Statistics I

What is Data? Consist of information coming from observations, counts, measurements, or responses. “People who eat three daily servings of whole grains have been shown to reduce their risk of…stroke by 37%.” (Source: Whole Grains Council) “Seventy percent of the 1500 U.S. spinal cord injuries to minors result from vehicle accidents, and 68 percent were not wearing a seatbelt.” (Source: UPI)

What is Statistics? The science of collecting, organizing, analyzing, and interpreting data in order to make decisions.

Data Sets Population The collection of all outcomes, responses, measurements, or counts that are of interest. Sample A subset of the population.

Example: Identifying Data Sets In a recent survey, 1500 adults in the United States were asked if they thought there was solid evidence for global warming. Eight hundred fifty-five of the adults said yes. Identify the population and the sample. Describe the data set. (Adapted from: Pew Research Center)

Solution: Identifying Data Sets The population consists of the responses of all adults in the U.S. The sample consists of the responses of the 1500 adults in the U.S. in the survey. The sample is a subset of the responses of all adults in the U.S. The data set consists of 855 yes’s and 645 no’s. Responses of adults in the U.S. (population) Responses of adults in survey (sample)

Parameter and Statistic P arameter A number that describes a population characteristic. Average age of all people in the United States S tatistic A number that describes a sample characteristic. Average age of people from a sample of three states

Example: Distinguish Parameter and Statistic Decide whether the numerical value describes a population parameter or a sample statistic. 1.A recent survey of a sample of college career centers reported that the average starting salary for petroleum engineering majors is $83,121. (Source: National Association of Colleges and Employers) Solution: Sample statistic (the average of $83,121 is based on a subset of the population)

Example: Distinguish Parameter and Statistic Decide whether the numerical value describes a population parameter or a sample statistic. 2.The 2182 students who accepted admission offers to Northwestern University in 2009 have an average SAT score of (Source: Northwestern University) Solution: Population parameter (the SAT score of 1442 is based on all the students who accepted admission offers in 2009)

Designing a Statistical Study 1. Identify the variable(s) of interest (the focus) and the population of the study. 2. Develop a detailed plan for collecting data. If you use a sample, make sure the sample is representative of the population. 3. Collect the data. 4. Describe the data using descriptive statistics techniques. 5. Interpret the data and make decisions about the population using inferential statistics. 6. Identify any possible errors.

Data Collection Observational study A researcher observes and measures characteristics of interest of part of a population. Researchers observed and recorded the mouthing behavior on nonfood objects of children up to three years old. (Source: Pediatric Magazine)

Data Collection Experiment A treatment is applied to part of a population and responses are observed. An experiment was performed in which diabetics took cinnamon extract daily while a control group took none. After 40 days, the diabetics who had the cinnamon reduced their risk of heart disease while the control group experienced no change. (Source: Diabetes Care)

Data Collection Survey An investigation of one or more characteristics of a population. Commonly done by interview, mail, or telephone. A survey is conducted on a sample of female physicians to determine whether the primary reason for their career choice is financial stability.

Example: Methods of Data Collection A study of the effect of eating oatmeal on lowering blood pressure is an example of experiment / observational study / survey ? Solution: Experiment (Measure the effect of a treatment – eating oatmeal)

Example: Methods of Data Collection A study of how fourth grade students solve a puzzle is an example of: experiment / observational study / survey? Solution: Observational study (observe and measure certain characteristics of part of a population)

Example: Methods of Data Collection A study of U.S. residents’ approval rating of the U.S president. is an example of: experiment / observational study / survey ? Solution: Survey (Ask “Do you approve of the way the president is handling his job?”)

Summary

Example 1A Classify Study Types A.Determine whether the situation describes a survey, an experiment, or an observational study. Then identify the sample, and suggest a population from which it may have been selected. MOVIES A retro movie theater wants to determine what genre of movies to play during the next year. They plan to poll 50 random area residents and ask them what their favorite movies are. Answer: This is a survey, because the data are collected from participants' responses to the poll. The sample is the 50 people area residents that are polled, and the population is all area residents.

Example 1B Classify Study Types B.Determine whether the situation describes a survey, an experiment, or an observational study. Then identify the sample, and suggest a population from which it may have been selected. DRIVING A driving school wants to determine the main issue drivers face while taking the driving test. They watch and record 30 random people taking the test. Answer: This is an observational study, because the school is going to observe the drivers without their being affected by the study. The sample is the 30 drivers selected, and the population is all drivers that may take the test.

