Statistics - Algebra 2 Teaching Suggestions
Unit 9
9.1 To identify and perform an appropriate method of gathering data (experiment, simulations, observational studies including sample surveys).
9.1A. SUGGESTION: List several examples of topics of study and have students identify whether census, sample survey, simulation, experiment or observational survey would be the best method for collecting the data.
9.1B. SUGGESTION: Have students bring in examples of experiments and observational studies to analyze. These can be found in online newspapers, magazines, research papers, scientific journals, etc. Determine if bias exists. If so, identify and suggest possible corrections of the bias. Make sure to discuss sampling errors (random sampling error, undercoverage, bad sampling – VRS, convenience) and non-sampling errors (processing errors, response bias, question wording and nonresponse bias) and make sure students know the difference between the two.
9.1C. SUGGESTION: Describe surveys and have students identify the sampling flaws.
The following website has examples of survey templates that can be used in the classroom and analyzed.
https://www.surveymonkey.com/mp/survey-templates/
9.1D. SUGGESTION: Have students design an experiment that minimizes sampling error. Students should be able to set up a simple experiment and blocking, blindness and matched pairs should be investigated.
9.1E. SUGGESTION: Have students create their own sample survey. They will develop their own questions (discussing question wording bias) and carry out the survey. Students will then report the results along with biases they encountered. Students can pick a topic that interests them and can pick their population (students at their school, high school teachers, adults in their neighborhood, etc.)
9.1F. SUGGESTION: Give students a list of questions and ask them what would be the best method for gathering the data choosing from: census, experiment, simulation observational study or sample survey. Possible questions: How many hours of television does a high school student watch per day? How much pressure can be exerted on a chicken egg before it breaks? What percentage of American League baseball players had a batting average about 0.300 this season? Is there a relationship between the amount of physical activity a person gets and his or her perceived level of stress? Do the seniors at your school do less homework than the sophomores? Are oral medications better than lotions in killing lice? Does smoking 2 packs of cigarettes a day vs 1 pack of cigarettes a day increase your chances of getting lung cancer? What is the likelihood of having no girls in 6 births?
9.2 To understand the importance of randomization and the difference between random sampling and random assignment.
9.2A. SUGGESTION: Have students distinguish between random sampling and random assignment. Random sampling occurs when a study is set up so that the sample that is selected will represent the population in all its diversity. Random assignment refers to when a scientific study (experiment) with a goal of inferring a cause and effect relationship is set up. Random assignment tends to produce treatment groups with the same mix of values for variables extraneous to the study.
9.2B. SUGGESTION: Have students use a random number generator (computer or graphing calculator) or random number table to draw a simple random sample (SRS).
9.2C. SUGGESTION: Have students do the Gettysburg Address activity. Provide students a copy of the Gettysburg Address and have them scan it for 5 seconds. Students should then try to estimate the average length of the words used in the speech. After collecting all of the students guesses, now have students randomly select 10 words from the passage (use a random generator on their calculator) and find the average length of those ten words. Compare these random averages to the guessed average. Our eyes are “biased” and will see longer words and not identify how many small words there are in the passage. The guessed average will always be higher than the actual, which is 4.3, found from finding the average of all 268 words of the Gettysburg Address. This illustrates how important randomization is. One formal write-up of this activity can be found at the website below. The pdf also includes other activities to help illustrate the importance of randomization and how it reduces bias.
http://express.lander.edu/stats/images/materials/rossmanmyrtlebeachhandout.pdf
9.2D. SUGGESTION: Have students discuss different ways of obtaining random samples (SRS, stratified, cluster, multistage, systematic) and the merits of each of those methods. Also have students look at samples that do not employ randomness and are biased like convenience and voluntary response samples. These types of samples systematically favor some parts of the population over others in choosing the sample.
