Common Core Math III Unit 1: Statistics

We will discuss the following four topics during this unit: 1. Normal Distributions 2. Sampling and Study Design 3. Estimating Population Parameters 4. Expected Value and Fair Game

Characteristics of Normal Distribution symmetric with respect to the mean mean = median = mode 100% of the data fits under the curve

parameterstatistic meanµ proportionp standard deviation σs Some New Symbols Parameter – populationStatistic - sample

The Normal Distribution Curve µ = 0 σ =

Z-Score The z-score is number of standard deviations (σ) a value is from the mean (µ) on the normal distribution curve.

What is the z-score of the value indicated on the curve?

What is the z-score of the value indicated on the curve?

What is the meaning of a positive z-score? What about a negative z-score? Instead of estimating, we are given a formula to help us find a precise z-score:

How do you use this? The mean score on the SAT is 1500, with a standard deviation of 240. The ACT, a different college entrance examination, has a mean score of 21 with a standard deviation of 6. If Bobby scored 1740 on the SAT and Kathy scored 30 on the ACT, who scored higher?

BobbyKathy z = 1 Kathy scored higher. Kathy’s z-score shows that she scored 1.5 standard deviations above the mean. Bobby only scored 1 standard deviation above the mean. z = 1.5

The Empirical Rule In statistics, the 68–95–99.7 rule, also known as the empirical rule, states that nearly all values lie within three standard deviations of the mean in a normal distribution.

68% of the data falls within ± 1 σ 68%

95% of the data falls within ± 2 σ 95%

99.7% of the data falls within ± 3 σ 99.7%

When you break it up… μμ+σμ+σμ+2σμ+3σμ-3σμ-2σμ-σμ-σ 34% 13.5% 2.35%.15%

Day 2

Warm Up -Take out signed syllabus & student info sheet -Take out homework & stamp sheet 1)A fifth grader takes a standardized achievement test (mean = 125, standard deviation = 15) and scores a 148. What is the student’s z-score? Interpret in context. 2)If another student had a z-score of -2, what was their score on the test?

Guided Notes Today’s Objectives: 1)To review z-scores & empirical rule 2)To use technology to calculate probabilities

Review: Z-scores & Percentages Work in your groups to complete problems 1 – 6. Be prepared to explain your answers!

what happens if you’re looking for scores that are not perfect standard deviations away from the mean? You might be wondering… normalcdf (lower bound, upper bound, µ, σ)

The scores on the CCM3 midterm were normally distributed. The mean is 82 with a standard deviation of 5. Create and label a normal distribution curve to model the scenario. How do you use this?

Draw the curve, add the mean, then add the standard deviations above and below the mean. 34% 13.5% 2.35%.15%

a. scored between 77 and 87 b. scored between 82 and 87 c. scored between 72 and 87 d. scored higher than 92 e. scored less than 77 68% 34% 81.5% 2.5% 16% Find the probability that a randomly selected person:

normalcdf (80, 90, 82, 5) = a. What’s the probability that a randomly selected student scored between 80 and 90? b. What’s the probability that a randomly selected student scored below 70? normalcdf (0, 70, 82, 5) = c. What’s the probability that a randomly selected student scored above 79? normalcdf (79, 100, 82, 5) = Or 1 – normalcdf(0,79,82,5) =.7257

invnorm (percent of area to left, ,  ) You can also work backward to find percentiles! d. What score would a student need in order to be in the 90 th percentile? invnorm (0.9, 82, 5) = 88.41, or 89

e. What score would a student need in order to be in top 20% of the class? invnorm (0.8, 82, 5) = 86.21, or 87

The average waiting time at Walgreen’s drive- through window is 7.6 minutes, with a standard deviation of 2.6 minutes. When a customer arrives at Walgreen’s, find the probability that he will have to wait a. between 4 and 6 minutes b. less than 3 minutes c. more than 8 minutes d. Only 8% of customers have to wait longer than Mrs. Jones. Determine how long Mrs. Jones has to wait minutes

Classwork – Page 3 in your packet #s 1 – 3 Pick 1 #s 4 & 5 Pick 1 # 6 & 7 Pick 1 Turn in when you are done

Questions about normal distribution?