Example 1A A.survey B.experiment C.observational study A restaurant manager provides a new entrée to 30 randomly selected tasters and observes their reactions. Determine whether the situation describes a survey, an experiment, or an observational study.

Example 2A Choose a Study Type A.Determine whether the situation calls for a survey, an experiment, or an observational study. Explain your reasoning. VIDEO GAMES A gaming company plans to test whether a new controller is preferable to the old one. A group of teens will be observed while using the controllers, to see which one they use the most. Answer: The teens will be observed without being affected by the study, so this is an observational study.

Example 2B Choose a Study Type A.Determine whether the situation calls for a survey, an experiment, or an observational study. Explain your reasoning. RESTAURANTS A restaurant wants to conduct an online study in which they will ask customers whether they were satisfied with their dining experience. Answer: This situation calls for a survey because members of the sample population are asked for their opinion.

A.Survey; members of the sample are observed and asked their opinions. B.Experiment; members of the sample are observed and affected by the study. C.Observational study; members of the sample are observed and unaffected by the study. D.Experiment; members of the sample are treated and affected by the study. Determine whether the situation calls for a survey, an experiment, or an observational study. Explain your reasoning. MOVIES A production studio played a movie for a test audience and watched their reactions.

Let’s Practice… Pg. 3 # 1, 2 Pg. 4 # 1, 2

Statistics II

Opening Routine

In General Samples vary in how well they reflect the entire population. Random Sample: When all members of the population are equally likely to be chosen.

When a part of a population is overrepresented or underrepresented in a sample. BIAS

Example 3 Identify Bias in Survey Questions Determine whether the survey question is biased or unbiased. If biased, explain your reasoning. A. What is your favorite type of music? Answer: This question is unbiased because it is clearly stated and does not encourage a certain response.

Example 3 Identify Bias in Survey Questions Determine whether the survey question is biased or unbiased. If biased, explain your reasoning. B. Do you think that poisons, such as pesticides, should be sprayed on crops? Answer: This question is biased because the term "poison" could cause a strong reaction from the respondent.

Example 3 A.unbiased B.Biased; the question is confusing. C.Biased; the question addresses more than one issue. D.Biased; the question encourages a certain response. Determine whether the survey question is biased or unbiased. If biased, explain your reasoning. Are you planning on watching the ultimate sporting event, the Super Bowl?

A.Biased; the question is confusing and wordy. B.Biased; the question causes a strong reaction. C.Biased; the question encourages a certain response. D.unbiased Determine whether the survey question is biased or unbiased. If biased, explain your reasoning. Shouldn’t Megan Fox win the Best Actress award this year?

Measures of Central Tendency Mean: Median is the number in the middle when the numbers in a set of data are arranged in ascending or descending order. If the number of numbers in a data set is even, then the median is the mean of the two middle numbers. Mode is the value that occurs most frequently in a set of data. is the most common measure of central tendency. It is simply the sum of the numbers divided by the number of numbers in a set of data. This is also known as average. It is often denoted by the lower case Greek letter mu μ.

Standard Deviation Standard Deviation shows the variation in data. If the data is close together, the standard deviation will be small. If the data is spread out, the standard deviation will be large. Standard Deviation is often denoted by the lowercase Greek letter sigma,.

Note that: Standard deviation measures the dispersion of data. The greater the value of the standard deviation, the further the data tend to be dispersed from the mean.

Let’s Practice… Page 3 # 3-5 all Page 4 # 3-6 all Page 5 Look at Example in “Activity 2” Page 6 # 5-7 all

Statistics III

The bell curve which represents a normal distribution of data shows what standard deviation represents. One standard deviation away from the mean ( ) in either direction on the horizontal axis accounts for around 68 percent of the data. Two standard deviations away from the mean accounts for roughly 95 percent of the data with three standard deviations representing about 99 percent of the data.