9.2E. SUGGESTION: Have students set up a random assignment for an experiment. Give them an example of an experiment and ask how they would randomly assign the treatments. Students might suggest putting all names in a hat, shaking the hat and then picking out the treatment groups or they might number all participants and use a random number table or random number generator to pick the groups. Discuss how these methods will give us random, equal (in number) groups for our experiment. Explain how this is different from using a coin or dice to randomize our treatments (this would not result in equal number of participants in the groups). Students should then discuss whether to block, do a matched pairs or a completely randomized design. Have them diagram the experiment showing each stage of the experiment, starting with random assignment and ending with the response variable measured at the end.
9.2F. SUGGESTION: Have students do the “Rolling Down the River” activity which has students explore the importance of random sampling by creating convenience, stratified and simple random samples. Find the activity at the website below.
http://courses.ncssm.edu/math/Stat_inst01/PDFS/river.pdf
9.3 To organize data.
9.3A. SUGGESTION: Students should be familiar with different graphical representations of data including histograms, bar graphs, circle graphs, boxplots, dotplots, and stem and leaf plots.
9.3B. SUGGESTION: Give students a list of variables, like the ones below. Have the students decide whether each variable is categorical (qualitative) or quantitative (numerical) and decide which graph type would be best to display the data. Exs: height, eye color, shoe size, GPAs, SAT scores, a team’s jersey numbers, amount of sleep (in hours), pulse rate, number of letters in first name.
9.3C. SUGGESTION: Have students work with spreadsheets and graphing calculators to have experience with scaling graphs and recognizing distortions or bias in graphs.
9.3D. SUGGESTION: Have students find measures of center and variability for data sets using the appropriate measures with respect to its shape.
9.3E. SUGGESTION: Have students describe the data distribution by noting its center, shape and spread and identifying any unusual features like outliers, gaps and clusters.
9.3F. SUGGESTION: Have students organize data in frequency tables and describe the distribution by finding mean, standard deviation and shape (uniform, skewed left, skewed right, symmetric)
9.4 To know and apply the empirical rule.
9.4A. SUGGESTION: Have students define standard deviation in the context of different situations. Students must have an in depth understanding of standard deviation as a measurement tool in order to apply the empirical rule.
9.4B. SUGGESTION: Have students complete the standard deviation activity below. This activity helps students develop a better intuitive understanding of what is meant by variability in statistics. https://www.causeweb.org/repository/StarLibrary/activities/delmas2001/
9.4C. SUGGESTION: Have students investigate data in the real world that take on approximately normal shapes like heights of adults, size of things produced by machines, errors in measurements, blood pressure, IQ scores, marks on a test, and SAT scores.
9.4D. SUGGESTION: Have students use the empirical rule to assess if the data is normally distributed. Find data that is approximately normally distributed and calculate the mean and standard deviation of the data set. Determine if approximately 68% of the data falls one standard deviation away from the mean, if 95% of the data falls two standard deviations away from the mean and if almost all of the data falls within three standard deviations.
9.4E. SUGGESTION: Have students place a variable trait into a graphing calculator. This variable could be their height, weight, shoe size, last test score, how many hours of sleep they got last night, pulse rate, etc. The data may or may not be approximately normal, but by applying the empirical rule, students will be able to evaluate whether or not the data is symmetric, unimodal and bell-shaped. If heights are used and are approximately normal, this could be illustrated visually by having students make a human histogram by lining up in the classroom.
9.5 To use and apply normal distributions when appropriate (z-score, percentile).
9.5A. SUGGESTION: Have students define normal distributions as unimodal, symmetric and bell-shaped. Look at normal distributions with the same mean and different standard deviations, noting their different shapes and placement of their inflection points.
9.5B. SUGGESTION: Remind students that not all symmetric or “bell-shaped” curves are normal, only those with the following equation:
9.5C. SUGGESTION: Have students use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages with z-scores. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets and tables to estimate areas under the normal curve and to work backward to find an observation for a given percentile.
