Statistical Reasoning in Sports - Central York School District

Unit 1: Testing Claims in Sports

Standards

The Common Core Standards for High School Statistics & Probability

CCSS.Math.Content.HSS.IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
CCSS.Math.Content.HSS.IC.A.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

Know

An athlete/team’s ability is a true but unknown value that describes what the athlete/team would do if given an infinite number of opportunities in a specific context.
An athlete/team’s performance is what the athlete/team actually did in a limited number of opportunities in a specific context.
We use an athlete's/team’s performance to estimate their ability.
Random chance describes the variation between an athlete/team’s performance and their ability.
A streak is a set of one or more consecutive outcomes that are all the same.
An athlete is streaky if he or she tends to have streaks longer than expected by random chance alone.
An athlete with a longer-than-expected streak of successes is said to have the hot hand.
The outcomes of an athlete’s attempts are independent if his or her ability to be successful is the same following a successful attempt and following an unsuccessful attempt.

Understanding/Key Learning

Do

Unit Essential Questions

Lesson Essential Questions

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Assessments

Unit 2: Analyzing Categorical Variables

Standards

The Common Core Standards for High School Statistics & Probability

CCSS.Math.Content.HSS.ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
CCSS.Math.Content.HSS.IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
CCSS.Math.Content.HSS.IC.A.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

Know

Individuals are the things described by a set of data (e.g. athletes, teams, attempts, games, seasons).
Individuals are sometimes called cases or observational units.
A variable is an attribute that takes on different values for different individuals.
A categorical variable assigns labels that place individuals into groups.
Values of categorical variables are typically expressed with words or labels.
Graphs of categorical data include bar charts, pie charts, and segmented bar charts.
A quantitative variable takes number values (measurements or counts) for which it makes sense to find a mean.
Values of quantitative variables are expressed with numbers.
The distribution of a variable identifies the possible values of a variable and how often the variable takes those values.
A bar chart displays the possible outcomes of a categorical variable as individual, equally wide bars, with the height of each bar showing the frequency or relative frequency of the corresponding outcome.
A pie chart displays the possible outcomes of a categorical variable as slices of a circular pie, with the area of each slice proportional to the frequency of the corresponding outcome.
A segmented bar chart displays the possible outcomes of a categorical variable as segments of a rectangle, with the area of each segment proportional to the frequency of the corresponding outcome.
The null hypothesis, denoted H0, describes an initial belief that there was no change or no difference in an athlete’s ability.
The alternative hypothesis, denoted Ha, describes what we want to establish or what we suspect is true about an athlete’s ability.
The sample size is equal to the total number of individuals that we are analyzing in a given situation.
In a sports context, the law of large numbers says that an athlete’s performance will vary less and less from his or her ability as the sample size grows larger.

Understanding/Key Learning

Do

Identify individuals and variables in a data set.
Distinguish categorical and quantitative variables.
Graph the distribution of a categorical variable using bar charts, pie charts, and segmented bar charts.
Identify what makes some graphs deceptive.
State the null hypothesis and the alternative hypothesis for a test about change in ability.
Describe evidence for the alternative hypothesis and provide two explanations for the evidence.
Use a spinner to simulate the distribution of the number (or proportion) of successful attempts.
Use the results of a simulation to determine if there is convincing evidence of a change in ability.
Use technology to generate a distribution of simulated performances.
Use the law of large numbers to explain the effect of sample size.
Test a claim about ability using the State-Simulate-Conclude process.

Unit Essential Questions

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Unit 3: Comparing Two Percentages

Standards

The Common Core Standards for High School Statistics & Probability

CCSS.Math.Content.HSS.ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
CCSS.Math.Content.HSS.IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
CCSS.Math.Content.HSS.IC.A.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
CCSS.Math.Content.HSS.CP.A.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified.

Know

A response variable measures the outcome of interest in a study.
An explanatory variable may help predict or explain changes in a response variable.
An explanatory variable is sometimes called a predictor variable.
A two-way table summarizes the relationship between two categorical variables.
Two variables have an association if knowing the value of one variable helps predict the value of the other variable.
A side-by-side bar chart can be used to display the association between two categorical variables.
For each category of the explanatory variable in a two-way table, each category of the response variable is given its own equally wide bar and the height of the bar corresponds with the percentage of times that category occurs.
Bars in a two-way table are typically grouped by the categories of the explanatory variable.
When the categories of the explanatory variable have different sample sizes, it is important to use relative frequencies rather than frequencies to compare the distribution of the response variable.
In a test for a difference in percentages, the null hypothesis specifies that there is no difference or no change in ability.
In a test for a difference in percentages, the alternative hypothesis specifies what we suspect is true.
In a test for a difference in percentages, the evidence for the alternative hypothesis is found by calculating the difference in the percentage of successes for the two contexts.
To determine if the evidence for the alternative hypothesis is convincing, conduct a simulation to determine what differences in performance are likely to happen by random chance, assuming the athlete/team’s ability is the same in both contexts.
In a test, failing to reject the null hypothesis is not the same as accepting the null hypothesis.
In a test, we never prove that the alternative hypothesis is true.

