NRHS AP Statistics Curriculum Number Lesson Requirement 1 What is statistics? 2 Given a set of data, identify the individuals and variables (categorical or quantitative). 3 Given a set of data, create a dot plot and time plot manually and using the TI-84 4 Create a histogram using class width, class limits, class boundaries, class midpoints, and relative frequency. 5 Interpret histograms. 6 Stem and Leaf display find the mean and median and mode. 7 How to find the Measures of Central Tendency (center): Mean, Median, Mode manually and using the TI-84. 8 Describe the difference between parameters and statistics with respect to the mean and standard deviation. 9 Have Students using their descriptive statistics sheets that will be given to them on the AP to compare formulas. 10 Given a density graph, Students should be able to calculate the area under the curve. 11 Describe the normal distribution use exactly one, two or three standard deviations away from the mean to find probabilities. π₯−µ 12 Use the normal curve using the formula π§ = π to find the exact probabilities of ANY value from the mean. Use the chart to locate the probability. 13 Given the probability or percentile, find π₯ (do inverse) 14 Define explanatory and response variables, graph, and interpret scatterplots. Describe outliers and influential points. 15 Define correlation and calculate the correlation (π) manually using statistics descriptive sheet and using TI-84. Describe strength and direction. Interpret coefficient of determination (π 2). 16 Calculate the slope and π¦ −intercept to obtain the πΏππ πΏ. 17 Using the calculated πΏππ πΏ find a predicted π¦ for a specified π₯. 18 Find residuals manually and graphically. Describe what is implied when the residuals form a pattern and are not scattered. 19 Calculate exponential growth. Find the πΏππ πΏ using logs. 20 Calculate power growth. Find the πΏππ πΏ. 21 Describe simple random sample [SRS] and random number table. 22 How to use the random digit table [RDT] on the AP and how to set up an experiment, which will be appropriate to use the RNT. Use TI-84 to conduct simulations (rand button). 23 Describe systematic sampling, cluster sampling, convenience Revised 2015 College Board Approved 2008 NRHS AP Statistics Curriculum 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 sampling. Describe a census, sample simulation, observational study, experimental study, placebo, randomized two treatments experiment, double-blind experiment, control group, confounding (lurking variables), randomization, and replication. Use scenarios to describe surveys nonresponse, voluntary response, hidden bias, and other bias generalizing results. What is probability? Law of large numbers? Discuss important facts. Given a sample space, find the probability of an event and the complement of the event. Use descriptive statistics sheet to show Students the formulas. Describe the difference between independent, dependent, and mutually exclusive events. Use Venn Diagram and formula sheet. Multiplication rule—combinations for probability Definition of discrete random variable and continuous random variable. Describe the difference between the probability of less than (<) and less than or equal to (≤) for continuous random variable. Find the mean ( πΈ(π₯) ) and standard deviation ( π )for discrete probability distribution. Use descriptive statistics sheet to show Students formula. Describe the features of a binomial experiment and use the formula for the binomial probability distribution and using the TI-84. Formula for the binomial probability distribution and using the TI-84. Find the mean and standard deviation of binomial probability distributions. Calculate the probability. Formula for the binomial probability distribution and using the TI-84. Describe the features of a geometric probability distribution Find the mean and standard deviation of geometric probability distributions. The central limit theorem: probability regarding π₯ and π₯Μ . Use z when π ≥ 30. Sampling distribution for the proportion p. Make inference about the population ππ>5 and ππ>5. Find the mean and standard deviation for the proportion. Estimating mu with large samples. Make inference about π and π . Have Students look up confidence intervals using chart. Construct a confidence interval for the population mean using Revised 2015 College Board Approved 2008 NRHS AP Statistics Curriculum samples. 42 Estimating mu with small samples. Students should know the difference between π§ and π‘ with respect to estimating π . Only use π for π‘-distribution. Have Students look up confidence intervals with π. π. = π − 1 43 Construct a confidence interval for the population mean using samples. Interpret a confidence interval. Estimating p for the proportion. Construct a confidence interval the mean p. Interpret a confidence interval. Given all information for a set of data with the exception of π, find an appropriate sample size (i.e. solve for n) Given two populations, estimate the difference between the mean and make inference about the population. Given two populations, estimate the difference between the proportion mean State the Null and Alternate hypotheses. Discuss Type I and Type II. Discuss level of significance. Discuss one or two tailed test. Test involving the mean mu (large samples). Use TI-84 to assess a test. Discuss critical values for testing, Critical region. Draw conclusions. Discuss statistical significance. The π −value in Hypothesis Testing identify the meaning of π −value for a statistical test. Compute the π −value for large sample tests of µ (mu). Draw test conclusions based on the π −values and the level of significance. Identify the