
(4) fifty citizens who do not have driver's licenses.
4. A survey completed at a large university asked 2,000 students to estimate the average
number of hours they spend studying each week. Every tenth student entering the library was
surveyed. The data showed that the number of hours that the students spending studying was
15.7 per week. Which characteristic of the study could create a biased result?
(1) the size of the sample
(2) the size of the population
(3) the method analyzing the data
(4) the method choosing the students who were surveyed
Examples 5 - 10: Suggest a method (population survey, sample survey, simulation, observation,
controlled experiment) that might be used to collect data for the study.
5. Number of texts sent a week by students in St. Francis Prep.
6. Amount of fertilizer to be used to produce a prize-winning watermelon.
7. Number of hours of sleep obtained by a whale living in the ocean.
8. Number of hours spent watching television by members of your COR.
9. Effect of an advertising campaign on "name recognition" of a new product.
10. Amount of time for the red, green, and yellow lights to reaming active in a busy
intersection.
Measures of Central Tendency:
Measures of central tendency are summary statistics that indicate where the center of a
collection of data lies. Three common measures of central tendency are mean, median, and
mode.
Mean:
The mean refers to the arithmetic average of the data and its symbolized as X , read as "x bar."
The mean is the most common measure of central tendency. The mean is the sum of all of the
data values divided by the number of data values.
Median:
The median is the middle number of a data set arranged in numerical order. Therefore,
whenever we are looking for the median, we must write our data in numerical order. To find
out which position is the median, add 1 to the total number of data values and divide by 2. The
value in that position, counting from the first or last term, is the median. If there is no middle
term, find the average of the middle two terms to arrive at your median.
Mode:
The mode is the most commonly repeated data value in a set. Some collections of data, often
referred to as distributions, will have no mode and others will have multiple modes.
Standard Deviation
The standard deviation shows how much variation there is from the "average" (mean). A low
standard deviation indicates that the data points tend to be very close to the mean, whereas
high standard deviation indicates that the data are spread out over a large range of values.
Finding Mean/Median/Mode and Standard Deviation using the calculator.
Press STAT 1.Edit
Enter the data into List 1.
If there is a frequency list or any second list, enter that data into List 2.
Press STAT , move the arrow to CALC, and press 1. 1-Var Stats , then Enter
If there is a frequency list, Press STAT CALC 1. 1-Var Stats 2nd 1, 2nd 2 Enter
A list of statistical information will be displayed:
x
mean of the data
x
sum of the values
sum of the squares of the values
x 2
Sx
sample standard deviation
population standard deviation (ALWAYS USE POPULATION FOR STANDARD
 x
n
number of items in sample
DEVIATION UNLESS THE PROBLEMS SAYS
min X lowest value
THE WORD SAMPLE)
Q1
first quartile
Med
median
Q3
third quartile
max X maximum value
(Many of these items, you will not use. Just copy the numbers needed to answer the question.
Also, mode is not given – you still have to find that by hand)
Model Problem 3:
Nine members of the basketball team played during all or part of the last game. The number
of points scored by each of the players was: 21, 15, 12, 9, 8, 7, 5, 2, 2. What is the mean?
Answer: mean = 9
Model Problem 4:
An English teacher recorded the number of spelling errors in the 40 essays written by
students. The table below shows the number of spelling errors and the frequency of the
number of errors, that is, the number of essays that contained that number of misspellings.
Find the mean number of spelling errors for these essays.
Errors
0 1 2 3 4 5 6 7 8 9 10
Frequency 1 3 2 2 6 9 7 5 2 1 2
Answer: mean = 5.1
Model Problem 5:
Find the standard deviation, to the nearest tenth, of the following set of data: 11, 17, 31, 18,
25, 12, 19, 13, 15.
Answer:
 x = 6.1
Model Problem 6:
Geoff stocked up on canned goods at the grocery store. The frequency table to the right
shows the price of the canned goods. Find the median price of the canned goods.
Price (in cents) 56 73 78 82 86
Frequency
4 9 7 5 5
Answer: The median is 78, but your answer is $0.78, since the data is given in cents.
Model Problem 7:
Find the mode of the following data: 67, 54, 91, 67, 83, 46, 72, 54, 91, 81, 75, 67, 54, 88.
Answer:
Since 67 and 54 both appear three times, both of these serve as modes. This data set is said
to be bimodal.
Model Problem 8:
In May 2008, the government predicted that gasoline prices would peak at a national
average of $3.73 a gallon in June. However, this prediction turned out to be incorrect. The
average weekly prices of gasoline in Rochester from May 7 to July 16, 2008 were: $3.85,
$3.92, $4.09, $4.12, $4.18, $4.20, $4.20, $4.20, $4.23, $4.24. Find the mean, median, and
mode for these data. (Round values to 2 decimal places) Determine the standard deviation
for this data
Answer: Mean: 4.12
Median: 4.19
Mode: the number that repeats the most often is $4.20
Standard Deviation: .13
Problems:
12. What is mean of 37, 54, 72, 89, 74, 83, 90, and 93?
11. The table below lists the scores for a multiple-choice test is Ms. A's class. Find the mean,
median, and mode for the data to the nearest hundredth.