Understanding/Key Learning

Do

Distinguish between explanatory and response variables.
Summarize the association between two categorical variables in a two-way table.
Construct a segmented bar chart to display association between categorical variables.
Describe the association between two categorical variables.
State hypotheses for testing a difference in percentages.
Describe evidence for the alternative hypothesis and provide two explanations for the evidence.
Explain how to simulate the distribution of a difference in percentages using note cards.
Estimate a p-value using the results of a simulation.
Determine if there is convincing evidence for the alternative hypothesis using a p-value.
Estimate a p-value using technology.
Test a claim about a difference in ability using the State-Simulate-Conclude process.

Unit Essential Questions

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Assessments

Unit 4: Experiments

Standards

The Common Core Standards for High School Statistics & Probability

CCSS.Math.Content.HSS.ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
CCSS.Math.Content.HSS.IC.B.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
CCSS.Math.Content.HSS.IC.B.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
CCSS.Math.Content.HSS.CP.A.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified.

Know

Two or more variables are confounded if it is impossible to determine which variable is causing a change in the response variable.
An observational study uses available data to determine if there is an association between the explanatory and response variables.
An experiment deliberately assigns treatments to subjects to see if changes in the explanatory variable cause changes in the response variable.
In an experiment, a control group is used to provide a baseline for comparing the effects of other treatments.
Depending on the purpose of the experiment, a control group may be given an inactive treatment or an active treatment.
In an experiment, random assignment is where treatments are assigned to subjects (or subjects are assigned to treatments) using a chance process.
The purpose of random assignment is to create groups that are roughly equivalent at the beginning of an experiment.
In an experiment, subjects are considered blind if they do not know what treatment they are receiving.
If the people evaluating the subject and collecting the data are also blind to the treatment, the experiment is called double-blind.
A placebo is a treatment that has no active ingredient, but is otherwise like other treatments.
The placebo effect describes the fact that some subjects in an experiment will respond favorably to any treatment, even an inactive treatment.
A Type I error occurs when we find convincing evidence for the alternative hypothesis, when in reality the null hypothesis is true.
A Type II error occurs when we do not find convincing evidence for the alternative hypothesis, when in reality the alternative hypothesis is true.
The significance level of a test is the boundary value we use to decide whether a difference in performance is unlikely to happen by random chance alone.
The most common significance level is 0.05.

Understanding/Key Learning

Do

Explain the concept of confounding and how it limits the ability to make cause-and-effect conclusions.
Distinguish observational studies from experiments.
Explain the purpose of comparison in an experiment.
Explain the purpose of random assignment in an experiment and how to perform the random assignment.
Identify variables that should be kept the same in an experiment.
Describe the placebo effect and the purpose of blinding in an experiment.
State hypotheses for testing a difference in percentages for an experiment, and state the evidence for the alternative hypothesis.
Use simulation to estimate a p-value for an experiment, and draw an appropriate conclusion.
Describe a Type I and a Type II error in context.
Explain the consequences of making each type of error.
Explain how to reduce the probability of each type of error.

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Unit 5: Conditional Probability and Strategy in Sports

Standards

The Common Core Standards for High School Statistics & Probability

CCSS.Math.Content.HSS.CP.A.1Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
CCSS.Math.Content.HSS.CP.A.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
CCSS.Math.Content.HSS.CP.A.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
CCSS.Math.Content.HSS.CP.A.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.
CCSS.Math.Content.HSS.CP.A.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
CCSS.Math.Content.HSS.CP.B.6 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.
CCSS.Math.Content.HSS.CP.B.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
CCSS.Math.Content.HSS.CP.B.8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)×P(B|A) = P(B)×P(A|B), and interpret the answer in terms of the model.
CCSS.Math.Content.HSS.MD.B.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Know

