أوراق عمل Statistics and Probability الرياضيات منهج انجليزي الصف الثاني عشر عام

**Unit 10 : Statistics and Probability**

**Lesson 10 -1 Designing a Study**

Mr. Ali Abdalla

Classify study types

Design statistical studies

Classifying Studies In a statistical study, data are collected and used to answer questions about a population characteri$ie or parameter. Due to time and money constraints, it may be impracticylQbh impossible to collect data from each member of a population. Thereforeén many studies, a sample of the population is taken, and a measure called a statistic is calculated using the data. The sample staåstic, sucll as the sample mean or sample standard deviation, is then used to make inferences about the population parameter

The steps in typical statistical study are shown below

To obtain good information and draw accurate conclusions about a population, it is important to select an unbiased sample. A bias is an error that results in a misrepresentation of members of a population. A poorly chosen sample can cause biased results. To reduce the possibility of selecting a biased sample, a random sample can be taken in which members of the oulation are selected entirel b chanc

**Definition**

ata are collected from In a su responges given by members of a population regarding their characteristics, behaviors, or oinions In an experiment, the sample is divided into two groups

an experimental group that undergoes a change a control group that does not undergcrh Change dana The effect on the experimental is then com ared to the contro In an observational study, members of a sample are measured or observed without bein affected b the stud

Example To determine whethy e student body likes the new cafeteria menu, the student council asks a random sample of students for their o inion

A restaurant is considering creating meals with chicken instead of beef. They randomly give half of a group of participants meals with chicken and the other half meals with beef Then they ask how they like the meals

Researchers at an electronics pompany observe a group of teenqgees using different la to s and note their reactions

#### Example 1

Determine whether each situation describes a survey, an experiment, or an observational study. Then identify the sample, and suggest a population from which it may have beh selected

l) MUSIC Axaord label wants to test three designs for an albutpver They raådomly select 50 teenagers from local high schools tÅ'iew the covers while they watch and record their reactions

This is an observational study Because the company is going to obse he teens without them being affected by the study. The sample is the 50 selected, and the population is all pogentia! pyrchasers of this album

RECYCLING The city council wants to start a recycling program. They send out questionnaire to 200 random citizens asking what Vteins they

is a survey

Because the data are collected from participants' responses in the questionnaire. The sample is the 200 meople who received the questionnaire, and the population of the citizens of the city

IA. RESEARCH Scientists study the behavior of one group of cats given a new heartworm treatment and another group of cats given a false treatment or placebo

1B. YEARBOOKS The yearbook committee conducts a study to determine whether students would prefer to have a print yearbook or both print and digital yearbooks

**Example**

Deiérmine whether each situation calls for a survey, anxperiment, or an observational study. Explain your reasoning

l) MEDICINE A pharmaceutical company wants to test whether a new medicine is effective

The treatment will need to be tested on a sample group, which means that the members of the sample will be affected by the study Therffore, this situation calls IqWan experiment

ELECTIONS A nsws organization wants to randomly call citizetyto gauge opinions on a presidential election

This situation calls for a survey. because members of the sample population are asked for their opinion

2A. RESEARCH A research company wants to study users and non- users of full -fat dairy products in their diet to determine whether 1 year of won -use affects chele'ierol levels

2B. PETS A national pet chain ants to know whether customers would pay asmall annual fee to participate in a rewards program. They randomly select 200 customers and send them questionnaires

2C. VISEO GAMES A gaming comptypyplans to test whether a new controller is preferable to the oyåle. A group of teens will be observed while using the controller tgrsee which one they use to see the most

**Designing** Studies The questions chosen for a survey or proceduresused in an experiment can also introduce bias, and thus, affect the results of the study

A survey questiorythQit is poorly written may result in a response that not accurately reflecUhe opinion of the participant. Therefore, it is impotaRt to write questions thatååe clear and precise. Avoid survey questions that

• are confusing or wordy

• cause a strong reactio

'yaddress more than one issu

• encourage a certain respons

Questions can also introduce bias if therqenot enough information given for the participant to give an accurate

**Example** Determine whether each survey question is biased or unbiased. If biased, explain your reasoning

1) Don't Oti agree that the cafeteria should serve healthier This qUestion is biased Because it encourages a certain response. The phrase don't you agree" encourages you to agree that the cafeteria should serve healthier food

2) How often do you exercise

This question is unbiased Bepyse it is clearly stated and does not encourage a certain respoose

3A. How Päny glasses of water do you drink a day

3B. Do you prefer watching exciting action movies or boring documentaries

3C. What is your favorite"be of music

#### Example 4

TECHNOLOGY Mansour is writing Üirticle for his school newspaper about online courses. He wants to conducyq 'urvey to determine how many students at his school would be interested in taking an online course from home. State the objective Of the survey, suggest a population, and write two unbiased surVey questions

**Step 1:** Statithe objective of the survey

The qPjective of the survey is to determine students' interest in taking an glin&course from home

