�1

ACT U1 - Percent, Stats, and Probability Name ________________________

Statistics - Mean, Median, and Mode

- In statistics, its all about the data and what the data is saying. There are three basic functions you can perform on said data. You can take the:

_______________ - the average of the set of numbers,

_______________ - the middle term of a set that is arranged in order, or the average of the two middle terms if there is an even number of terms. Or,

_______________ - the item that occurs most frequently.

1. Find the mean of these scores: 34, 78, 62, 19, 71, 53, 86, 23, 78.

2. Below is a list of times it took Simon to run 1 mile on 7 different days. What was the average time for the 1-mile run on these days?

Simon’s times for a 1-mile run

7 minutes6.5 minutes7.5 minutes7.25 minutes6.75 minutes7.5 minutes6.5 minutes

3. The mean of a list 25.6 and there are 12 items. What is the sum of the items?

4. The sum of the items in a set is 761.8 and the mean is 58.6. How many items are there?

5. In one county, 6 farms have an average crop of 47 bushels of corn per acre. Another 9 farms have an average crop of 49 bushels of corn per acre. What is the average number of bushels of corn per acre for these farms?

6. In a group of 15 students, 6 students are 61 inches tall, 5 students are 60 inches tall, and the remaining students are 59 inches tall. What is the average height of these students?

�27. On a recent quiz, 3 students received 100, 4 students received 90, 6 students received 80, 5 students received 70, and 2 students received 60. What was the average of all the grades in the class?

8. Tim drove an average of 27.8 miles a day for 7 days. How many miles did he drive in that week?

9. From Monday to Friday, Brenda spends an average of 1.5 hours a day on the phone with her customers. On Saturday and Sunday, Benda spends an average of 1 hour a day on the phone. What is the average amount of time that Brenda spends on the phone each day for the whole week, rounded to the nearest hundredth?

10. Find the median of these scores: 34, 78, 62, 19, 71, 53, 86, 23, 78.

11. Find the median of these scores: 23, 98, 45, 16, 72, 20, 106, 72.

12. Find the mode of these numbers: 23, 98, 45, 16, 72, 20, 106, 72, 81, 45, 72.

13. In which of the following lists of numbers are the mean, median, and mode all equal?

A. 5, 3, 6, 4, 7, 5, 2, 8, 5B. 7, 2, 5, 9, 2, 4, 8, 2C. 6, 9, 4, 7, 6, 3, 5D. 1, 6, 3, 7, 3, 8E. 1, 9, 3, 7, 2, 8, 7

14. Find the mean, median, and mode for each set of data. Round to the nearest tenth.

a. 2, 5, 8, 9, 4, 3, 6, 11, 6 b. 80, 94, 64, 69, 79, 94 c. 458, 361, 467, 297, 467, 178, 621

mean: __________ mean: __________ mean: __________median: __________ median: __________ median: __________mode: __________ mode: __________ mode: __________

�3d. 5, 9, 9, 5, 4, 3, 2, 4, 10 e. 4.0, 2.4, 6.1, 5.0, 2.5, 6.0, 4.9 f. 589.7, 400, 475, 65.4, 475, 523.9

mean: __________ mean: __________ mean: __________median: __________ median: __________ median: __________mode: __________ mode: __________ mode: __________

g. 18, 37, 56, 29.8, 17.4, 24.2, 1.1, 41.6, 32.8, 37, 37.1, 29.2

mean: __________median: __________mode: __________

ACT-TYPE Problems (mean, median, and mode)1. What is the sum of the mean, median, and

mode of the numbers 4, 9, 4, 7, 1, 3, 10, 2?

A. 11B. 12C. 13D. 14E. 15

2. Below is a list of 5 people and the number of glasses of milk they drink each day:

Person # of glasses of milkBeth 3 glassesChris 5 glassesMike 2 glassesEmily 6 glassesJulia 8 glasses

What is the average number of glasses that these people drink each day, rounded to the ones place?

F. 3G. 4H. 4.8J. 5K. 5.8

3. Evan scored a total of 93 points, with an average of 18.6 points per game. How many games did Evan play?

A. 4B. 5C. 6D. 7E. 8

4. The average weight of 4 men is 172 pounds and the average weight of 6 women is 134 pounds. What is the average weight of all 10 people?

F. 145G. 148.6H. 149.2J. 151K. 153.4

5. Ed was able to do an average of 56.5 push-ups during 16 different attempts. What is the total number of push-ups that Ed completed?

A. 875B. 880C. 895D. 900E. 904

�4

Normal Distribution

- Now that you know and understand what a mean is, let’s take it a step further. In stats, you deal with probability in many forms, some easy and some not so easy. We will first look at the not so easy form first, which is _______________ _______________ .

- This type of probability is based strictly on a normal _______________ _______________ . The general form of this curve is shown below.

- The very middle of the bell curve is the mean, � . The areas under this curve represents _______________ from normal distribution. Every decimal represents a _______________ of the area under the curve. The total area under the curve is _____ .

- The typical difference between the mean and a data value is called the ______________ ______________ , which is represented by � .