The probability of an event is a number between 0 and 1 that describes the relative frequency of an event in a very large number of repetitions of a chance process.
The probability of an event is denoted as P (event).
The complement of event A, denoted AC, is the event that A does not occur.
The complement rule says P (AC) = 1 – P (A).
The general addition rule says that for any two events A and B, P (A or B) = P (A) + P(B) – P(A and B).
Events A and B are mutually exclusive if they cannot occur together.
Mathematically, events A and B are mutually exclusive if P(A and B) = 0.
If events A and B are mutually exclusive, P(A or B) = P(A) + P(B).
A conditional probability describes the probability that an event occurs, given that another event is known to have occurred.
Conditional probabilities are expressed in the form P(A | B).
P(A | B) is read “The probability that event A occurs, given that event B is known to have occurred.”
Two events are independent if knowing the outcome of one event does not change the probability of the other event.
Mathematically, events A and B are independent if P (A | B) = P (A | BC) = P (A)
A tree diagram organizes conditional probabilities that describe a sequence of events.
The general multiplication rule says that for any two events A and B, P (A and B) = P(A) × P (B | A)
The multiplication rule for independent events says that for two independent events A and B, P (A and B) = P (A) × P (B)
A team’s win probability measures the proportion of games a team would win if they could replay the game many, many times in the same context.
To evaluate strategies in sports, pick the option that results in the greatest win probability.

Understanding/Key Learning

Do

Interpret the probability of an event as a long-run relative frequency.
Use basic probability rules, including the complement rule, to calculate probabilities from a two-way table.
Use the general addition rule to calculate probabilities.
Calculate conditional probabilities from a two-way table.
Justify if two events are independent.
Use a tree diagram to model a chance process involving a sequence of outcomes.
Calculate probabilities using the general multiplication rule.
Calculate conditional probabilities using a tree diagram.
Calculate probabilities using the multiplication rule for independent events.
Calculate a win probability in a particular context.
Use win probability to determine when a strategy will be effective.

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Unit 6: Analyzing Quantitative Variables

Standards

The Common Core Standards for High School Statistics & Probability
CCSS.Math.Content.HSS.ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
CCSS.Math.Content.HSS.ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
CCSS.Math.Content.HSS.ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Know

Use a dotplot, histogram, or boxplot to display the distribution of a quantitative variable.
In a dotplot, we graph each data value as a dot above its location on a number line, stacking the dots if a value occurs more than once.
A histogram displays each interval of values as a bar.
The heights of the bars in a histogram show the frequencies or relative frequencies of values in each interval.
A boxplot is a visual representation of the five-number summary of the distribution of a quantitative variable.
The five-number summary consists of the minimum, Q1, median, Q3, and maximum.
To describe a distribution or compare distributions, address four characteristics: shape, center, variability, and outliers.
There are several common shapes that distributions of quantitative variables can take.
A distribution is symmetric if the left side of the graph is roughly a mirror image of the right side of the graph.
A distribution is skewed to the right if the right side of the graph is more spread out than the left side of the graph.
A distribution is skewed to the left if the left side of the graph is more spread out than the right side of the graph.
A distribution can also be single-peaked, double-peaked, or roughly uniform (no peaks).
Boxplots do not reveal peaks or gaps between values.
Two commonly used measures of center are the mean and the median.
The mean of a data set is the average value.
To compute the mean, add all the values and divide by the number of individuals.
The median of a data set is the middle value when the values are arranged in order from smallest to largest.
If there are two middle values, use the mean of these two values as the median.
When a distribution is roughly symmetric, the mean and median should be approximately the same.
If a distribution is skewed to the right, the mean will typically be greater than the median.
If a distribution is skewed to the left, the mean will typically be less than the median.
Two commonly used measures of variability are the range and the interquartile range.
The range is a single number that measures the distance between the minimum value and the maximum value of a distribution.
Range = max – min.
The interquartile range (IQR) is a single number that measures the range of the middle 50% of a distribution.
IQR = Q3 – Q1.
The first quartile Q1 is the value that separates the smallest 25% of the values from the largest 75%.
The third quartile Q3 is the value that separates the smallest 75% of the values from the largest 25%.
Outliers are unusually large or unusually small values that fall outside the pattern of the rest of the distribution.
The 1.5×IQR rule states that low outliers ≤ Q1 – 1.5×IQR and high outliers ≥ Q3 + 1.5×IQR.
A measure of center or variability is resistant if it isn’t affected very much by skewness or outliers in a distribution.
The median is a resistant measure of center and the IQR is a resistant measure of variability.

