PRE-CALCULUS DATA AND STATISTICS UNIT Measures of Central Tendency and Variance

Mean: Median: Mode: The following is the number of days that it takes UPS from mailing to delivery of a package:

{𝟔, 𝟒, 𝟑, 𝟒, 𝟐, 𝟓, 𝟑, 𝟒, 𝟓, 𝟐, 𝟑, 𝟒} Mean: Median: Mode: Calculating the Expected Value (Weighted Mean) The probability distribution for the number of games played in each World Series.

Number of Games 4 5 6 7

Probability of this number of games

5/27 5/27 6/27 11/27

Calculate the Expected Value:

𝟒 ( ) + 𝟓 ( ) + 𝟔 ( ) + 𝟕 ( )

Box and Whisker Plots 5 significant values needed for a Box and Whisker Plot: Array: Median: Max/Min Q1/Q3: Interquartile Range Make a Box and Whisker Plot: The number of complaints about the weather heard in 2 weeks:

{𝟓, 𝟑, 𝟗, 𝟐, 𝟏𝟒, 𝟔, 𝟖, 𝟓, 𝟖, 𝟏𝟑, 𝟑, 𝟏𝟓, 𝟕, 𝟒} Array: Median: Q1/Q3: Range:

Measures of Variance

A measure of _________________________ is a value that determines how ___________________ out the data is from the center: 3 Main Types: Range: Variance: Standard Deviation:

Low Standard Deviation means: How to Find the Variance and Standard Deviation:

1. 2. 3. 4.

Find the variance and standard deviation of the following: {𝟏𝟎, 𝟏𝟐, 𝟏𝟒, 𝟏𝟓, 𝟏𝟖, 𝟐𝟎, 𝟐𝟑}

Using a Calculator to Find Statistics 1. Press STAT to enter values in the L1 column. 2. Press STAT, then scroll to CALC and select 1 (1-Var Stats) Use the calculator to verify your calculations above.

Outliers An outlier is an ________________________ value that is much greater or much less than the other data. Outliers have a strong effect on both the ________________ and __________________ _________________ An outlier is a data value that is more than _____ standard deviations from the mean.

|𝑽𝒂𝒍𝒖𝒆 − 𝒎𝒆𝒂𝒏|

𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒅𝒆𝒗𝒊𝒂𝒕𝒊𝒐𝒏 >

The following numbers represent the number of points a PW basketball player scored in each game.

{𝟐, 𝟔, 𝟒, 𝟐, 𝟔, 𝟔, 𝟏𝟎, 𝟑, 𝟏𝟗, 𝟒, 𝟒, 𝟐, 𝟑} Use a calculator to verify that 19 is an outlier.

Homework for Measures of Central Tendency and Variance 1. The probability distribution of the number of accidents in a week at a busy intersection is as given:

Number of Accidents 0 1 2 3

Probability 0.75 0.15 0.08 0.02

Calculate the expected number of accidents for one week. 2. The following is the probability distributions for the payout of a lottery ticket:

Value $0 $1 $5 $20 $100 $1000

Probability 0.9359 0.05 0.01 0.003 0.001 0.0001

Calculate the expected value of the ticket: 3. Make a box and whisker plot of the following:

{𝟏𝟑, 𝟏𝟒, 𝟏𝟖, 𝟏𝟑, 𝟏𝟐, 𝟏𝟕, 𝟏𝟓, 𝟏𝟐, 𝟏𝟑, 𝟏𝟗, 𝟏𝟏, 𝟏𝟒, 𝟏𝟒, 𝟏𝟖, 𝟐𝟐, 𝟐𝟑} Array: Median: Q1: Q3: Range: 4. Find the variance and standard deviation without using the STAT key.