components of a small sample test of µ. Use degrees of freedom and the π‘ −table to find critical values. Compute the sample test statistic. Estimate the π −value for a small sample test of µ. Draw conclusions using the level of significance alpha and the π −value. Test involved paired differences (dependent samples)identify paired data and dependent samples. Explain the advantages of paired data tests. Compute differences and the sample test statistic. Use degree of freedom and the t table to find critical values. Estimate the P value and conclude the test. Testing Difference of Two Means identify independent sample and sampling distribution. Compute the sample test statistic and π −value for µ1 πππ µ2 (large samples). Identify independent sample and sampling distribution. Compute the test statistic and π −value for µ1 πππ µ2 for small samples. Compute the test statistic, critical value, and π −value for 44 45 46 47 48 49 50 51 52 53 54 55 56 57 Revised 2015 College Board Approved 2008 NRHS AP Statistics Curriculum 58 59 60 61 62 63 64 65 66 67 π1 πππ π2. Use the level of significance α and the π −value to conclude test. Review how to find the LSRL and correlation (π) in the calculator. Use πππ₯, πππ¦, πππ₯π¦, which is equivalent to the variance of π₯, π¦, πππ π₯π¦ respectively. Use the formula to calculate the standard error of estimate. Calculate the margin of error for π¦ −hat at a specified π₯. Construct a confidence interval for π¦ −hat. Find a confidence interval level for the slope of the πΏππ πΏ. Test the slope of πΏππ πΏ to determine if there is a linear relationship between the explanatory variable and the response variable. Students should be able to read minitab output(computer software). Compute Chi-Square with one set of data to test “goodness of fit.” Use a contingency table to compute the expected frequency of Chi-Square. Find the number of degrees of freedom(π − 1) (πΆ − 1) in a contingency table. Use TI-84 to determine if there is an association between two events. **** Review for AP Exam using practice exams and AP free response problems***** 68 69 70 71 72 73 74 75 76 Testing and Estimating a Single Variance or Standard Deviation. Set up a test for a single variance. Compute the Chi Square statistic Use the Chi-Square statistic distribution to complete the test and estimate a P value. Compute the confidence intervals for variance and standard deviation. Set up a test for two variances. Use sample variances to compute the sample F statistic. Use the F distribution to complete the test and estimate a P value. One way ANOVA learn about the risk alpha of a type I error when we test several means at once Compute mean squares between groups within groups. Compute the sample F statistic Use the F distribution to conclude the test and estimate a P value. Learn the notation and setup for two-way ANOVA tests Learn about the three main types of deviations and how they break into additional effects. Revised 2015 College Board Approved 2008 NRHS AP Statistics Curriculum Topic by Topic I. ο· ο· ο· ο· ο· ο· ο· ο· ο· II. ο· ο· ο· ο· ο· ο· ο· ο· Linear Regression Students will be able to create scatter diagrams and determine the correct correlation given multiple choices. Students will be able to determine if a correlation is positive, negative or zero. Students will be able to use their TI-84 plus to enter in their data and find the line of best. Students will be able to use their TI-84 plus to find the standard deviations of both π₯ and π¦ and their slope of the line. They will then plug that information into the formula to find π (the correlation) Students will be able to use the line of best fit to predict values. They will then use the residual formula and find the residuals of the data. They will be able to then determine if a line of best fit should be used according to the pattern. If a line of best fit should not be used, students will explore quadratics, exponentials, logarithmic and other possible regressions using their TI 84 plus Students will be working backwards given the prediction point and the residual to find the actual and vice versa. Students will explore what happens to the correlation coefficient when they switch their x and y values and explore the result of the πΏππ πΏ. Students will then work with the formula of the correlation on the formula sheet and discuss why this happens. Students will learn about R2 (the ratio of explained variation over total variation). Students will be given multiple questions on linear regression from Practice AP books. Z-scores Students will begin by given data and finding the mean, median, and mode. Students will then be taught how to find the standard deviation using the formula and what is meant by a standard deviation. Students will be given questions based on the empirical rule (68-95-99) and will be asked to interpret the information given. Students will be given questions that ask about percent that do not fall one, two, or three standard deviations away Students will be given charts on z-scores and z-scores will be explained to the Students. Students will be begin by just finding the corresponding probability to z-scores and finding the probability above or below the curve Students will then be able to use their TI-84 plus to find the standard deviation along with the mean median and mode. Students will have questions where they must convert their information to a z-score, look up the z-score and find the probability. Revised 2015 College Board Approved 2008 NRHS AP Statistics Curriculum ο· ο· ο· ο· ο· ο· Students will then have to work backwards given the probability, the must find the z-score, place it back into the equation given the population