- What percent of the area under a normal curve lies within:

- 1 standard deviation of the mean?

- 2 standard deviation of the mean?

- 3 standard deviation of the mean?1. A survey shows that the time spent by shoppers in supermarkets is normally distributed with a mean of 45

minutes and a standard deviation of 12 minutes.

a. What percent of the shoppers at a supermarket will spend between 33 and 57 minutes in the supermarket?

b. What is the probability that a random chosen shopper will spend between 45 and 69 minutes in the supermarket?

x

σ

�52. According to a survey by the National Center for Health Statistics, the heights of adult men in the United

States are normally distributed with a mean of 69 inches and a standard deviation of 2.75 inches. If you randomly choose 3 adult men, what is the probability that all three are 71.75 inches or taller?

3. A normal distribution has a mean of 10 and a standard deviation of 1. Find the probability that a randomly selected x-value is in the given interval.

a. between 8 and 12

b. between 7 and 13

c. between 8 and 11

d. at most 10

e. at lest 12

f. at most 9

4. Give the percent of the area under a normal curve represented by the shaded region.

a. b.

c. d.

�6

Data Collection, Representation, and Interpretation

- As we stated before, statistics is all about the data. And on the ACT, you will have to read and interpret all kinds of tables and graphs. Each type of graph or table is used for a specific reason.

Representation Uses_____ table a. compare data_____ bar graph b. brings all the data together_____ line graph c. shows max, min, median, and quartile data points_____ circle graph d. shows relative size, usually with percents_____ stem-and-leaf e. shows change_____ box-and-whisker f. group data

1. A group of 20 students was asked to give the time it takes them, in minutes, to get to school each day. The responses were as follows: 21, 10, 12, 6, 20, 17, 9, 14, 6, 8, 12, 3, 9, 25, 29, 15, 22, 23, 14, 5. Arrange the responses of these 20 students in a stem-and-leaf diagram.

Use this data for questions 2-3.

The weights, in kilograms, of 20 students are given below.

60 43 72 81 76 58 75 53 71 48 57 61 84 55 62 47 88 79 56 82

2. Use the data to create a stem-and-leaf diagram.

3. Use the data to create a box-and-whisker plot.

�7Use this data for questions 4-5.

Given below is the amount of money Chad spent on his car from April through August.

4. Create a bar graph.

5. Create a line graph.

Use this data for question 6-7.

Roman is a car salesman. Given the below are the number of cars Roman sold during a six month period.

6. Create a line graph.

7. Create a bar graph.

�88. Use the stem-and-leaf diagram below, write the values for the minimum, first quartile, median, third quartile, and maximum.

9. Draw a box-and-whisker plot that incorporates the data from the previous problem.

10. Below is a bar graph comparing the number of miles 5 friends jog each week. Who jogs the second-farthest each week?

11. The circle graph below displays the amount of time Danielle spends doing a specific activity each day. On which activity does Danielle spend the least amount of time?

�9

ACT-TYPE Problems (Tables, Charts, and Graphs)

1. Which of the following is greater than 75% of the scores shown in the box-and-whisker plot shown below?

A. 9B. 12C. 14D. 26E. 28

2. A store manager receives hourly updates about the sales in cash registers throughout the store. The stem-and-leaf display below shows the sales from the most recent hour. What is the median sale?

F. 128.5G. 143H. 143.5J. 158K. 159

3. Alex likes to eat salads for lunch. Alex makes his salads with lettuce, tomatoes, onions, mushrooms, and peppers. Alex uses 2 times as many mushrooms as peppers, the same amount of onions as peppers, 3 times as many tomatoes as peppers, and 4 times as much lettuce as peppers. Which of the following circle graphs best fits this information?

A. B.

C. D.

E.

4. The bar graph below shows the scores on a recent math test. What percentage of the class scored in the 80’s or higher?

F. � G. �

H. �

I. �

J. �

5. During an August heat wave, Wendy kept track of the high temperature every day for a week. She graphed her results on a line graph. To the nearest degree, what was the average high temperature that week?

A. � B. � C. � D. � E. �

8%25%

4123%

66 23%

9123%

95°96°97°98°100°

�10

Probability

- _______________ of an event is the likelihood that it will occur. If an event will never occur, the probability is _____ . If an event will always occur, the probability is _____ . All other probabilities fall between 0 and 1. Use fractions to write the probability of an event. So, you should learn to appreciate fractions.

Think of rolling a fair die.

1. What is the probability of rolling a 3? __________2. What is the probability of rolling an even number? __________3. What is the probability of rolling a 7? __________4. What is the probability of rolling a number less than 10? __________

5. In a jar there are 3 blue marbles and 5 red marbles. If you reach into the jar and pick out a marble without looking, what is the probability that you will pick out a blue marble?

6. What is the probability of choosing, without looking, a red face card from a regular deck of cards? (A face card is a jack, a queen, or a king.)

You have a fair penny.

7. What is the probability of flipping a tail? __________8. What is the probability of flipping a head? __________9. What is the probability of flipping a head or a tail? __________10. You flip the penny 10 times and get 10 heads. What is the probability of getting a head on the next flip?