{𝟏𝟎, 𝟏𝟐, 𝟏𝟒, 𝟏𝟓, 𝟏𝟖, 𝟐𝟎, 𝟐𝟑}

Verify using the STAT function. 5. Suppose you have 10% chance of winning $100, a 30% chance of losing $2 (-2), and a 60% chance of breaking even. What is the expected value? 6. The Detroit Lions scored 24, 16, 9, 17, 17, 23, 20, 26, 17, 14, 58, 27, 28, 14, 17 and 7 in their 16 games last year. Find the mean and the standard deviation. Determine if 58 is an outlier and determine by how much it affected the mean and standard deviation.

PRE-CALCULUS DATA AND STATISTICS UNIT Gathering Data

A __________________________________ is the entire group of people or objects that you want information about. A ______________________________ is a survey of an entire population. A _________________________ is a part of the population surveyed. A _______________________ sample is one in which every member of the population has an __________ chance of being selected for a sample. A _____________________ sample is not representative of a population. A _________________________________ is a number that describes a sample. A _______________________________ is number that describes a population. Many times you can use a statistic to estimate a parameter. Example: In a survey of 50 students at a high school, 32 students said that they plan on attending the homecoming dance. The school has 325 students. Which number is a statistic? Which number is a parameter? Predict the number of students who plan on attending.

Data Collection Methods An ___________________________________ imposes a treatment on individuals to collect data on their response to the treatment. An ________________________________ _______________ observes individuals and measures variables without any controls on the individual in any way. A __________________________ will ask participants questions about their opinions or behavior. A researcher asks students the average number of hours of sleep each got per night to see if the amount of sleep affects student grades. What is it? A park employee wants to know if latex paint is more durable than oil-based paint. He paints 50 benches with each type. What it is? In a controlled experiment, the groups are divided into 2 groups: The _______________________________ group receives the treatment. The _______________________________ group receives In an unbiased experiment, groups are divided ______________________________. Also, a good experiment is conducted with a ________________________-blind technique.

Homework for Gathering Data 1. A researcher is considering 3 methods of evaluating two different types of cold medicines. Identify each type and which is most reliable.

Method A – Choose 50 people at random. Ask which cold medicines they have taken in the past and how effective they were.

Method B – Monitor 50 people with colds, and measure the length of the symptoms for the individuals who choose to take each type of medicine.

Method C – Randomly divide a group of 50 people with colds into two groups. Give each group a different medicine, and measure the length of the symptoms. 2. Which method is the most representative of the population? The Student Council wants to know if the students would like to have a wider selection of foods on the salad bar in the cafeteria.

Method 1 – Survey the first 30 students who walk into the school. Method 2 – Survey every student who buys a lunch on Friday. Method 3 – Survey every student who buys a meal at least 3 times a week.

3. In a survey of 30 employees at a company, 25 employees said that they were satisfied with their jobs. The company has 210 employees. Predict the number of employees that should look for a new job.

Biased or Unbiased? (Explain why it is unbiased) 4. On the first day of school, all incoming freshmen attend an orientation program. The principal wants to

learn the opinions of the freshmen regarding the orientation program. He decides to ask the first 25 freshmen that he sees.

5. The manager of an apartment building wants to know if the residents are satisfied with his service. He writes each apartment number on a piece of paper and places them in a hat. He draws 10 slips of paper and goes and asks those residents. 6. The members of a drama club want to know how much students are willing to pay for watching a performance. They decide that each member should ask five friends what they are willing to pay. 7. The manager of a city bus system wants to know if the people who ride the buses are satisfied with the service. He decides to post mail-in surveys on each of the city’s buses.

Experimental or Observational Study? 8. Does using a certain brand of cleaning product put people who use it frequently at greater risk of respiratory problems? 9. Does a certain shampoo work to reduce dandruff? 10. Is a certain chemical effective at killing a certain bacteria commonly found in household kitchens and bathrooms? 11. Do dogs kept as pets, live longer if they are allowed to run in a yard every day? 12. Does working a job for which half or more of the tasks involve use of computers increase the risk of certain cancers? Classify each method as a survey, experiment or observational study. Which is most reliable? 13. Randomly choose 50 people to exercise 3 hours a week and 50 people to do some other activity, then monitor their health. 14. Randomly choose 100 people. Ask how many hours a week they exercise, and how healthy they are. 15. Choose 50 people who exercise regularly and 50 who do not, then monitor their health.