Sampling and Study Design

Main Questions What is bias and how does it affect the data you collect? What are the different ways that a sample can be collected? What’s the difference between an experiment and an observational study? When is a sample considered random?

There are three ways to collect data: 1. Surveys 2. Observational Studies 3. Experiments

Surveys most often involve the use of a questionnaire to measure the characteristics and/or attitudes of people. ex. asking students their opinion about extending the school day Surveys

Individuals are observed and certain outcomes are measured, but no attempt is made to affect the outcome. Observational Studies

Treatments are imposed prior to observation. Experiments are the only way to show a cause-and-effect relationship. Remember: Correlation is not causation! Experiments

Observational Study or Experiment? Fifty people with clinical depression were divided into two groups. Over a 6 month period, one group was given a traditional treatment for depression while the other group was given a new drug. The people were evaluated at the end of the period to determine whether their depression had improved. Experiment

To determine whether or not apples really do keep the doctor away, forty patients at a doctor’s office were asked to report how often they came to the doctor and the number of apples they had eaten recently. Observational Study Observational Study or Experiment?

To determine whether music really helped students’ scores on a test, a teacher who taught two U. S. History classes played classical music during testing for one class and played no music during testing for the other class. Experiment Observational Study or Experiment?

Types of Sampling In order to collect data, we must choose a sample, or a group that represents the population. The goal of a study will determine the type of sampling that will take place.

All individuals in the population have the same probability of being selected, and all groups in the sample size have the same probability of being selected. Simple Random Sample (SRS)

Putting 100 kids’ names in a hat and picking out 10 - SRS Putting 50 girls’ names in one hat and 50 boys’ names in another hat and picking out 5 of each – not a SRS

If a researcher wants to highlight specific subgroups within the population, they divide the entire population into different subgroups, or strata, and then randomly selects the final subjects proportionally from the different strata. Stratified Random Sample

The researcher selects a number at random, n, and then selects every nth individual for the study. Systematic Random Sample

Subjects are taken from a group that is conveniently accessible to a researcher, for example, picking the first 100 people to enter the movies on Friday night. Convenience Sample

The entire population is divided into groups, or clusters, and a random sample of these clusters are selected. All individuals in the selected clusters are included in the sample. Cluster Sample

The names of 70 contestants are written on 70 cards, the cards are placed in a bag, the bag is shaken, and three names are picked from the bag. Name that sample! Simple random sampleStratified sample Convenience sample Cluster sample Systematic sample

To avoid working late, the quality control manager inspects the last 10 items produced that day. Simple random sampleStratified sample Convenience sample Cluster sample Systematic sample Name that sample!

A researcher for an airline interviews all of the passengers on five randomly selected flights. Simple random sampleStratified sample Convenience sample Cluster sample Systematic sample Name that sample!

A researcher randomly selects and interviews fifty male and fifty female teachers. Simple random sampleStratified sample Convenience sample Cluster sample Systematic sample Name that sample!

Every fifth person boarding a plane is searched thoroughly. Simple random sampleStratified sample Convenience sample Cluster sample Systematic sample Name that sample!

Warm Up – Quiz Review 1) Two students take equivalent stress tests. Which is the highest relative score ? A score of 90 on a test with a mean of 86 & a standard deviation of 18. A score of 18 on a test with a mean of 15 & a standard deviation of 5. 2) The AFM Final exam scores were normally distributed with a mean of 77 and standard deviation of 3. a)Draw & label a normal curve b)What range of scores was in the middle 68% of the data? c)What percent of students scored higher than an 87%? d)What grade would a student need to be in the 85 th percentile?

Homework Check – pg 8 1) systematic2) cluster 3) stratified4) cluster 5) vol response 6) convenience 7) stratified 8) cluster 9) convenience 10) simple random 11) stratified 12) systematic 13) simple random 14) cluster

Day 4 – Bias in Sampling

Types of Bias in Survey Questions Bias occurs when a sample systematically favors one outcome.

The use of double negatives in this question caused confusion in the way people responded to the survey. 22% of those surveyed said that it was possible that the holocaust did not occur. This is an example of question wording bias! In 1992, a Roper poll conducted for the American Jewish Community of the Holocaust asked: “Does it seem possible or does it seem impossible to you that the Nazi extermination of the Jews never happened?”