Example 1 Use the Empirical Rule to Analyze Data A. A normal distribution has a mean of 45.1 and a standard deviation of 9.6. Find the values that represent the middle 99.7% of the distribution. μ = 45.1 and σ = 9.6 The middle 99.7% of data in a normal distribution is the range from μ – 3σ to μ + 3σ – 3(9.6) = (9.6) = 73.9 Answer:

Example 1 Use the Empirical Rule to Analyze Data A. A normal distribution has a mean of 45.1 and a standard deviation of 9.6. Find the values that represent the middle 99.7% of the distribution. μ = 45.1 and σ = 9.6 The middle 99.7% of data in a normal distribution is the range from μ – 3σ to μ + 3σ – 3(9.6) = (9.6) = 73.9 Answer: Therefore, the range of values in the middle 99.7% is 16.3 < X < 73.9.

Concept

z -scores A z -score reflects how many standard deviations above or below the mean a raw score is. The z-score is positive if the data value lies above the mean and negative if the data value lies below the mean.

z -score formula Where x represents an element of the data set, the mean is represented by and standard deviation by.

Example: Finding a z-Score Given an Area Find the z-score that corresponds to a cumulative area of z 0 z Solution:

Solution: Finding a z-Score Given an Area Locate in the body of the Standard Normal Table. The values at the beginning of the corresponding row and at the top of the column give the z-score. The z-score is –0.35.

Example: Finding a z-Score Given an Area Find the z-score that has 10.75% of the distribution’s area to its right. z0 z Solution: 1 – = Because the area to the right is , the cumulative area is

Solution: Finding a z-Score Given an Area Locate in the body of the Standard Normal Table. The values at the beginning of the corresponding row and at the top of the column give the z-score. The z-score is 1.24.

Example: Finding an x-Value A veterinarian records the weights of cats treated at a clinic. The weights are normally distributed, with a mean of 9 pounds and a standard deviation of 2 pounds. Find the weights x corresponding to z-scores of 1.96, –0.44, and 0. Solution: Use the formula x = μ + zσ z = 1.96:x = (2) = pounds z = –0.44:x = 9 + (–0.44)(2) = 8.12 pounds z = 0:x = 9 + (0)(2) = 9 pounds Notice pounds is above the mean, 8.12 pounds is below the mean, and 9 pounds is equal to the mean.

Example: Finding a Specific Data Value Scores for the California Peace Officer Standards and Training test are normally distributed, with a mean of 50 and a standard deviation of 10. An agency will only hire applicants with scores in the top 10%. What is the lowest score you can earn and still be eligible to be hired by the agency? Solution: An exam score in the top 10% is any score above the 90 th percentile. Find the z- score that corresponds to a cumulative area of 0.9.

Solution: Finding a Specific Data Value From the Standard Normal Table, the area closest to 0.9 is So the z-score that corresponds to an area of 0.9 is z = 1.28.

Solution: Finding a Specific Data Value Using the equation x = μ + zσ x = (10) = 62.8 The lowest score you can earn and still be eligible to be hired by the agency is about 63.

Let’s Practice…

Statistics IV

Concept

z -score formula Where x represents an element of the data set, the mean is represented by and standard deviation by.

Analyzing the data Suppose SAT scores among college students are normally distributed with a mean of 500 and a standard deviation of 100. If a student scores a 700, what would be her z -score? Answer Now

Analyzing the data Suppose SAT scores among college students are normally distributed with a mean of 500 and a standard deviation of 100. If a student scores a 700, what would be her z -score? Her z -score would be 2 which means her score is two standard deviations above the mean.

Analyzing the data A set of math test scores has a mean of 70 and a standard deviation of 8. A set of English test scores has a mean of 74 and a standard deviation of 16. For which test would a score of 78 have a higher standing? Answer Now

Analyzing the data To solve: Find the z -score for each test. A set of math test scores has a mean of 70 and a standard deviation of 8. A set of English test scores has a mean of 74 and a standard deviation of 16. For which test would a score of 78 have a higher standing? The math score would have the highest standing since it is 1 standard deviation above the mean while the English score is only.25 standard deviation above the mean.

Analyzing the data What will be the miles per gallon for a Toyota Camry when the average mpg is 23, it has a z value of 1.5 and a standard deviation of 2? Answer Now

Analyzing the data What will be the miles per gallon for a Toyota Camry when the average mpg is 23, it has a z value of 1.5 and a standard deviation of 2? Using the formula for z-scores: The Toyota Camry would be expected to use 26 mpg of gasoline.

Analyzing the data A group of data with normal distribution has a mean of 45. If one element of the data is 60, will the z -score be positive or negative? Answer Now

Analyzing the data A group of data with normal distribution has a mean of 45. If one element of the data is 60, will the z -score be positive or negative? The z -score must be positive since the element of the data set is above the mean.

