Worked solution P23-Easy-1. Use one-variable description. The data structure and parameter match this procedure directly. The mathematical anchor is $$\overline{x}=\sum x_i/n.$$ Symbols must be defined from the stated variables and population. Interpretation: A complete response should describe shape, center, variability, and unusual features in context. Validity: Before calculation, check shape and unusual values before choosing resistant or nonresistant summaries. Error to reject: Do not substitute binomial model merely because both methods contain familiar statistical notation; its data structure is a fixed number of independent binary trials.

Worked solution P23-Easy-2. Use two-way table analysis. The data structure and parameter match this procedure directly. The mathematical anchor is $$P(A\mid B)=P(A\cap B)/P(B).$$ Symbols must be defined from the stated variables and population. Interpretation: A complete response should compare conditional distributions without claiming causation. Validity: Before calculation, use the conditioning group's total as the denominator. Error to reject: Do not substitute sampling distribution for a mean merely because both methods contain familiar statistical notation; its data structure is repeated random samples of size n.

Worked solution P23-Easy-3. Use normal probability. The data structure and parameter match this procedure directly. The mathematical anchor is $$z=(x-\mu )/\sigma .$$ Symbols must be defined from the stated variables and population. Interpretation: A complete response should translate standardized area back to the original context. Validity: Before calculation, justify the normal model before using normal areas. Error to reject: Do not substitute one-proportion z interval merely because both methods contain familiar statistical notation; its data structure is one random sample with a binary response.

Worked solution P23-Easy-4. Use binomial model. The data structure and parameter match this procedure directly. The mathematical anchor is $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}.$$ Symbols must be defined from the stated variables and population. Interpretation: A complete response should interpret probability over repetitions of the full n-trial process. Validity: Before calculation, verify BINS, including constant p. Error to reject: Do not substitute two-proportion z test merely because both methods contain familiar statistical notation; its data structure is two independent groups with binary responses.

Worked solution P23-Easy-5. Use sampling distribution for a mean. The data structure and parameter match this procedure directly. The mathematical anchor is $$S{E}_{\overline{X}}=\sigma /\sqrt{n}.$$ Symbols must be defined from the stated variables and population. Interpretation: A complete response should distinguish variation of sample means from variation of individuals. Validity: Before calculation, check independence and population shape or CLT support. Error to reject: Do not substitute one-sample t interval merely because both methods contain familiar statistical notation; its data structure is one quantitative sample with unknown sigma.

Worked solution P23-Easy-6. Use one-proportion z interval. The data structure and parameter match this procedure directly. The mathematical anchor is $$\widehat{p}\pm z^{*}\sqrt{\widehat{p}(1-\widehat{p})/n}.$$ Symbols must be defined from the stated variables and population. Interpretation: A complete response should state confidence as a property of the interval method. Validity: Before calculation, check randomization, independence, and large observed counts. Error to reject: Do not substitute paired t test merely because both methods contain familiar statistical notation; its data structure is matched or before-after quantitative measurements.

Worked solution P23-Easy-7. Use two-proportion z test. The data structure and parameter match this procedure directly. The mathematical anchor is $$z=({\widehat{p}}_1-{\widehat{p}}_2)/S{E}_0.$$ Symbols must be defined from the stated variables and population. Interpretation: A complete response should conclude about evidence for a population difference. Validity: Before calculation, pool only under the equal-proportions null. Error to reject: Do not substitute chi-square independence test merely because both methods contain familiar statistical notation; its data structure is one population classified by two categorical variables.

Worked solution P23-Easy-8. Use one-sample t interval. The data structure and parameter match this procedure directly. The mathematical anchor is $$\overline{x}\pm t^{*}s/\sqrt{n}.$$ Symbols must be defined from the stated variables and population. Interpretation: A complete response should report the interval in measurement units. Validity: Before calculation, check randomization, independence, and robustness to shape. Error to reject: Do not substitute least-squares regression merely because both methods contain familiar statistical notation; its data structure is two quantitative variables.

