Inference for Categorical Data: Proportions · ⁨الاستدلال للبيانات التصنيفية: النسب⁩

Enduring Understanding (VAR-1): Given that variation may be random or not, conclusions are uncertain.

Learning Objective VAR-1.H: Identify questions suggested by variation in the shapes of distributions of samples taken from the same population. [Skill 1.A]

• VAR-1.H.1 Variation in shapes of data distributions may be random or not.

Enduring Understanding (UNC-4): An interval of values should be used to estimate parameters, in order to account for uncertainty.

Learning Objective UNC-4.A: Identify an appropriate confidence interval procedure for a population proportion. [Skill 1.D]

• UNC-4.A.1 The appropriate confidence interval procedure for a one-sample proportion for one categorical variable is a one sample $z$-interval for a proportion.

Learning Objective UNC-4.B: Verify the conditions for calculating confidence intervals for a population proportion. [Skill 4.C]

• UNC-4.B.1 In order to make assumptions necessary for inference on population proportions, means, and slopes, we must check for independence in data collection methods and for selection of the appropriate sampling distribution.
• UNC-4.B.2 In order to calculate a confidence interval to estimate a population proportion, $p$, we must check for independence and that the sampling distribution is approximately normal.
  • a. To check for independence:
    • i. Data should be collected using a random sample or a randomized experiment.
    • ii. When sampling without replacement, check that $n \leq 10\%N$, where $N$ is the size of the population.
  • b. To check that the sampling distribution of $\hat{p}$ is approximately normal (shape):
    • i. For categorical variables, check that both the number of successes, $n\hat{p}$, and the number of failures, $n(1-\hat{p})$ are at least 10 so that the sample size is large enough to support an assumption of normality.

Learning Objective UNC-4.C: Determine the margin of error for a given sample size and an estimate for the sample size that will result in a given margin of error for a population proportion. [Skill 3.D]

• UNC-4.C.1 Based on sample data, the standard error of a statistic is an estimate for the standard deviation for the statistic. The standard error of $\hat{p}$ is $SE_{\hat{p}} = \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$.
• UNC-4.C.2 A margin of error gives how much a value of a sample statistic is likely to vary from the value of the corresponding population parameter.
• UNC-4.C.3 For categorical variables, the margin of error is the critical value ($z^*$) times the standard error (SE) of the relevant statistic, which equals $z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$ for a one sample proportion.
• UNC-4.C.4 The formula for margin of error can be rearranged to solve for $n$, the minimum sample size needed to achieve a given margin of error. For this purpose, use a guess for $\hat{p}$ or use $\hat{p} = 0.5$ in order to find an upper bound for the sample size that will result in a given margin of error.

Learning Objective UNC-4.D: Calculate an appropriate confidence interval for a population proportion. [Skill 3.D]

• UNC-4.D.1 In general, an interval estimate can be constructed as point estimate ± (margin of error). For a one-sample proportion, the interval estimate is $\hat{p} \pm z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$.
  • Clarifying statement: Formulas for interval estimates do not appear explicitly on the AP Statistics Formula Sheet provided with the AP Statistics Exam. However, these formulas do not need to be memorized, as they can be constructed based on the general test statistic formula and the relevant standard error formulas that are provided on the formula sheet.
• UNC-4.D.2 Critical values represent the boundaries encompassing the middle C% of the standard normal distribution, where C% is an approximate confidence level for a proportion.

Learning Objective UNC-4.E: Calculate an interval estimate based on a confidence interval for a population proportion. [Skill 3.D]

• UNC-4.E.1 Confidence intervals for population proportions can be used to calculate interval estimates with specified units.

Enduring Understanding (UNC-4): An interval of values should be used to estimate parameters, in order to account for uncertainty.