__________

You have six same-size balls numbered 1, 2, 4, 5, 7, and 8 in a box. You pick one without looking. What is the probability of picking

11. an 8? __________12. an even number? __________13. a number greater than 6? __________14. a number divisible by 6? __________15. a multiple of 4? __________

You have a standard deck of 52 cards. What is the probability of picking

16. a 7? ___________17. a red card? __________18. a heart? __________19. a non-face card? __________

20. a 2, 3, or 4? __________21. a 9 or 10? __________22. a jack, or a queen, or a king? __________

�11

ACT-TYPE Problems (Probability)

1. In a drawer there are 3 brown socks, 2 blue socks, and 5 red socks. You pick a sock without looking. What is the probability of choosing a brown sock?

A. �

B. �

C. �

D. �

E. �

2. What is the probability of rolling a 1 or a 5 with a single six-sided die?

F. �

G. �

H. �

J. �

K. �

3. From a regular deck of 52 cards, what is the probability of choosing a card that is not an ace or a face card, but that is divisible by 2?

A. �

B. �

C. �

D. �

E. �

4. If a king and a jack are removed from a regular 52-card deck, what is the probability of picking face card?

F. �

G. �

H. �

J. �

K. �

5. There are 20 colored pencils in a box: 4 red, 5 green, 10 blue, and 1 black. You pick a pencil without looking. What is the sum of the probability of picking a red pencil and the probability of picking a black pencil?

A. �

B. �

C. �

D. �

E. �

23310121537

1613122356

32611314713513

1513232535

710151201412

�12

Counting

- There are some elementary techniques that can help you to count very efficiently. We will talk about three of them: _______________ , _______________ , and _______________ .

*Note: Calculators are your best friend when working counting problems.

Product - just multiply.

1. You are buying a frozen yogurt cone. The yogurt store has three different types of cones, six different flavors, and eight kinds of toppings. Multiply to find how many types of yogurt cones you can buy.

Permutations - multiply in a specific order. (factorials)

2. Three students, Alex, Bonnie, and Charles, line up single file. How many different ways can they line up? Feel free to make a list.

3. Nine students are going to line up single file for movie tickets. In how many different ways can the students line up?

4. Four students in a club are interested in two positions, president and vice president. How many ways can the students be chosen for two positions?

5. There are five cars and three parking spaces. In how many different ways can the cars be parked in the spaces?

�13Combinations - an arrangement of a certain number of items in which order does not matter.

6. There are just two parking spaces, but four cars — red, blue, yellow, and green. In how many different ways can two cars be parked? Order doesn’t matter.

7. There are three job openings for Level I computer technicians, and six applicants. How many different ways could these six applicants be chosen to fill the three job openings?

Practice of All Types

8. How many different ways are there to place 5 books on 5 different shelves if you can place only one book on each shelf?

9. If Andy has 7 shirts, 6 pants, and 8 ties, how many different outfits can he make with the pants, shirts, and ties?

10. There are 4 coach seats left on an airplane and 7 people waiting for those seats. How many different ways can you choose people to fill the 4 seats?

11. There are 4 people running for 4 different positions: President, Vice President, Secretary, and Treasurer. How many different ways are there to place the 4 people into these different positions?

12. For breakfast, Jane has cereal, orange juice, an apple, and a piece of toast. Jane keeps 4 different types of cereal, 2 different kinds of orange juice, 3 different types of apples, and 2 types of bread. How many different breakfast choices does she have?

�1413. There are 6 puppies born, but the mother always feed only 4 puppies at a time. How many different groups

of puppies can she feed at one time?

14. There are 5 windows in a room, and Rich bought 5 different curtains to go on the windows. How many different ways can Rich place the curtains on the windows?

15. Steve knows 5 notes on the guitar and plays 4 notes in a row. How many different ways will Steve be able to arrange his notes?

16. There are 9 different things to drink in Lisa’s house, but Lisa has only 3 glasses. How many different ways are there to put 3 different drinks in the glasses. (You can put only one type of drink into each glass.)

17. A 7-digit phone number uses all the digits from 3 to 9. How many different possible phone numbers are there?

�15ACT-TYPE Problems

1. Eight students try out for two openings on the debate team. In how many different ways can these two openings be filled?

A. 2B. 8C. 28D. 56E. 84

2. A store manager has four different gifts to give to the first four people who enter the store. In how many different ways can he distribute the gifts?

F. 4G. 8H. 16J. 24K. 32

3. A license plate has three letters (A-Z) followed by three digits (0-9). How many different license plates can be produced?

A. 11,232,000B. 12,654,720C. 12,812,904D. 15,600,000E. 17,576,000

4. A newspaper editor needs five stories for the front page of the next issue. There are six reporters available to write the stories. How many ways can they be assigned if each story is written by only one reporter?

F. 5G. 6H. 30J. 120K. 720

5. How many 3-digit area codes can be created if the first digit cannot be zero?

A. 100B. 300C. 729D. 900E. 999