Test Grade
100
95
90
85
80
75
70
60
Frequency
1
2
2
7
6
5
3
2
13. What is the mean of the data in the accompanying table?
Scores
25
20
11
Frequency
3
2
5
(1) 11
(2) 14.5
(3) 15
(4) 16
10
4
14. The term “snowstorms of note” applies to all snowfalls over 6 inches. The snowfall
amounts for snowstorms of note in Utica, New York, over a four-year period are as follows: 7.1,
9.2, 8.0, 6.1, 14.4, 8.5, 6.1, 6.8, 7.7, 21.5, 6.7, 9.0, 8.4, 7.0, 11.5, 14.1, 9.5, 8.6
What are the mean and population standard deviation for these data, to the nearest hundredth?
(1) mean = 9.46; standard deviation = 3.74
(3) mean = 9.45; standard deviation = 3.74
(2) mean = 9.46; standard deviation = 3.85
(4) mean = 9.45; standard deviation = 3.85
13. Andrew wanted a raise in his allowance for doing yard work. His father said he should find
out the "average" payment others in the neighborhood received. Andrew surveyed the other
families on the block and discovered that they paid the following prices for having their lawns
cut and raked: $18.00, $22.00, $17.50, $15.50, $25.00, $17.50, $20.00, $15.00, $26.00, $28.00.
a. Which "average" should Andrew use to promote his increase in his allowance?
b. Which "average" might his father use to rebut Andrew's argument?
15. For the data 14, 18, 21, 19, 27, 23, 17, which statement is true?
(1) mean = median (2) mean < median (3) mean > median
(4) median = mode
3. From 1984 to 1995, the winning scores for a golf tournament were 276, 279, 279, 277, 278,
278, 280, 282, 285, 272, 279, and 278. Find the Sample Standard Deviation for this data.
14. One hundred senior girls were interviewed about their price limit for the "perfect" prom
dress. Their responses were summarized in the table below. Determine the mean, median, and
mode for the set of data.
Maximum Price
$100
$150
$200
$250
$300
(prom dress)
# of Girls
23
22
26
14
15
7. The scores on a mathematics test are 42, 51, 58, 64, 70, 76, 76, 82, 84, 88, 88, 90, 94,
94, 94, and 97. For this set of data, find the standard deviation to the nearest tenth.
6. Using the scores in the accompanying table, find the standard deviation to the nearest
hundredth. Find the mean to the nearest tenth.
Scores
60
65
70
75
80
Frequency
2
6
4
8
5
8. Find, to the nearest tenth, the standard deviation of this set of data.
Find the mean.
xi
87
89
91
93
95
fi
3
4
3
6
2
BINOMIAL EXPANSION
How to find a one term in a binomial expansion such as find the 5th term of (3x – 3)12
Let’s call the term we are looking for r (or the rth term)
To find the rth term of (a + b)n we use:
n
Ex: Find the 5th term of (3x – 4)12
n
C r-1 an - (r - 1) br -1
C r an - (r - 1) br -1 for (ax + b)n
n = 12, r = 5, a = 3x and b = (-4)
write the formula
C 5-1 (3x)12 - (5 - 1) (-4)5 -1
substitute n, r, x and b
12
(4)
4
(-4)
simplify
12 C 4 (3x)
8(-4)4
C
(3x)
simplify
12 4
12
Now you use your calculator to solve
to get 12 C 4 : type 12, then
then hit
to “PRB”
for “nCr”, then type 4
your screen will say 12 nCr 4 hit
and it equals 495.
Then you need to find 3 to the 8th power = 6561 (get by typing
You then find (-4) to the fourth power = 256 (get by typing
Now substitute those into the formula.
(495)(6561x8)(256)
Multiply all of the numbers and variable to get your answer 831,409,920x8
Ex 2 :Find the 3rd term of (x + 3y)9
nCr-1 an-(r-1) br-1
9C3-1
(x)9-(3-1) (3y)3-1
n = 9, r = 3, a = x and b = 3y
write the formula
substitute n, r, x and b
(9C2)(x)9-2(3y)2
simplify
7
2
(36)(x) (3y)
use calculator to find 9C2
(36)(x7)(9y2) = 324x7y2
multiply coefficients for final answer
Find the rth root of each of the following binomials.
