Statistics: Easter Break Assignmentalgebra II & Trig

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Statistics: Easter Break AssignmentAlgebra 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.

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 , 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:

mean of the data

sum of the values

sum of the squares of the values

sample standard deviation

population standard deviation (ALWAYS USE POPULATION FOR STANDARD

number of items in sample DEVIATION UNLESS THE PROBLEMS SAYS

lowest value THE WORD SAMPLE)

first quartile

median

third quartile

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)

Problems:

11. What is mean of 37, 54, 72, 89, 74, 83, 90, and 93?

12. 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.

13. What is the mean of the data in the accompanying table?

(1) 11(2) 14.5(3) 15(4) 16

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

15. 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?

16. 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

17. 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.

18. 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.

19. 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.

20. Using the scores in the accompanying table, find the standard deviation to the nearest

hundredth. Find the mean to the nearest tenth.

21. Find, to the nearest tenth, the standard deviation of this set of data.

Find the mean.

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 C r - 1 an - (r - 1) br -1 for (ax + b)n

Ex: Find the 5th term of (3x – 4)12n = 12, r = 5, a = 3x and b = (-4)

n C r-1 an - (r - 1) br -1write the formula

12 C 5-1 (3x)12 - (5 - 1) (-4)5 -1substitute n, r, x and b

12 C 4 (3x)12 - (4) (-4)4simplify

12 C 4 (3x)8(-4)4simplify

Now you use your calculator to solve

to get 12 C 4 : type 12, then to “PRB”

then hit 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)9n = 9, r = 3, a = x and b = 3y

nCr-1 an-(r-1) br-1write the formula

9C3-1 (x)9-(3-1) (3y)3-1substitute n, r, x and b

(9C2)(x)9-2(3y)2simplify

(36)(x)7(3y)2use calculator to find 9C2

(36)(x7)(9y2) = 324x7y2multiply coefficients for final answer

Find the rth root of each of the following binomials.

22) Find the 3rd term of (x + y)623) Find the 6th term of (x – 5)8

24) Find the 4th term of (2x + 7y)825) Find the 4th term of (1 – x)7

26) Find the 10th term of (x + 3)1227) Find the 3rd term of (3x – 2)10

PROBABILITY: BERNOULLI EXPERIMENTS

nCr sr f n-r

n = number of trials

r = exactly # of successes used for "exactly"

s = probability of a success r successes

f = probability of a failure

n  r = number of failures

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 = 10nCr sr f n-r

r = 6

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?

Solution: n = 100nCr sr f n-r

r = 25

n – r = 75

0.0098258819

The probability of rolling 4 exactly 25 times is approximately 0.010

EXERCISES:

28. 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.

29. 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.

30. 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?

31. 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.

32. 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

33. 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?

34. 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) (2) (3) (4)

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:

35. 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?

36. The probability that a planted watermelon seed will sprout is . If Peyton plants seven

seeds from a slice of watermelon, find, to the nearest ten-thousandth, the probability that at least five will sprout.

37. 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?

38. 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.

39. 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.