9.5D. SUGGESTION: Use technology with the normalcdf and invNorm commands on the graphing calculator to find the area under the normal curve or to find the observation that matches a particular percentile.
9.5E. SUGGESTION: Have students graph approximately normal data as a histogram with technology and compare that graph with the normal probability plot of that distribution. If the data is approximately normal, the normal probability plot (NPP) should be very close to linear. Also plot skewed data that is not normal and notice the differences in the normal probability plots. If the data as skewed right, the NPP will “drop off to the right” from the linear pattern and if the data is skewed left, it will “drop off to the left”.
9.5F. SUGGESTION: Have students work on the “Number of Hours of Sleep” activity. Students work in groups and answer how many hours they slept in the past 24 hours. Students answer questions based upon their amount of sleep answers and a previously conducted survey regarding the number of hours of sleep of high school students. The activity and worksheet can be found at the following website: http://www.regentsprep.org/Regents/math/algtrig/ATS2/NormalResource.htm
9.6 To know the difference between a parameter and a statistic.
9.6A. SUGGESTION: Have students distinguish between parameters and statistics. A population is a big word for a “group of things we are studying”. A population can be big or small, but it is the group we want information about. Parameters are the “things we want to measure about our group – the population”. Note parameter and population both start with a “p”. Since it is usually impossible to get information about the population (if it is possible, then we can do a census), it is important that we do proper random sampling of our population. The data we get from this sample is called a statistic, and will accurately represent the population parameter if we have a random, unbiased sample. Sample and statistic both begin with an “s”, helping to keep the terms identified correctly. Our sample is a subset of the population chosen to represent that group.
9.6B. SUGGESTION: Have students identify the population, the parameter of interest, the sample and the sample statistic for the following reports.
Exs: 1) A magazine asked all subscribers whether they had used alternative medical treatments and, if so, whether they had benefited from them. For almost all of the treatments, approximately 20% of those responding reported cures or substantial improvement in their condition. 2) A company packaging snack foods maintains quality control by randomly selecting 10 cases from each day’s production and weighing the bags, and then inspecting the contents. The weight of a case should be 2 lbs. One day they found that the weight of the 10 cases was 2.08 lbs. 3) State police set up a roadblock to estimate the percentage of cars with up-to-date registration and insurance. They found problems with 10% of the cars they stopped. 4) The Environmental Protection Agency took soil samples at 20 locations near a former industrial waste dump and checked each for evidence of toxic chemicals. The found no elevated levels of any harmful substances. 5) The telephone company says that 62% of all residential phone numbers in Los Angeles are unlisted. A telephone survey contacts a random sample of 1000 Los Angeles telephone numbers, of which 58% are unlisted.
9.7 To understand that inference is drawing a conclusion about a population parameter based on a random sample from that population.
9.7A. SUGGESTION: Have students perform a hands-on activity to investigate a claim about a population proportion. Students are divided into small groups and presented with M & M’s to check the claim that the company produces 16% green M & M’s in its packages. Students will be exploring the relationships between a population, population parameters, random samples and statistics. By the end of the investigation, students will be able to informally relate a sample statistic to a known population parameter. The activity can be found at the following website: http://www.amstat.org/education/STEW/pdfs/PopulationParameterswithMMs.pdf For data on M&Ms and in depth study of colors of 48 packages, visit: http://joshmadison.com/2007/12/02/mms-color-distribution-analysis/
9.7B. SUGGESTION: Have students investigate the Reese’s Pieces applet which can be set with a known parameter and then draws random samples from that population to illustrate the conclusions that can be made about parameters based on unbiased statistics. This is also an informal introduction to a sampling distribution. http://www.rossmanchance.com/applets/Reeses3/ReesesPieces.html
9.7C. SUGGESTION: Have students find sample means and sample proportions from data collected for a study or experiment. Calculated the estimated population proportion or mean by calculating the sample proportion or mean and discuss how this estimate will closely resemble the parameter if the sample is unbiased.