1) Find the 3rd term of (x + y)6
2) Find the 6th term of (x – 5)8
th
8
3) Find the 4 term of (2x + 7y)
4) Find the 4th term of (1 – x)7
5) Find the 10th term of (x + 3)12
6) Find the 3rd term of (3x – 2)10
)
)
PROBABILITY: BERNOULLI EXPERIMENTS
nC r s
r f n-r
n = number of trials
r = exactly # of successes
s = probability of a success
f = probability of a failure
n  r = number of failures
used for "exactly"
r successes
Ex 1: A test consists of 10 multiple choice questions with five choices for each question. As an
experiment, you GUESS on each and every answer without even reading the questions.
What is the probability, the thousandth, of getting exactly 6 questions correct on this test?
Solution: n = 10
r=6
nC r s
r f n-r
n–r=4
0.005505024
The probability of getting exactly 6 questions correct is about 0.006
Ex 2: When rolling a die 100 times, what is the probability of rolling a 4 exactly 25 times?
r n-r
Solution: n = 100
nC r s f
r = 25
n – r = 75
0.0098258819
The probability of rolling 4 exactly 25 times is approximately 0.010
EXERCISES:
1. If the probability that it will rain on any given day this week is 60%, find the probability that
it will rain exactly 3 out of 7 days this week.
2. The Coolidge family’s favorite television channels are 3, 6, 7, 10, 11, and 13. If the
Coolidge family selects a favorite channel at random to view each night, what is the
probability that they choose exactly three even numbered channels in five nights? Express
your answer as a fraction or as a decimal rounded to four decimal places.
4. Ginger and Mary Anne are planning a vacation trip to the island of Capri, where the
probability of rain on any day is 0.3. What is the probability that during their five days on
the island, they have no rain on exactly three of the five days?
3. During a recent survey, students at Franconia College were asked if they drink coffee in the
morning. The results showed that two-thirds of the students drink coffee in the morning and
the remainder do not. What is the probability that of six students selected at random,
exactly two of them drink coffee in the morning? Express your answer as a fraction or as a
decimal rounded to four decimal places.
5. Which fraction represents the probability of obtaining exactly eight heads in ten
tosses of a fair coin?
(1) __45__
(2) __64__
(3) __90__
(4) __180__
1024
1024
1024
1024
6. The probability that Kyla will score above a 90 on a mathematics test is 4/5. What is
the probability that she will score above a 90 on three of the four tests this quarter?
7. The Hiking Club plans to go camping in a State park where the probability of rain on
any given day is 0.7. Which expression can be used to find the probability that it will
rain on exactly three of the seven days they are there?
(1) 7 C3 (0.7) 3 (0.3) 4
(2) 7 C3 (0.3) 3 (0.7) 4 (3) 4 C3 (0.7) 3 (0.7) 4 (4) 4 C3 (0.4) 4 (0.3) 3
AT LEAST/AT MOST
Some questions will ask for “at least” or “at most” a certain amount of favorable outcomes:
There are five questions on a test, the probability of getting a question correct is 0.25.
a) What is the probability of getting at least 3 questions correct?
b) What is the probability of getting at most 3 questions correct?
a) At least 3 correct means you can get exactly 3 correct, exactly 4 correct, or exactly 5 correct.
Use Bernoulli’s Formula to find the Probability of exactly 3, then exactly 4 and exactly 5
and we add the 3 probabilities together
P (at least 3 correct) = P(exactly 3) + P(exactly 4) + P(exactly 5)
+
+
= 0.087890625 + 0.0146484375 + 0.00097656525
= 0.10351515625
b) At most 3 correct means you can get exactly 0, exactly 1, exactly 2, or exactly 3 correct.
P(at most 3 correct) = P(exactly 0) + P(exactly 1) + P(exactly 2) + P(exactly 3)
+
+
+
= 0.237346875 + 0.3955078125 + 0.263671875 + 0.087890625
= 0.9844171875
EXERCISES:
1. Tim Parker, a star baseball player, hits one home run for every ten times he is at bat. If
Parker goes to bat five times during tonight’s game, what is the probability that he will hit at
least four home runs?
3
2. The probability that a planted watermelon seed will sprout is . If Peyton plants seven
4
seeds from a slice of watermelon, find, to the nearest ten-thousandth, the probability that
at least five will sprout.
3. On mornings when school is in session in January, Sara notices that her school bus is late
one-third of the time. What is the probability that during a 5-day school week in January
her bus will be late at least three times?
4. Dr. Glendon, the school physician in charge of giving sports physicals, has compiled his
information and has determined that the probability a student will be on a team is 0.39.
Yesterday, Dr. Glendon examined five students chosen at random.
a) Find, to the nearest hundredth, the probability that at least four of the five students will
be on a team.
b) Find, to the nearest hundredth, the probability that exactly one of the five students will
not be on a team.
5. On any given day, the probability that the entire Watson family eats dinner together is
2/5. Find the probability that, during any 7-day period, the Watsons eat dinner
together exactly 3 times.