Learning Objective UNC-4.F: Interpret a confidence interval for a population proportion. [Skill 4.B]

• UNC-4.F.1 A confidence interval for a population proportion either contains the population proportion or it does not, because each interval is based on random sample data, which varies from sample to sample.
• UNC-4.F.2 We are C% confident that the confidence interval for a population proportion captures the population proportion.
• UNC-4.F.3 In repeated random sampling with the same sample size, approximately C% of confidence intervals created will capture the population proportion.
• UNC-4.F.4 Interpreting a confidence interval for a one-sample proportion should include a reference to the sample taken and details about the population it represents.
  • Illustrative examples for UNC-4.F.4: For interpreting a 99% confidence interval of (0.268, 0.292), based on the proportion of a nationally representative sample of twelfth-grade students who answered a particular multiple choice question correctly: "We are 99 percent confident that the interval from 0.268 to 0.292 contains the population proportion of all United States twelfth-grade students who would answer this question correctly" (2011 FRQ 6(a)).

Learning Objective UNC-4.G: Justify a claim based on a confidence interval for a population proportion. [Skill 4.D]

• UNC-4.G.1 A confidence interval for a population proportion provides an interval of values that may provide sufficient evidence to support a particular claim in context.

Learning Objective UNC-4.H: Identify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population proportion. [Skill 4.A]

• UNC-4.H.1 When all other things remain the same, the width of the confidence interval for a population proportion tends to decrease as the sample size increases. For a population proportion, the width of the interval is proportional to $\dfrac{1}{\sqrt{n}}$.
• UNC-4.H.2 For a given sample, the width of the confidence interval for a population proportion increases as the confidence level increases.
• UNC-4.H.3 The width of a confidence interval for a population proportion is exactly twice the margin of error.

Enduring Understanding (VAR-6): The normal distribution may be used to model variation.

Learning Objective VAR-6.D: Identify the null and alternative hypotheses for a population proportion. [Skill 1.F]

• VAR-6.D.1 The null hypothesis is the situation that is assumed to be correct unless evidence suggests otherwise, and the alternative hypothesis is the situation for which evidence is being collected.
• VAR-6.D.2 For hypotheses about parameters, the null hypothesis contains an equality reference (=, ≥, or ≤), while the alternative hypothesis contains a strict inequality (<, >, or ≠). The type of inequality in the alternative hypothesis is based on the question of interest. Alternative hypotheses with < or > are called one-sided, and alternative hypotheses with ≠ are called two-sided. Although the null hypothesis for a one-sided test may include an inequality symbol, it is still tested at the boundary of equality.
• VAR-6.D.3 The null hypothesis for a population proportion is: $H_0 : p = p_0$, where $p_0$ is the null hypothesized value for the population proportion.
• VAR-6.D.4 A one-sided alternative hypothesis for a proportion is either $H_a : p < p_0$ or $H_a : p > p_0$. A two-sided alternate hypothesis is $H_a : p_1 \neq p_2$.
• VAR-6.D.5 For a one-sample $z$-test for a population proportion, the null hypothesis specifies a value for the population proportion, usually one indicating no difference or effect.

Learning Objective VAR-6.E: Identify an appropriate testing method for a population proportion. [Skill 1.E]

• VAR-6.E.1 For a single categorical variable, the appropriate testing method for a population proportion is a one-sample $z$-test for a population proportion.

Learning Objective VAR-6.F: Verify the conditions for making statistical inferences when testing a population proportion. [Skill 4.C]

• VAR-6.F.1 In order to make statistical inferences when testing a population proportion, we must check for independence and that the sampling distribution is approximately normal:
  • a. To check for independence:
    • i. Data should be collected using a random sample or a randomized experiment.
    • ii. When sampling without replacement, check that $n \leq 10\%N$.
  • b. To check that the sampling distribution of $\hat{p}$ is approximately normal (shape):
    • i. Assuming that $H_0$ is true $(p = p_0)$, verify that both the number of successes, $np_0$, and the number of failures, $n(1-p_0)$ are at least 10 so that that the sample size is large enough to support an assumption of normality.

Enduring Understanding (VAR-6): The normal distribution may be used to model variation.