PRE-CALCULUS DATA AND STATISTICS UNIT Significance of Experimental Results

Hypothesis Testing- used to determine if the difference in two groups is likely to be caused by chance. (Or is something actually causing the difference) Think about flipping a coin- what do you expect? What if you flipped 20 times and got only 1 head? Hypothesis Testing begins with an assumption called the ______ hypothesis. This is a statement that there is ____ difference between the two groups. The null hypothesis is the _________________ of what the experimenter wants to prove. Rejecting the null hypothesis is the goal of most experiments. Example: A pre-calculus teacher wants to know if students in his morning class do better on a test than students in his afternoon class. He compares the test scores of 10 randomly chosen students in each class. Morning Class: 76, 81, 71, 80, 88, 66, 79, 67, 85, 68 Afternoon Class: 80, 91, 74, 92, 80, 80, 88, 67, 75, 78 a. State the null hypothesis: b. Compare the results. Does the teacher have enough evidence to reject the null hypothesis?

z-Tests If a sample contains at least 30 individuals, an experimenter can use a _____________ to compare the _____________ of the sample to the ___________ of the population.

𝒛 = �̅� − 𝝁

𝝈

√𝒏

𝒘𝒉𝒆𝒓𝒆 𝝁 𝒊𝒔 𝒕𝒉𝒆 𝒎𝒆𝒂𝒏 𝒐𝒇 𝒕𝒉𝒆 𝒑𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏, 𝒙 ̅ 𝒊𝒔 𝒕𝒉𝒆 𝒎𝒆𝒂𝒏 𝒐𝒇 𝒕𝒉𝒆 𝒔𝒂𝒎𝒑𝒍𝒆.

A common measure for z-tests is known as the ___% confidence level. If |𝒛| > 𝟏. 𝟗𝟔, then you can reject the null hypothesis with ____% certainty If |𝒛| < 𝟏. 𝟗𝟔, then you do not have enough evidence to reject the null hypothesis. Example: A tax preparer claims an average refund of $3000. In a random sample of 40 clients, the average refund was $2600 and the standard deviation was $300.

What is the null hypothesis? Is there enough evidence to reject his claim?

Homework for Significance of Experimental Results 1. Westphalia is conducting a water quality study. It is measuring trace levels of

certain chemicals in the blood of the residents with the levels in the blood of the residents of Pewamo who already know their water quality is high. The blood levels of the substance in micrograms per milliliter are shown in the chart.

Pewamo 9 5 4 7 5 8 7 3 4 6 8 5 Westphalia 2 4 8 5 9 7 9 5 4 10 5 2

a. State the null hypothesis for the experiment. b. Compare the two towns using a box and whisker plot for each. Do you think there is enough

evidence for the researcher to reject the null hypothesis?

2. An investment consultation firm claims that it will increase the returns on its clients’ investments to an average of 20% of the original with little risk involved. In a random sample of 50 clients, that average return on investments was 18.5% with a standard deviation of 4%. State the null hypothesis: Calculate the z-value to the nearest hundredth. Is there enough evidence to reject the claim? 3. The publishers of an AP Psychology study guide claims that their book will increase a user’s AP score to 85%. In a random sample of 20 students who used the study guide, the AP score was 88% with a standard deviation of 8%.

State the null hypothesis:

Calculate the z-value to the nearest hundredth. Is there enough evidence to reject the claim?

PRE-CALCULUS DATA AND STATISTICS UNIT Normal Distributions

Other names fo a Normal Distributions:

The area under the curve is __. About ___% lie within ___ standard deviation of the mean. About ___% lie within ___ standard deviation of the mean. About ___% lie within ___ standard deviation of the mean. Another look at the Curve….