Understanding/Key Learning

There are two basic types of variables: categorical and quantitative.
Distributions of quantitative variables are displayed with dotplots, histograms, and boxplots.
Distributions of quantitative variables are summarized with measures of center and measures of variability.
A complete description of the distribution of a quantitative variable includes information about shape, outliers and/or unusual features, center, and variability.

Do

Construct a well-labeled dotplot.
Describe the shape of a distribution of quantitative data.
Construct a well-labeled histogram.
Calculate the mean of a distribution of quantitative data.
Calculate the median of a distribution of quantitative data.
Calculate the range of a distribution of quantitative data.
Calculate the quartiles and the IQR of a distribution of quantitative data.
Describe the effect of outliers and skewness on measures of center and variability.
Identify outliers in a distribution of quantitative data using the 1.5 3 IQR rule.
Construct a boxplot to summarize a distribution of quantitative data.
Compare the shape, center, variability, and outliers for two or more distributions of quantitative data.

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Assessments

Unit 7: Comparing Two Means or Medians

Standards

The Common Core Standards for High School Statistics & Probability

CCSS.Math.Content.HSS.ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
CCSS.Math.Content.HSS.ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
CCSS.Math.Content.HSS.IC.B.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
CCSS.Math.Content.HSS.IC.B.6 Evaluate reports based on data.

Know

For quantitative variables, ability is the mean (or median) value of the variable in a very large number of observations.
When one or both distributions of performances is skewed or has outliers, consider a test for a difference in medians rather than a difference in means.
When testing for a difference in means (or medians), the null hypothesis says that the athlete’s or team’s ability is the same in both contexts.
When testing for a difference in means (or medians), the alternative hypothesis says that the athlete’s or team’s ability is greater in one context than the other.
To simulate a difference in means (or medians), write each performance on a note card, shuffle the cards, place them at random into two piles that match the sizes of the original groups, calculate the difference in means (or difference in medians), and then record this difference on a dotplot.
A p-value is the probability of finding evidence for the alternative hypothesis as strong as or stronger than the observed evidence when the null hypothesis is true.
When the p-value is small (less than 0.05), there is convincing evidence for the alternative hypothesis.
When the p-value is not small (greater than or equal to 0.05), there is not convincing evidence for the alternative hypothesis.

Understanding/Key Learning

Do

Explain the concept of ability when analyzing quantitative data.
State the hypotheses for testing a difference in ability, including the evidence for the alternative hypothesis.
Explain how to simulate the distribution of a difference in means using note cards.
Estimate a p-value using the results of a simulation.
Determine if there is convincing evidence for the alternative hypothesis.
Determine if a cause-and-effect conclusion is appropriate.
Estimate a p-value using technology.
Test a claim about a difference in ability using the State-Simulate-Conclude (SSC) process.
Explain why comparing medians might be more appropriate than comparing means.
Explain how to simulate the distribution of a difference in medians using note cards.

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Unit 8: Exploring Paired Data

Standards

The Common Core Standards for High School Statistics & Probability

CCSS.Math.Content.HSS.ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
CCSS.Math.Content.HSS.ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
CCSS.Math.Content.HSS.IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
CCSS.Math.Content.HSS.IC.A.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

Know

Paired data result from recording two values of the same quantitative variable for each individual or each pair of similar individuals.
Paired data can arise in experiments or observational studies.
The best way to distinguish paired data from unpaired data is to consider how the data were collected.
Data tables showing paired data often have a column that labels the values in each pair.
If two data sets have different sample sizes, the data cannot be paired.
Analyze paired data by finding the difference for each pair, constructing a dotplot of the differences, counting the number of dots on each side of 0, and calculating the mean difference.
When testing hypotheses involving paired data, the null hypothesis is that athletes or teams have the same ability in two different contexts.
Use the mean difference as evidence for the alternative hypothesis.
To simulate the distribution of a mean difference when the null hypothesis is true, use two note cards for each pair of values. For the first pair, write each of the two values on a different note card, shuffle the cards, and assign one to the first treatment/ context and assign the second card to the other treatment/context. Do this for all remaining pairs, then find the simulated difference for each pair and the simulated mean difference using all the pairs. Record the simulated mean difference and repeat many times.
Use the results of the simulation to estimate the p-value by calculating the probability of getting a mean difference as big as or bigger (or as small as or smaller) than the actual mean difference.
If the p-value is small, there is convincing evidence for Ha.
If the p-value is not small, there is not convincing evidence for Ha.
Cause and effect can’t be definitively concluded from an observational study due to the possibility of confounding.
The benefit of using paired data is that we can account for a source of variability in the response variable, making a Type II error less likely to occur than if the data were unpaired.
Other ways to decrease the probability of a Type II error include increasing the sample size, increasing the significance level, and accounting for other sources of variability (e.g., by using paired data).