Learning Objective VAR-6.G: Calculate an appropriate test statistic and $p$-value for a population proportion. [Skill 3.E]

• VAR-6.G.1 The distribution of the test statistic assuming the null hypothesis is true (null distribution) can be either a randomization distribution or when a probability model is assumed to be true, a theoretical distribution ($z$).
• VAR-6.G.2 When using a $z$-test, the standardized test statistic can be written: $\text{test statistic} = \dfrac{\text{sample statistic} - \text{null value of the parameter}}{\text{standard deviation of the statistic}}$. This is called a $z$-statistic for proportions.
• VAR-6.G.3 The test statistic for a population proportion is: $z = \dfrac{\hat{p} - p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}$.
  • Clarifying statement: The formulas for test statistics do not appear explicitly on the AP Statistics Formula Sheet provided with the AP Statistics Exam. However, these formulas do not need to be memorized, as they can be constructed based on the general test statistic formula and the relevant standard error formulas that are provided on the formula sheet.
• VAR-6.G.4 A $p$-value is the probability of obtaining a test statistic as extreme or more extreme than the observed test statistic when the null hypothesis and probability model are assumed to be true. The significance level may be given or determined by the researcher.

Enduring Understanding (DAT-3): Significance testing allows us to make decisions about hypotheses within a particular context.

Learning Objective DAT-3.A: Interpret the $p$-value of a significance test for a population proportion. [Skill 4.B]

• DAT-3.A.1 The $p$-value is the proportion of values for the null distribution that are as extreme or more extreme than the observed value of the test statistic. This is:
  • a. The proportion at or above the observed value of the test statistic, if the alternative is >.
  • b. The proportion at or below the observed value of the test statistic, if the alternative is <.
  • c. The proportion less than or equal to the negative of the absolute value of the test statistic plus the proportion greater than or equal to the absolute value of the test statistic, if the alternative is ≠.
• DAT-3.A.2 An interpretation of the $p$-value of a significance test for a one-sample proportion should recognize that the $p$-value is computed by assuming that the probability model and null hypothesis are true, i.e., by assuming that the true population proportion is equal to the particular value stated in the null hypothesis.

Enduring Understanding (DAT-3): Significance testing allows us to make decisions about hypotheses within a particular context.

Learning Objective DAT-3.B: Justify a claim about the population based on the results of a significance test for a population proportion. [Skill 4.E]

• DAT-3.B.1 The significance level, $\alpha$, is the predetermined probability of rejecting the null hypothesis given that it is true.
• DAT-3.B.2 A formal decision explicitly compares the $p$-value to the significance level, $\alpha$. If the $p$-value $\leq \alpha$, reject the null hypothesis. If the $p$-value $> \alpha$, fail to reject the null hypothesis.
• DAT-3.B.3 Rejecting the null hypothesis means there is sufficient statistical evidence to support the alternative hypothesis. Failing to reject the null means there is insufficient statistical evidence to support the alternative hypothesis.
• DAT-3.B.4 The conclusion about the alternative hypothesis must be stated in context.
• DAT-3.B.5 A significance test can lead to rejecting or not rejecting the null hypothesis, but can never lead to concluding or proving that the null hypothesis is true. Lack of statistical evidence for the alternative hypothesis is not the same as evidence for the null hypothesis.
• DAT-3.B.6 Small $p$-values indicate that the observed value of the test statistic would be unusual if the null hypothesis and probability model were true, and so provide evidence for the alternative. The lower the $p$-value, the more convincing the statistical evidence for the alternative hypothesis.
• DAT-3.B.7 $p$-values that are not small indicate that the observed value of the test statistic would not be unusual if the null hypothesis and probability model were true, so do not provide convincing statistical evidence for the alternative hypothesis nor do they provide evidence that the null hypothesis is true.
• DAT-3.B.8 A formal decision explicitly compares the $p$-value to the significance $\alpha$. If the $p$-value $\leq \alpha$, then reject the null hypothesis, $H_0 : p = p_0$. If the $p$-value $> \alpha$, then fail to reject the null hypothesis.
• DAT-3.B.9 The results of a significance test for a population proportion can serve as the statistical reasoning to support the answer to a research question about the population that was sampled.

Enduring Understanding (UNC-5): Probabilities of Type I and Type II errors influence inference.

Learning Objective UNC-5.A: Identify Type I and Type II errors. [Skill 1.B]

• UNC-5.A.1 A Type I error occurs when the null hypothesis is true and is rejected (false positive).
• UNC-5.A.2 A Type II error occurs when the null hypothesis is false and is not rejected (false negative).
  • Table of Errors: With Actual Population Value across the top ($H_0$ true; $H_a$ true) and Decision down the side (Reject $H_0$; Fail to Reject $H_0$): Reject $H_0$ when $H_0$ true = Type I Error; Reject $H_0$ when $H_a$ true = Correct Decision; Fail to Reject $H_0$ when $H_0$ true = Correct Decision; Fail to Reject $H_0$ when $H_a$ true = Type II Error.