Z-Scores: Example: The ACT test is designed to have a mean of 20 with a standard deviation of 5. What is the probability that a student at PW will score above a 25? A 30? There are 75 students who took the ACT at PW. How many scored between 15 and 25? How many scored less than a 25?

Using a table:

Z -2.5 -2 -1.5 -1 -.5 0 .5 1 1.5 2 2.5

Area 0.01 0.02 0.07 0.16 0.31 .5 0.69 0.84 0.93 0.98 0.99

Calculating Probabilities using a Standard Normal Curve

Use this formula to calculate the Z score: 𝒛 = 𝒙− 𝝁

𝝈

Example: Scores on a normally distributed test have a mean of 75 and a standard deviation of 8. Estimate the probability that a student scored less than 87. Estimate the probability that a student received between a 71 and a 75. Calculate the z for each: Subtract the two z’s:

Skewed Data Sets Not all data sets are normally distributed. The “tail” determines how a graph is skewed. Normal Curve: Skewed Left: Skewed Right:

Homework for Normal Distributions 1. The amount of coffee in a can has a mean of 350 g and a standard deviation of 4 g.

a. What % of cans have less than 338 g of coffee? b. What is the probability that a can has between 342 g and 350 g? c. What is the probability that a can has less than 342 g or more than 346 g of coffee?

2. The time that the track team is done with practice is normally distributed with a mean finishing time of 4:30 pm and a standard deviation of 15 minutes.

a. Find the probability that practice ends after 4:45 pm. b. Find the probability that the practice time ends between 4:15 pm and 5:00 pm. 3. Temperatures for a popular spring break destination are normally distributed with a mean of 76 degrees and a standard deviation of 6 degrees. Using the table below, find the probabilities.

Z -2.5 -2 -1.5 -1 -.5 0 .5 1 1.5 2 2.5

Area 0.01 0.02 0.07 0.16 0.31 .5 0.69 0.84 0.93 0.98 0.99

a. A randomly selected day will be below 64 degrees. b. A randomly selected day will be above 70 degrees. c. A randomly selected day will be between 85 and 91 degrees. d. A randomly selected day will be between 73 and 79 degrees.

PRE-CALCULUS DATA AND STATISTICS UNIT Analyzing Decisions

In experiments, the ____________________ Value (EV) is the weighted average of the outcomes. To find the EV, simply multiply each possible outcome by the probability. Example: What is the EV when a dice is thrown?

Value of Side 1 2 3 4 5 6

Probability 1/6

𝑬𝑽 = 𝟏 (

) + 𝟐 (

) + 𝟑 (

) + 𝟒 (

) + 𝟓 (

) + 𝟔 (

)

What is the EV of rolling 2 dice? Simply just _____ the EV’s together…. EV’s are used in making decisions in business, insurance settings, traffic intersections, etc. Example: Road Rage Reggie has two choices when he goes to work. Route A always takes 15 minutes. Route B takes only 12 minutes, unless there is a traffic jam, which will make his trip last 20 minutes and ruin his day. If the chance of a traffic jam is 15%, which route will be the best for him to have the greatest chance of him keeping a great attitude. EV(A) = ____ minutes. EV(B) = 0.15(____) + .85(_____) = ______ minutes. Which route should he take? The Monty Hall Problem- 3 Doors- One has a car behind the other 2, a goat. Suppose the contestant selects Door A. The host then reveals a goat behind one of the two remaining doors.

Contents Behind Each Door Results Door A Door B Door C Result if switched If not switched

Car Goat Goat

Goat Car Goat

Goat Goat Car

Most people are convinced that once the choice is made and a door is revealed with a goat behind, that there is now a 50% chance of winning a car. That is incorrect! Do the math.