Worked solution P23-Easy-9. Use paired t test. The data structure and parameter match this procedure directly. The mathematical anchor is $$t=\overline{d}/(s_d/\sqrt{n}).$$ Symbols must be defined from the stated variables and population. Interpretation: A complete response should conclude about the mean within-pair change. Validity: Before calculation, analyze independent pair differences. Error to reject: Do not substitute randomized experiment merely because both methods contain familiar statistical notation; its data structure is experimental units assigned to treatments.

Worked solution P23-Easy-10. Use chi-square independence test. The data structure and parameter match this procedure directly. The mathematical anchor is $${\chi }^2=\sum (O-E{)}^2/E.$$ Symbols must be defined from the stated variables and population. Interpretation: A complete response should conclude about association rather than causation. Validity: Before calculation, check randomization, independent observations, and expected counts. Error to reject: Do not substitute one-variable description merely because both methods contain familiar statistical notation; its data structure is one quantitative variable.

Worked solution P23-Easy-11. Use least-squares regression. The data structure and parameter match this procedure directly. The mathematical anchor is $$\widehat{y}=a+bx.$$ Symbols must be defined from the stated variables and population. Interpretation: A complete response should interpret slope and predictions within the observed x range. Validity: Before calculation, inspect form, residual pattern, and influential points. Error to reject: Do not substitute two-way table analysis merely because both methods contain familiar statistical notation; its data structure is two categorical variables.

Worked solution P23-Easy-12. Use randomized experiment. The data structure and parameter match this procedure directly. The mathematical anchor is $$\text{effect estimate}={\overline{x}}_T-{\overline{x}}_C.$$ Symbols must be defined from the stated variables and population. Interpretation: A complete response should support causation while limiting population generalization to the sampling design. Validity: Before calculation, use random assignment, comparison, control, and replication. Error to reject: Do not substitute normal probability merely because both methods contain familiar statistical notation; its data structure is one quantitative variable modeled as normal.

Worked solution P23-Easy-13. Use one-variable description. The data structure and parameter match this procedure directly. The mathematical anchor is $$\overline{x}=\sum x_i/n.$$ Symbols must be defined from the stated variables and population. Interpretation: A complete response should describe shape, center, variability, and unusual features in context. Validity: Before calculation, check shape and unusual values before choosing resistant or nonresistant summaries. Error to reject: Do not substitute binomial model merely because both methods contain familiar statistical notation; its data structure is a fixed number of independent binary trials.

Worked solution P23-Easy-14. Use two-way table analysis. The data structure and parameter match this procedure directly. The mathematical anchor is $$P(A\mid B)=P(A\cap B)/P(B).$$ Symbols must be defined from the stated variables and population. Interpretation: A complete response should compare conditional distributions without claiming causation. Validity: Before calculation, use the conditioning group's total as the denominator. Error to reject: Do not substitute sampling distribution for a mean merely because both methods contain familiar statistical notation; its data structure is repeated random samples of size n.

Worked solution P23-Easy-15. Use normal probability. The data structure and parameter match this procedure directly. The mathematical anchor is $$z=(x-\mu )/\sigma .$$ Symbols must be defined from the stated variables and population. Interpretation: A complete response should translate standardized area back to the original context. Validity: Before calculation, justify the normal model before using normal areas. Error to reject: Do not substitute one-proportion z interval merely because both methods contain familiar statistical notation; its data structure is one random sample with a binary response.

Worked solution P23-Tough-1. Choose one-variable description for one quantitative variable; choose binomial model for a fixed number of independent binary trials. The deciding feature is the design and variable structure, not which calculator menu looks familiar. For the selected method, use $$\overline{x}=\sum x_i/n.$$ The neighboring method instead centers on $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}.$$ Interpretation: For one-variable description, describe shape, center, variability, and unusual features in context; for the neighboring method, interpret probability over repetitions of the full n-trial process. Validity: Selected-method check: check shape and unusual values before choosing resistant or nonresistant summaries. Neighboring-method check: verify BINS, including constant p. Error to reject: A method can produce a number even when its parameter and data structure are wrong, so calculator output is not validation.