Later, a new survey was conducted in which the question was rephrased: “Does is seem possible to you that the Nazi extermination of the Jews never happened, or do you feel certain that it happened?” In the new survey, only 1% of those surveyed stated that it was possible that the holocaust never occurred.

1. Question wording bias The wording of the question may be loaded in some way to unduly favor one response over another. (also called leading questions) 2. Undercoverage bias occurs when the sample is not representative of the population. 3. Response bias occurs when survey respondents lie or misrepresent themselves. (also called social desirability) 4. Nonresponse bias occurs when an individual is chosen to participate, but refuses. 5. Voluntary response bias occurs when people are asked to call or mail in their opinion.

On the twelfth anniversary of the death of Elvis Presley, a Dallas record company sponsored a national call-in survey. Listeners of over 1000 radio stations were asked to call a number (at a charge of $2.50) to voice an opinion concerning whether or not Elvis was really dead. It turned out that 56% of the callers felt Elvis was alive. Voluntary response bias Name that bias!

In 1936, Literary Digest magazine conducted the most extensive public opinion poll in history to date. They mailed out questionnaires to over 10 million people whose addresses they had obtained from vehicle registration lists. More than 2.4 million people responded, with 57% indicating that they would vote for Republican Alf Landon in the upcoming Presidential election. However, Democrat Franklin Roosevelt won the election, carrying 63% of the popular vote. Undercoverage bias Name that bias!

Do you think the city should risk an increase in pollution by allowing expansion of the Northern Industrial Park? Why is this question biased? Can you rephrase it to remove the bias?

If you found a wallet with $100 in it on the street, would you do the honest thing and return it to the person or would you keep it? Can you rephrase it to remove the bias? Why is this question biased?

Questions about sampling?

Estimating Population Parameters Vocabulary for this lesson is important! Parameter a value that represents a population Statistic a value based on a sample and used to estimate a parameter

parameterstatistic meanµ proportionp standard deviation σs population sample

Homework & Stamp sheets out! Get ready for the Quiz!

Homework Check – pg No; only teenagers who read Teen and choose to write would respond. This does not represent all teens. 2. No; the sample only includes students. Students are most likely young, and would not be representative of “average American adults”, which should include all age groups. 3. No; men would probably be better represented than women (more men probably read this than women); only people who read Consumer Reports and choose to respond would be represented. Voluntary response samples tend to over-represent the people who have strong opinions. 4. yes, this is a random sample and she is sampling enough to have a good idea of the quality. 5. No; 1/6 of the chosen sample was eliminated which is not representative to the entire school 6. While this does represent a simple random sample, the sample is really not representative of the class because only boys are represented. A better way to create a sample would be to used a stratified sample. Sort students by gender and choose the same amount of males & females to allow equal representation of both genders.

Day 5 – Margin of Error

You poll 1,000 people and ask who they will vote for president. 500 say they'll vote for Obama, while 450 say they'll vote for Romney. 500/1000 = 50% 450/1000 = 45% Would you think that overall, Obama leads 50% to 45%? Would you think overall Romney is down 5 points? If you said, "yes" to either, then you are not correct. Now you might ask, why?

You only sampled 1000 people! Even if you picked a sample that was a perfect representation of the population, the outcome might not accurately reflect the population. There will be some error!

Finding a Margin of Error – pg 11 Margin of error is a “cushion” around a statistic. ME = n = sample size

Suppose that 900 American teens were surveyed about their favorite event of the Winter Olympics. Ski jumping was the favorite of 20% of those surveyed. This result can be used to predict the proportion of ALL American teens who favor ski jumping. We can confidently state that the true percentage of American teens who favor ski jumping falls between 17% and 23%.

What sample size produces a given margin of error? If you want your margin of error to be 2%, what size sample will you need? 2500

How does sample size affect margin of error? If your sample size is 400 and you wish to cut the margin of error in half, what will your new sample size be? 1600 So, if you want half of.05, we need to solve

Warm Up 1) The Raleigh Times newspaper interviewed 1200 adults in Wake County. The surveys found that 682 of the adults thought the national minimum wage should be raised. The newspaper stated that “0.57 ± of adults believe that minimum wage should be increased.” Discuss what this interval means, in context. 2) A normal distribution has a mean of 120 and a standard deviation of 20. For this distribution, what score corresponds to the 90th percentile?