mean and standard deviation to find the mean. (i.e. a student scored 77% on the math portion of the SATs, if the math portion of the SAT scores are normally distributed with a µ= 510 and σ = 50, what did the student score?) Students will learn how to find probabilities and means using their TI-84 calculator Students will learn how to find probabilities for averages (the central limit theorem). Students will be able to find the probability using a z-score given a sample mean (π₯Μ ), population mean (µ), population standard deviation (σ) and sample size (π). Students will then learn about binomial distribution. They will be given the same data and use the formula µ=ππ and σ= √πππ. Students will learn about proportions and will then be able to find the mean and π ππ standard deviation of the sampling distribution by using the formula π and √ π to ο· ο· ο· III. find the mean and standard deviation, respectively. Students will work on various questions to find the mean and standard deviations. They will then find z-scores and the probabilities corresponding to them. Students will then be able to compare the central limit theorem to the binomial distribution as well as sampling distributions for proportions. Students will use their TI-84 to find the probabilities and compare. Estimations ο· ο· ο· ο· ο· ο· Students will begin this topic by talking about what happens when µ is unknown but π₯Μ is given. We will discuss confidence intervals and their meanings. We will discuss the meaning of 90%, 95%, 98%, 99% confidence intervals and levels. Students will then take the π§ −score formula and place it between two critical π values to form π₯Μ ± π§ π √ π Students will solve for µ and will be left with π₯Μ ± π§ π. Students will discuss the √ margin of error and what it means in terms of the population. Students will begin working on problems and creating confidence intervals so they can estimate where µ (when σ is known) and how confident we are we then will turn to estimating mu when σ is unknown. We will replace the z- critical value with t-critical value. Students already know how to find the t-value. We will continue to work on problems when sample sizes are small and the sample standard deviation is unknown. Students will move on and find out what is the appropriate sample size to choose given the margin of error that is wanted. Students will do interpretations of both confidence intervals and confidence levels and have multiple choice questions on what is the correct interpretation of confidence intervals and confidence levels. Revised 2015 College Board Approved 2008 NRHS AP Statistics Curriculum ο· ο· Students will distinguish between independent samples (such as basketball players salaries and football players salaries) and dependent samples (such as amount of fish before a fire and after a fire). Students will compute an interval for the difference between µ1-µ2 when the standard deviations are known (π§ −score) and the standard deviations are π2 π2 1 2 unknown (π‘ −score). Students will use the formula π₯Μ 1 − π₯Μ 2 ± zc √π + π ο· ο· ο· IV. Students will create a confidence interval and interpret based on the intervals signs (i.e. if the interval is positive, one can conclude µ1 > µ2 and if the interval is negative µ1 < µ2 Students will also be able to also work on two proportions using the formulas. Use the TI-84 go to STAT –TEST and we will work on π§ −interval, π‘ −interval, 2−sample π§ −interval 2-sample π‘ −interval, 1 −prop π§ −interval, 2 −prop π§ −interval Students will work on various questions for AP review books Hypothesis Testing ο· ο· ο· ο· ο· ο· ο· Students will understand the rationale for statistical tests and that it is designed to assess the strength of the evidence (data) Students will begin by stating the null hypothesis (the statement under investigation or being tested). They will understand that this is the statement that says things are the same. Students will then state an alternative hypothesis (this is the statement Students will use to try to reject the null hypothesis Students will learn about type I and type II errors based on examples such as courtroom where a judge makes the correct decision (no error) or if it was true and rejected (type I error) and if it was false and not rejected (type II error) or false and rejected was a correct decision Students will learn about the level of significance alpha is the probability of rejecting null hypothesis when it is true. Students will learn the three types of tests (left-tailed that is the parameter is less than the claimed null hypothesis, the right-tailed is the parameter that is more than the claimed null hypothesis, and two-tailed that the claim is different) Students will learn about 90 95 98 99 percent corresponding z scores of 1.645, 1.96, 2.33, 2.58 and then find the P values that correspond to it. P-values of 5% or 1% will be explained. Students will see how the critical values go hand-in-hand with the p-values. Students will state a p-value first then begin the problem given. The will find a zscore and look up the z-score to find the corresponding probability. They will then compare this to the p-value. If the probability that was just found is smaller than the p-value, they will reject the null hypothesis. Otherwise there is not sufficient evidence to reject. Students will also have to memorize the 1% and 5% z-scores for left, right and two-sided tail. They will begin the problem and find a z test. Revised 2015 College Board Approved 2008 NRHS AP Statistics Curriculum ο· ο· ο· Then just compare it to 1.645 1.96 2.33 2.58 then determine if the null hypothesis is rejected or failed to reject. Students will then do the same for a