Understanding/Key Learning

Do

Identify paired data in experiments and observational studies.
Use a graph to analyze the distribution of differences in a paired data set.
Calculate and interpret the mean difference in a paired data set.
Identify if a cause-and-effect conclusion is appropriate.
State hypotheses for testing a difference in ability using paired data, including the evidence for the alternative hypothesis.
Explain how to simulate the distribution of a mean difference using note cards.
Estimate a p-value using the results of a simulation.
Determine if there is convincing evidence for the alternative hypothesis.
Explain the benefit of using paired data.
Estimate a p-value using technology
Test a claim about a difference in ability using the State-Simulate- Conclude (SSC) process.

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Assessments

Unit 9: More Measures of Variability

Standards

The Common Core Standards for High School Statistics & Probability

CCSS.Math.Content.HSS.ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
CCSS.Math.Content.HSS.ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
CCSS.Math.Content.HSS.ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Know

An athlete or team is consistent if their performances do not vary much from their ABILITY.
There are four ways to measure consistency (variability): range, interquartile range (IQR), mean absolute deviation (MAD), and standard deviation (SD).
The mean absolute deviation (MAD) measures the average distance the values in a distribution are from their mean.
The MAD is calculated by finding the absolute value of the deviation from the mean for each data value, adding these absolute deviations, and dividing by the number of data values.
MAD = value - xn
In sports, the standard deviation measures the typical distance between an athlete’s performances and his or her ability.
The standard deviation is calculated by finding the squared deviation from the mean for each data value, adding these squared deviations, and dividing by 1 less than the number of data values.
SD = value - x2n - 1
Both MAD and SD are not resistant to outliers.
Values that are more than 1 MAD from the mean increase the value of the MAD and values that are more than 1 SD from the mean increase the value of the SD.
An athlete’s true standard deviation is an unknown value that describes what the standard deviation would be after an infinite number of performances in a particular context.
An athlete’s observed standard deviation is what we calculate based on a limited number of performances in a particular context.
We use an athlete’s observed standard deviation to estimate his or her true standard deviation.
In a test for the difference of two standard deviations, the null hypothesis is that the true standard deviations are the same in both contexts.
To determine which differences in standard deviations are likely to occur by random chance alone when the null hypothesis is true, conduct a simulation.
In a simulation for a difference in standard deviations, the p-value by calculating the probability of getting a difference in standard deviations as big as or bigger (or as small as or smaller) than the difference in observed standard deviations.
If the p-value is small, there is convincing evidence for Ha
If the p-value is not small, there is not convincing evidence for Ha.

Understanding/Key Learning

Do

Use graphs to compare consistency for two or more distributions of quantitative data.
Calculate and interpret the mean absolute deviation (MAD) for a distribution of quantitative data.
Explain the properties of the MAD, including how it is affected by individual values.
Calculate and interpret the standard deviation (SD) for a distribution of quantitative data.
Explain the properties of the SD, including how it is affected by individual values.
State the hypotheses for testing a difference in standard deviations, including the evidence for the alternative hypothesis.
Explain how to simulate the distribution of a difference in standard deviations using note cards.
Estimate a p-value using the results of a simulation
Determine if there is convincing evidence for the alternative hypothesis.
Estimate a p-value using technology.

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Unit 10: Standardized Scores and Normal Distributions

Standards

The Common Core Standards for High School Statistics & Probability

CCSS.Math.Content.HSS.ID.A.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. 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.