Example 1 Use the Empirical Rule to Analyze Data B. A normal distribution has a mean of 45.1 and a standard deviation of 9.6. What percent of the data will be greater than 54.7? The value 54.7 is 1σ more than μ. Approximately 68% of the data fall between μ – σ and μ + σ, so the remaining data values represented by the two tails covers 32% of the distribution. We are only concerned with the upper tail, so 16% of the data will be greater than Answer:

Example 1 Use the Empirical Rule to Analyze Data B. A normal distribution has a mean of 45.1 and a standard deviation of 9.6. What percent of the data will be greater than 54.7? The value 54.7 is 1σ more than μ. Approximately 68% of the data fall between μ – σ and μ + σ, so the remaining data values represented by the two tails covers 32% of the distribution. We are only concerned with the upper tail, so 16% of the data will be greater than Answer: 16%

Example 1 A.0.3% B.2.5% C.5% D.97.5% A normal distribution has a mean of 38.3 and a standard deviation of 5.9. What percent of the data will be less than 26.5?

Example 1 A.0.3% B.2.5% C.5% D.97.5% A normal distribution has a mean of 38.3 and a standard deviation of 5.9. What percent of the data will be less than 26.5?

Example 2 Use the Empirical Rule to Analyze a Distribution A. PACKAGING Students counted the number of candies in 100 small packages. They found that the number of candies per package was normally distributed, with a mean of 23 candies per package and a standard deviation of 1 piece of candy. About how many packages have between 22 and 24 candies? 22 and 24 are 1σ away from the mean. Therefore, about 68% of the data are between 22 and 24. Since 100 × 68% = 68 we know that about 68 of the packages will contain 22 to 24 pieces. Answer:

Example 2 Use the Empirical Rule to Analyze a Distribution A. PACKAGING Students counted the number of candies in 100 small packages. They found that the number of candies per package was normally distributed, with a mean of 23 candies per package and a standard deviation of 1 piece of candy. About how many packages have between 22 and 24 candies? 22 and 24 are 1σ away from the mean. Therefore, about 68% of the data are between 22 and 24. Since 100 × 68% = 68 we know that about 68 of the packages will contain 22 to 24 pieces. Answer: about 68 packages

Example 2 Use the Empirical Rule to Analyze a Distribution B. PACKAGING Students counted the number of candies in 100 small packages. They found that the number of candies per package was normally distributed, with a mean of 23 candies per package and a standard deviation of 1 piece of candy. What is the probability that a package selected at random has more than 25 candies? Values greater than 25 are more than 2σ from the mean. The values that are more than 2σ from the mean cover two tails and 5% of the distribution. We are only concerned with the upper tail, so 2.5% of the data will be greater than 25. Answer:

Example 2 A.17% B.34% C.68% D.81.5% DRIVER’S ED The number of students per driver’s education class is normally distributed, with a mean of 26 students per class and a standard deviation of 3 students. What is the probability that a driver’s education class selected at random will have between 23 and 32 students?

Example 2 A.17% B.34% C.68% D.81.5% DRIVER’S ED The number of students per driver’s education class is normally distributed, with a mean of 26 students per class and a standard deviation of 3 students. What is the probability that a driver’s education class selected at random will have between 23 and 32 students?

Example 3 Use z-Values to Locate Position Find σ if X = 28.3, μ = 24.6, and z = Indicate the position of X in the distribution. Formula for z-Values X = 28.3,  = 24.6, z = 0.63 Divide each side by Simplify. 0.63σ = 3.7Multiply and subtract.

Example 3 Use z-Values to Locate Position Answer: σ is Since z is 0.63, X is 0.63 standard deviations greater than the mean.

Example 3 A B C D Find μ if X = 19.2, σ = 3.7, and z = –1.86.

Example 3 A B C D Find μ if X = 19.2, σ = 3.7, and z = –1.86.

Example 4 A.22% B.28% C.72% D.78% INSECTS The lifespan of a specific insect is normally distributed with a mean of 12.3 days and a standard deviation of 3.9 days. Find P(X > 10).

Example 4 A.22% B.28% C.72% D.78% INSECTS The lifespan of a specific insect is normally distributed with a mean of 12.3 days and a standard deviation of 3.9 days. Find P(X > 10).

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