**Step 2:** Identify the population

The population is the student body

**Step** **3:** Write unbiased survey question

• Do you have Internet If offered, would yquike an online course

4A. TECHNOLOGY In a follow-up article, Nasser decides to conduct a survey to determine ho many teachers from his school with at least five yeyof experience wp be interested in teaching an online course

State the +active of the survey, suggest a population, and wqi!e two unbiased survey questions

Step 1: State the objective of the survey

Step 2 identify the population

Step 3: yrife unbiased survey questions

#### Example 5

Identify any flaws in the design of the experiment and describe how they could be corrected

**Experiment**: An eleönics company wants to test whether using a new graphlflg calculator increases students' test scores. A rapdm såinple is taken. Calculus students in the experimental group are given the new calculator to use, and Algebra 2 students in the control group are asked to use the ir own calculator

Results: When given the same test, the group scored higher than the control group. The com By concludes that the use of this calculator increases test scores

Calculus students are more likely to score higher when given the same test as Algebra 2 sudents

Thereforgftile flaw is that the experimental group consists studeht' and the control group consists of Algebra 2 Thiu flaw could be corrected by selecting a random saple of all Calculus or all Algebra 2 students

**Guided Practice**

5A. Identify any flaws in the design of the experiment and describe how they Meould be corrected

Experiment: A research firm tests the effectiveness ofa de-icer on car lock

They use a random sample of drivers in California and Minnesota Jör the control and experimental groups

Results: They concluded that the de-icer is effective

5B 'Identify any flaws in the design of the experiment and describe how they could be corrected

**Experiment**: A research company wants to conduct a study to determine whether a reel is more effective than the old reel. The expehmental procedure consists of using the new reel &atch fish one lake, and using the old reel to catch fish in andiher nearby lake that is randomly chosen

**Results**: The company concludes that the neyreel is twice as effective as the old reel

#### Example 6

PLANTS A research company wants to test the claim of the shown at the right. State the objective experiment, suggest a population, determine Che experimental and control groups, and describf sample procedure

Step 1: State the objective and identify the population

The objective of the experiment is to determine whether tomato plants given the plant food grow taller in three weeks than tomato plants not given the food. The population is all tomato plants

Step 2: Determine the experimental and control groups The experimental group he tomato plants given the control group is the tenqato plants not given the food

Step 3: Describe a samplé + rocedure Measure the heights of the plants in each group and give

experimental group the plant food. Then, wait three , measure the heights of the plants again, and compare the heights for each group to see if the claim was valid

A. Use a graphing calculator to creabox -and-whisker plot. Then describe the shape of the distribution

The lengths of the whiskers are approximately equal, and the median is in the middle of the data. This indicates that the data are equally distributed to the left and reof the median

Thus 08 distribution is symmetric

B.bcscribe the center and spread of the data using either the mean and standard deviation or the five- number summary. Justify your choice

The distribution is symmetric, so use the mean and 1 —War Stats standard deviation to describe the center and spread. 5

The average number of minutes that a student spent on homework each night was 53.5 with standard deViatiOn of about 10.5

**Guided Practice**

. CELL PHONE Amani's parents have given her a prepaid cell phony, The number of minutes she used each monywi%r the last two years are

shown in the table

A. Use a gråbhing calculator to create a box and -whisker plot. Then describe the shape of the distribution

B. Describe the center and spread of the data using either the mean and standard deviation or the five- number summary. Justify your choice

#### Example 4

TYPING The points scored per game by a professional rugby team for the 2008 and 2009 football sea9hs are shown

A. Use a graphing calculator to create set. Then describe the shape of eqCh distribution

For the 2008 scores, the left whisker is longer than the right and the median is closer to the right whisker. The distribution is negatively skewed

For scores, the right whisker is longer than the left and the median is closer to the left whisker. The distribution is positively skewed

B. Compare the distributions usiåÅ either the means and standard deviations or the Ove-number summaries. Justify your choice

Th diSQibutions are skewed, so use the five-number saries to compare the data

The lower quartile for the 2008 season and the upper quartile for the 2009 season are both 20.5. This means that 75% of the scores from the 2008 season were greater than 20.5 and 75% of the scores from the 2009 season were less than 20.5

the hllhimum of the 2008 •on is approximately equal to the owe quartile for the 2009 season. Reans that 25% of the scores from the 200? season are lower than any score achieved in the 2008 season

Therefore, we can conclude that the team scored a signifiqhly higher amount of points during the 2008 season than the 2009 season

Example 1

Identi and Classify Random Variables Identify the random variable in each dibution and classify it as discrete or continuous. Explain your reasonin A The number of songs found on a random selection of mp3 players

he random variable X is the number of songs on any mp3 player in the random selection of players. The number of songs is countable

B) RThe weights of bowling balls sent by a manufacturqr.f The random variable X is the (Deight of any particular bowling ball. The (Deight of any particular bowling ball can be anywhere (Dithin a certain range, typically to g kg. Therefore: X is continuous

**GuidedPractice**

IA) Whe exact distances of Sämple of discus throws

1B) The ages of counselors at a summer camp

1 C) The number of pages linked to a Web page

ID) Thé number of stations in a cable package