Learning Objective UNC-5.B: Calculate the probability of a Type I and Type II errors. [Skill 3.A]

• UNC-5.B.1 The significance level, $\alpha$, is the probability of making a Type I error, if the null hypothesis is true.
• UNC-5.B.2 The power of a test is the probability that a test will correctly reject a false null hypothesis.
• UNC-5.B.3 The probability of making a Type II error $= 1 - power$.

Learning Objective UNC-5.C: Identify factors that affect the probability of errors in significance testing. [Skill 4.A]

• UNC-5.C.1 The probability of a Type II error decreases when any of the following occurs, provided the others do not change:
  • i. Sample size(s) increases.
  • ii. Significance level ($\alpha$) of a test increases.
  • iii. Standard error decreases.
  • iv. True parameter value is farther from the null.

Learning Objective UNC-5.D: Interpret Type I and Type II errors. [Skill 4.B]

• UNC-5.D.1 Whether a Type I or a Type II error is more consequential depends upon the situation.
• UNC-5.D.2 Since the significance level, $\alpha$, is the probability of a Type I error, the consequences of a Type I error influence decisions about a significance level.

Enduring Understanding (UNC-4): An interval of values should be used to estimate parameters, in order to account for uncertainty.

Learning Objective UNC-4.I: Identify an appropriate confidence interval procedure for a comparison of population proportions. [Skill 1.D]

• UNC-4.I.1 The appropriate confidence interval procedure for a two-sample comparison of proportions for one categorical variable is a two-sample $z$-interval for a difference between population proportions.

Learning Objective UNC-4.J: Verify the conditions for calculating confidence intervals for a difference between population proportions. [Skill 4.C]

• UNC-4.J.1 In order to calculate confidence intervals to estimate a difference between proportions, we must check for independence and that the sampling distribution is approximately normal:
  • a. To check for independence:
    • i. Data should be collected using two independent, random samples or a randomized experiment.
    • ii. When sampling without replacement, check that $n_1 \leq 10\%N_1$ and $n_2 \leq 10\%N_2$.
  • b. To check that sampling distribution of $\hat{p}_1 - \hat{p}_2$ is approximately normal (shape).
    • i. For categorical variables, check that $n_1\hat{p}_1$, $n_1(1-\hat{p}_1)$, $n_2\hat{p}_2$, and $n_2\left(1-\hat{p}_2\right)$ are all greater than or equal to some predetermined value, typically either 5 or 10.

Learning Objective UNC-4.K: Calculate an appropriate confidence interval for a comparison of population proportions. [Skill 3.D]

• UNC-4.K.1 For a comparison of proportions, the interval estimate is $(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$.
  • Clarifying statement: Formulas for interval estimates do not appear explicitly on the AP Statistics Formula Sheet provided with the AP Statistics Exam. However, these formulas do not need to be memorized, as they can be constructed based on the general test statistic formula and the relevant standard error formulas that are provided on the formula sheet.

Learning Objective UNC-4.L: Calculate an interval estimate based on a confidence interval for a difference of proportions. [Skill 3.D]

• UNC-4.L.1 Confidence intervals for a difference in proportions can be used to calculate interval estimates with specified units.

Enduring Understanding (UNC-4): An interval of values should be used to estimate parameters, in order to account for uncertainty.

Learning Objective UNC-4.M: Interpret a confidence interval for a difference of proportions. [Skill 4.B]

• UNC-4.M.1 In repeated random sampling with the same sample size, approximately C% of confidence intervals created will capture the difference in population proportions.
• UNC-4.M.2 Interpreting a confidence interval for difference between population proportions should include a reference to the sample taken and details about the population it represents.

Learning Objective UNC-4.N: Justify a claim based on a confidence interval for a difference of proportions. [Skill 4.D]

• UNC-4.N.1 A confidence interval for difference in population proportions provides an interval of values that may provide sufficient evidence to support a particular claim in context.