Know

A standardized score, or z-score, measures how many standard deviations a performance is from the mean.
A positive z-score indicates that the performance is above the mean and a negative z-score indicates that the performance is below the mean.
z = performance - mean/standard deviation
Standardized scores are useful for comparing performances that come from distributions
with different amounts of variability, performances that are measured
with different units, or performances that come from different eras.
Standardized scores for several variables can be combined to calculate a combined rating for
athletes.
A normal distribution is a symmetric, single-peaked, and bell-shaped distribution.
We can model a distribution of data that is roughly normal with a normal curve.
The area under a normal curve is equal to 1 and the expected proportion of data values between two boundaries is equal to the area under the normal curve between those two boundaries.
When a distribution is roughly normal, the 68–95–99.7 rule states that about 68% of the values are within 1 standard deviation of the mean, about 95% of the values are within 2 standard deviations of the mean, and about 99.7% of the values are within 3 standard deviations of the mean.
The standard normal table provides the area under a standard normal curve and to the left of a particular z-score.
The standard normal distribution has mean = 0 and standard deviation = 1.
When representing areas under normal curves, draw a normal distribution with the horizontal axis labeled, scale the axis using the mean and standard deviation, mark the boundary value(s) clearly, and shade the area of interest.
When calculating areas under normal curves, standardize each boundary value and use the standard normal table or technology to find the appropriate area.
The pth percentile of a distribution is the boundary value such that p% of the values in the distribution are less than the boundary.
To find an unknown boundary value in a normal distribution, draw a normal distribution with the horizontal axis labeled and scaled using the mean and standard deviation, the area of interest shaded and labeled, and the unknown boundary value clearly marked.

Understanding/Key Learning

Do

Calculate and interpret a z-score.
Compare performances on different scales using z-scores.
Use z-scores to calculate a combined rating from performances on different scales.
Sketch a normal distribution with a given mean and standard deviation.
Use the 68–95–99.7 rule to estimate the percentage of values between two boundaries.
Assess normality with graphs and the 68–95–99.7 rule.
Use the standard normal table or technology to find the percentage of values in a specified interval.
Use the standard normal table or technology to find the boundary value for a given percentile.

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Unit 11: Sampling with Confidence

Standards

Know

The population in a statistical study is the entire group of individuals we want information about. Individuals can be people, animals, or things.
A census collects data from every individual in the population.
A sample is a subset of individuals in the population from which we collect data.
Selecting a random sample from the population allows you to generalize about the population because every member of the population has the same chance to be selected.
To select a random sample, give each individual in the population a distinct numerical label from 1 to N (the population size), use a random number generator to obtain n (the sample size) different integers from 1 to N, and then choose the individuals for the sample that correspond to the randomly selected integers.
The design of a sample survey shows bias if it is very likely to underestimate or very likely to overestimate the parameter you want to know. Bias can arise in different ways.
Convenience sampling selects individuals from the population who are easy to reach.
Voluntary response sampling allows people to choose to be included in the sample by responding to a general invitation.
Undercoverage occurs when some members of the population are less likely to be chosen or cannot be chosen in a sample.
Nonresponse occurs when an individual chosen for the sample can’t be contacted or refuses to participate.
Response bias occurs when there is a systematic pattern of inaccurate answers to a survey question.
Sampling variability refers to the fact that different samples of the same size from the same population produce different estimates.
When estimating ability, sampling variability refers to the fact that an athlete’s performance varies from his or her ability because of random chance.
The standard error of an estimated percentage or estimated mean measures how far the estimate typically varies from the population percentage or population mean by random chance in many samples of the same size from the same population.
The standard error can be estimated using a simulation with technology.
The margin of error of an estimated percentage or estimated mean describes how far, at most, we expect the estimate to vary from the population percentage or population mean.
A confidence interval gives an interval of plausible values for a parameter, such as a population percentage or a population mean.
The general form of a confidence interval is: single-value estimate ± margin of error
To calculate the margin of error for a 95% confidence interval, multiply the standard error by 2.
The formula for a 95% confidence interval is: single-value estimate ± 2 (standard error).
In general, when the sample size increases, the margin of error decreases.
Increasing the sample size typically takes more time and money.
The margin of error accounts only for sampling variability.
The margin of error does not account for any bias in the data collection process.
If there is bias in the data collection process, you should not be 95% confident that a “95% confidence” interval includes the true value.
If we assume that a set of performances is a random sample from the population of all possible performances an athlete could have had, we can use a confidence interval to estimate the ability of an athlete.

Understanding/Key Learning

Do

Identify the population and sample in a sample survey.
Describe how convenience sampling and voluntary response sampling can lead to bias.
Explain how random sampling can help to avoid bias.
Describe how to obtain a random sample using note cards or technology.
Explain how other aspects of a sample survey can lead to bias.
Explain the concept of sampling variability and the difference between performance and ability.
Use simulation to approximate the standard error of a sample percentage and interpret the standard error.
Calculate and interpret a 95% confidence interval for a percentage
Use a confidence interval to assess a claim.
Describe the effect of changing the sample size on the margin of error
Use simulation to approximate the standard error of a sample mean and interpret the standard error.
Calculate and interpret a 95% confidence interval for a mean.