Homework Check – Margin of Error ± ± ± ± ± ± ± ± a. ± b ± = to 0.177

Simulation Simulation is a way to model random events, so that simulated outcomes closely match real-world outcomes. Some situations may be difficult, time-consuming, or expensive to analyze. In these situations, simulation may approximate real-world results while requiring less time, effort, or money. Why run a simulation?

Carole and John are playing a dice game. Carole believes that she can roll six dice and get each number, one through six, on a single roll. John knows the probability of this occurrence is low. He bets Carole that he will wash her car if she can get the outcome she wants in twenty tries.

BEFORE YOU START! You are running 20 trials, so make 20 blanks on your paper. This will keep you from losing count of how many trials you’ve run. It also makes recording easy! __ __ __ __ __ __ __ __ __ __ 

What is the problem that we are simulating? Can Carole get one of each number in a roll of six dice? What random device will you use to simulate the problem and how will you use it? We will use the calculator to generate random numbers.

Seeding Since a calculator is a type of computer, it can never be truly random. For this reason, we can configure our calculators to give everyone the same set of “random” data (so we can all work together!). The process of calibrating our calculators in this way is called seeding. To seed your calculator: 5, STO , MATH, PRB, #1RAND, ENTER

To seed your calculator: 5, STO , MATH, PRB, #1RAND, ENTER To run the simulation: MATH, PRB, #5RandInt(1, 6, 6), ENTER

Your calculator should look like this:

How will you conduct each trial? How many trials will you conduct? I will use the randInt( command in my calculator to generate random integers. randInt(min value, max value, number of data in set) randInt(1, 6, 6)

What are the results of these trials? What predictions can be made based on these results? We received all 6 numbers only 1 out of 20 times. There’s approximately a 5% chance of this occurring. The more trials you run, the closer you will get to the theoretical probability (Law of Large Numbers).

Let 1-45 represent people with type O blood. On a certain day the blood bank needs 4 donors with type O blood. If the hospital brings in groups of five, what is the probability that a group would arrive that satisfies the hospitals requirements, assuming that 45% of the population has type O blood? Let represent people with other blood types. Remember to seed the calculator to 5! Then, run RandInt(1, 100, 5) twenty times. Record how many trials satisfy the hospital’s requirements.

To seed your calculator: 5, STO , MATH, PRB, #1RAND, ENTER To run the simulation: MATH, PRB, #5RandInt(1, 100, 5), ENTER Five of the twenty groups have four or more members with type O blood. Therefore, there is a 25% chance that they hospital will get the Type O blood they need. Reset the memory on your calculator: 2 nd +, 7, 1, 2,

Questions about simulations?

Expected Value and Fair Games

Expected Value Expected value is the weighted average of all possible outcomes. For example, a trial has the outcomes 10, 20 and 60. The average of 10, 20, and 60 = 30 This assumes an even distribution: 

Sometimes, outcomes will not have equal likelihoods. X123 P(X).5.25 E(X) =.5(1) +.25(2) +.25(3) = 1.75

You play a game in which you roll one fair die. If you roll a 6, you win $5. If you roll a 1 or a 2, you win $2. If you roll anything else, you lose. Create a probability model for this game: X P(X) 6 1, 2 3, 4, 5 $5 $2 $0 What would you be willing to pay to play? E(X) = 5(1/6) + 2(1/3) + 0(1/2) = /6 1/3 1/2 A price of $1.50 makes this a fair game.

At Tucson Raceway Park, your horse, My Little Pony, has a probability of 1/20 of coming in first place, a probability of 1/10 of coming in second place, and a probability of ¼ of coming in third place. First place pays $4,500 to the winner, second place $3,500 and third place $1,500. Is it worthwhile to enter the race if it costs $1,000? E(X) =.05(3500) +.1(2500) +.25(500)+.6(-1000) = -$50. 1 st 2 nd 3 rd Other X$3500$2500$500-$1000 P(X)

What does an expected value of -$50 mean? Its important to note that nobody will actually lose $50—this is not one of the options. Over a large number of trials, this will be the average loss experienced. This is the Law of Large Numbers! Insurance companies and casinos build their businesses based on the law of large numbers.

Questions about expected value?

Any questions about statistics?