Homework for Analyzing Decisions 1. Find the Expected Value of each number cube.

a. b. 2. Find the Expected Value of each spinner.

a. b. 3. Laidback Larry can take two routes to work. Route A takes 14 minutes without traffic, but 25 minutes with traffic. Route B takes 10 minutes without traffic, and 30 minutes with traffic. He estimates a 20% chance of traffic on Route A and a 40% chance on Route B. Which route would you recommend Larry take? Route A Route B 4. In a game show, a contestant must choose a question from one of 3 categories. Category A is worth $500 with a penalty of $100 for an incorrect answer. Category B questions are worth $100, with a $20 penalty. Category C questions are worth $50, with no penalty. The probabilities for each category are 0.1, 0.3 and 0.6 respectively. Calculate the EV’s for each category Category A Category B Category C

PRE-CALCULUS DATA ANALYSIS AND STATISTICS REVIEW 1. Make a box and whisker plot for the ages of the members of a men’s league basketball team. {𝟏𝟕, 𝟐𝟑, 𝟏𝟖, 𝟐𝟐 𝟒𝟓, 𝟐𝟖, 𝟐𝟏, 𝟐𝟓} 2. The following set of data shows the amount of money spent (to the nearest dollar) by shoppers at the local Walmart. {𝟑𝟓, 𝟏𝟖, 𝟒𝟗, 𝟓𝟓, 𝟐𝟖𝟎, 𝟐𝟗, 𝟒𝟐, 𝟔𝟏, 𝟏𝟗, 𝟖𝟎, 𝟑𝟑, 𝟒𝟓, 𝟔𝟕, 𝟐𝟖, 𝟕𝟏, 𝟑𝟕, 𝟒𝟖, 𝟓𝟎, 𝟑𝟏,𝟐𝟐} a. Use a calculator to find the mean and standard deviation. b. Identify any outliers (numerically) and describe its effect on the mean and standard deviation. 3. The probability distribution for the number of substitute teachers needed at PW on any given Friday is shown below. Find the expected number of substitutes in the building on a Friday.

# of Substitutes 0 1 2 3 4

P(Substitute) 0.05 0.38 0.41 0.08 0.08

4. Jumping-Jack Inserts claims to increase one’s vertical leap by about 2 inches. The data below shows the vertical leaps in inches for several players with and without the inserts.

W/O Inserts 23 27 33 31 36 30 35 27

W/ Inserts 24 30 27 36 28 31 35 36

a. State the null hypothesis of the study. b. Compare the results of the two groups using the medians of each group. Is there enough evidence to reject the null hypothesis? 5. Classify each as survey (S), observational study (OS) or experimental study(ES). ____ Method A – Randomly divide a group of 100 people with minor cuts into two groups. Have each group use a different ointment, and record how long it takes their cuts to heal. ____ Method B – Choose 100 people at random. Ask which ointment they have used in the past and how quickly their cuts have healed with each ointment. ____ Method C – Monitor 100 people who are currently treating minor cuts with an ointment of their choosing, and record how long it takes for them to heal.

Which one would be most reliable?

6. Beth has a quarter, a dime and a nickel in her pocket. She pulls out two coins at random. What is the expected value of the coins? 7. Scores on a test are normally distributed with a mean of 78 and a standard deviation of 6. Use the table to find the probabilities of:

Z -2.5 -2 -1.5 -1 -.5 0 .5 1 1.5 2 2.5

Area 0.01 0.02 0.07 0.16 0.31 .5 0.69 0.84 0.93 0.98 0.99

a. A randomly selected student scored below 80. b. A randomly selected student scored between 74 and 80. 8. What is the expected value of the number cube below.

9. A basketball teams scores are normally distributed with the average being 52 and the standard deviation of 6. Label the normal curve below with the percentages for each interval.

a. What is the probability that the team scores over 58? b. What percentage of games did the team score between 40 and 58 points? Be able to properly identify the parts of an experiment. (Hypothesis, experimental group, control group, treatment (IV))