Enduring Understanding (VAR-6): The normal distribution may be used to model variation.

Learning Objective VAR-6.H: Identify the null and alternative hypotheses for a difference of two population proportions. [Skill 1.F]

• VAR-6.H.1 For a two-sample test for a difference of two proportions, the null hypothesis specifies a value of $0$ for the difference in population proportions, indicating no difference or effect.
• VAR-6.H.2 The null hypothesis for a difference in proportions is: $H_0 : p_1 = p_2$, or $H_0 : p_1 - p_2 = 0$.
• VAR-6.H.3 A one-sided alternative hypothesis for a difference in proportions is $H_a : p_1 < p_2$, or, $H_a : p_1 > p_2$. A two-sided alternative hypothesis for a difference of proportions is $H_a : p_1 \neq p_2$.

Learning Objective VAR-6.I: Identify an appropriate testing method for the difference of two population proportions. [Skill 1.E]

• VAR-6.I.1 For a single categorical variable, the appropriate testing method for the difference of two population proportions is a two-sample $z$-test for a difference between two population proportions.

Learning Objective VAR-6.J: Verify the conditions for making statistical inferences when testing a difference of two population proportions. [Skill 4.C]

• VAR-6.J.1 In order to make statistical inferences when testing a difference between population proportions, we must check for independence and that the sampling distribution is approximately normal:
  • a. To check for independence:
    • i. Data should be collected using two independent, random samples or a randomized experiment.
    • ii. When sampling without replacement, check that $n_1 \leq 10\%N_1$ and $n_2 \leq 10\%N_2$.
  • b. To check that the sampling distribution of $\hat{p}_1 - \hat{p}_2$ is approximately normal (shape):
    • i. For the combined sample, define the combined (or pooled) proportion, $\hat{p}_c = \dfrac{n_1\hat{p}_1 + n_2\hat{p}_2}{n_1 + n_2}$. Assuming that $H_0$ is true $(p_1 - p_2 = 0$ or $p_1 = p_2)$, check that $n_1\hat{p}_c$, $n_1\left(1-\hat{p}_c\right)$, $n_2\hat{p}_c$, and $n_2\left(1-\hat{p}_c\right)$ are all greater than or equal to some predetermined value, typically either 5 or 10.

Enduring Understanding (VAR-6): The normal distribution may be used to model variation.

Learning Objective VAR-6.K: Calculate an appropriate test statistic for the difference of two population proportions. [Skill 3.E]

• VAR-6.K.1 The test statistic for a difference in proportions is: $z = \dfrac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}_c(1-\hat{p}_c)}\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$, where $\hat{p}_c = \dfrac{n_1\hat{p}_1 + n_2\hat{p}_2}{n_1 + n_2}$.
  • Clarifying statement: The formulas for test statistics do not appear explicitly on the AP Statistics Formula Sheet provided with the AP Statistics Exam. However, these formulas do not need to be memorized, as they can be constructed based on the general test statistic formula and the standard error formulas for each of the relevant test statistics that are provided on the formula sheet.

Enduring Understanding (DAT-3): Significance testing allows us to make decisions about hypotheses within a particular context.

Learning Objective DAT-3.C: Interpret the $p$-value of a significance test for a difference of population proportions. [Skill 4.B]

• DAT-3.C.1 An interpretation of the $p$-value of a significance test for a difference of two population proportions should recognize that the $p$-value is computed by assuming that the null hypothesis is true, i.e., by assuming that the true population proportions are equal to each other.

Learning Objective DAT-3.D: Justify a claim about the population based on the results of a significance test for a difference of population proportions. [Skill 4.E]

• DAT-3.D.1 A formal decision explicitly compares the $p$-value to the significance $\alpha$. If the $p\text{-value} \leq \alpha$, then reject the null hypothesis, $H_0 : p_1 = p_2$, or $H_0 : p_1 - p_2 = 0$. If the $p$-value $> \alpha$, then fail to reject the null hypothesis.
• DAT-3.D.2 The results of a significance test for a difference of two population proportions can serve as the statistical reasoning to support the answer to a research question about the two populations that were sampled.