proportion π. Students will continue on tests involving paired differences (dependent samples). They will place their data into the calculator and subtract L2 from L1 and place those answers in L3. They will find the mean, sample standard deviation and n and compute a t-statistic. They will then determine whether they reject or fail to reject the null hypothesis. Example: 10 students taking the SATs then taking a prep course then taking the SATs again. Find the difference in the exams and determine if the course made a difference or not. Students will then work on tests involving independent samples. They will use the π₯Μ −π₯Μ formula π§= π12 π22 √ ο· ο· ο· ο· V. + π1 π2 Students will then be able to determine if they should reject the null hypothesis or not. The same is done with proportions. Students will make the right interpretation at the end. (if the example was two types of teaching methods and the p-value was less than 5% we can concluded at the 5% level of significance there is sufficient evidence to show that the second teaching method increased the population mean score on the exam. Students' notebooks are divided by topics. Students now will have to do the same questions using confidence intervals. They will look at µ and determine if µ is in the interval. They will see how confidence intervals and hypothesis testing are interchangeable, and how using a confidence interval, students could reject or fail to reject the null hypothesis. With smaller samples students will learn about pooled standard deviation. Example: suppose the sample values of standard deviations π 1 πππ π 2 are sufficiently close and that there is reason to believe the population standard deviations are equal then you can use the pooled standard deviation. Chi-squared ο· ο· ο· Students will set up a test to investigate whether a distribution is a “good fit” or not. Students will have an overview of chi-squared distribution (π 2 ) (learn how to read the chart, right-skewed, degrees of freedom, etc.). Students will learn about contingency tables and will have observed values in a single row. They will then find the expected value and place it in the row below it. π΄(πππ πππ£ππ−πΈπ₯ππππ‘ππ)2 ο· Students will find the π 2 test statistic using the formula ο· Students will also work on examples for the “goodness of fit” such as what is most important (vacation, salary, etc) to workers during the year 2008 and determining if that importance stayed the same—a “good fit” the year 2012. They will then look at the chart with π − 1 degrees of freedom and determine if the two events are independent (example could be keyboard arrangement and learning times) πΈπ₯ππππ‘ππ Revised 2015 College Board Approved 2008 NRHS AP Statistics Curriculum Students will then move to π X π contingency tables where the expected for each cell is π ππ€ πππ‘ππ∗πΆπππ’ππ πππ‘ππ found by using the formula ( πππ‘ππ ππππ’πππ‘πππ ). Students will then do π 2 then compare to the π −value. (An example could be types of soda students drink and the school level –high school or elementary school). Students will also have to find between what two π −values the π 2 falls between. VI. Randomization ο· ο· ο· ο· Students will set up learn about randomization and be given a random digit table [RDT]. Students will learn how to number things 0 − 9, 01 − 10, 001 − 100 etc. and read off digits to choose their samples randomly. We will then set up experiments so randomness is used. Example 1: Imagine you have to use three homes for questioning on a street that has 10 homes. Students will number the homes 0-9 and read off single digits until they find the first three homes. Those three homes will be used for the interview. Example 2: Let’s assume that we wanted to see if the NYC homes have deadbolts and we knew that 75% of homes in NYC had deadbolts. We wanted to see if 10 homes we went to had deadbolts. Students will learn that they can number homes as follows: 00-74 are homes with deadbolts and 75-99 without. They will read from the RDT and find the first ten homes and see who has the deadbolts and compare it to the actual. Students will learn many terminologies such as double-blinding, SRS, experiments, response variables, explanatory variables, voluntary response, question wording etc. Students will be given paragraphs of certain situations and will have to locate bias in the paragraphs. They will also have to rewrite paragraphs and tell me what they could do to make it as unbiased as possible VII. Probability ο· ο· ο· ο· ο· Students will learn about Probability as what we want divided by total. They will learn about compliment (the probability of rain is .4 the probability of not rain is .6) Students will be introduced to two or more events Students will learn about “or” and “and” statements. Students will learn about independent and dependent probabilities. Students will learn about the following formulas: π(π΄ π π΅) = π (π΄) + π(π΅) − π(π΄ ∩ π΅). π(π΄ ∩ π΅) = π (π΄) ∗ π(π΅) if the events are independent. π(π΄ ∩ π΅) = 0 if the events are mutually exclusive. Revised 2015 College Board Approved 2008 NRHS AP Statistics Curriculum π(π΄ ∩ π΅) = π(π΄)π(π΅|π΄) for a conditional probability. ο· ο· Students will learn about mutually exclusive (picking a red marble and a green marble at the same time, passing and failing the same test) Students will deal with given statements. They will make tree diagrams and find probabilities. Examples will be testing positive for a disease is 30% and actually having the disease is 98%. If someone doesn’t have the disease, the probability having it is 3%. Find the probability of having the disease given the test positive. Revised 2015 College Board Approved 2008