Worked solution P23-Tough-2. Choose two-way table analysis for two categorical variables; choose sampling distribution for a mean for repeated random samples of size n. The deciding feature is the design and variable structure, not which calculator menu looks familiar. For the selected method, use $$P(A\mid B)=P(A\cap B)/P(B).$$ The neighboring method instead centers on $$S{E}_{\overline{X}}=\sigma /\sqrt{n}.$$ Interpretation: For two-way table analysis, compare conditional distributions without claiming causation; for the neighboring method, distinguish variation of sample means from variation of individuals. Validity: Selected-method check: use the conditioning group's total as the denominator. Neighboring-method check: check independence and population shape or CLT support. Error to reject: A method can produce a number even when its parameter and data structure are wrong, so calculator output is not validation.

Worked solution P23-Tough-3. Choose normal probability for one quantitative variable modeled as normal; choose one-proportion z interval for one random sample with a binary response. The deciding feature is the design and variable structure, not which calculator menu looks familiar. For the selected method, use $$z=(x-\mu )/\sigma .$$ The neighboring method instead centers on $$\widehat{p}\pm z^{*}\sqrt{\widehat{p}(1-\widehat{p})/n}.$$ Interpretation: For normal probability, translate standardized area back to the original context; for the neighboring method, state confidence as a property of the interval method. Validity: Selected-method check: justify the normal model before using normal areas. Neighboring-method check: check randomization, independence, and large observed counts. Error to reject: A method can produce a number even when its parameter and data structure are wrong, so calculator output is not validation.

Worked solution P23-Tough-4. Choose binomial model for a fixed number of independent binary trials; choose two-proportion z test for two independent groups with binary responses. The deciding feature is the design and variable structure, not which calculator menu looks familiar. For the selected method, use $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}.$$ The neighboring method instead centers on $$z=({\widehat{p}}_1-{\widehat{p}}_2)/S{E}_0.$$ Interpretation: For binomial model, interpret probability over repetitions of the full n-trial process; for the neighboring method, conclude about evidence for a population difference. Validity: Selected-method check: verify BINS, including constant p. Neighboring-method check: pool only under the equal-proportions null. Error to reject: A method can produce a number even when its parameter and data structure are wrong, so calculator output is not validation.

Worked solution P23-Tough-5. Choose sampling distribution for a mean for repeated random samples of size n; choose one-sample t interval for one quantitative sample with unknown sigma. The deciding feature is the design and variable structure, not which calculator menu looks familiar. For the selected method, use $$S{E}_{\overline{X}}=\sigma /\sqrt{n}.$$ The neighboring method instead centers on $$\overline{x}\pm t^{*}s/\sqrt{n}.$$ Interpretation: For sampling distribution for a mean, distinguish variation of sample means from variation of individuals; for the neighboring method, report the interval in measurement units. Validity: Selected-method check: check independence and population shape or CLT support. Neighboring-method check: check randomization, independence, and robustness to shape. Error to reject: A method can produce a number even when its parameter and data structure are wrong, so calculator output is not validation.

Worked solution P23-Tough-6. Choose one-proportion z interval for one random sample with a binary response; choose paired t test for matched or before-after quantitative measurements. The deciding feature is the design and variable structure, not which calculator menu looks familiar. For the selected method, use $$\widehat{p}\pm z^{*}\sqrt{\widehat{p}(1-\widehat{p})/n}.$$ The neighboring method instead centers on $$t=\overline{d}/(s_d/\sqrt{n}).$$ Interpretation: For one-proportion z interval, state confidence as a property of the interval method; for the neighboring method, conclude about the mean within-pair change. Validity: Selected-method check: check randomization, independence, and large observed counts. Neighboring-method check: analyze independent pair differences. Error to reject: A method can produce a number even when its parameter and data structure are wrong, so calculator output is not validation.

Worked solution P23-Tough-7. Choose two-proportion z test for two independent groups with binary responses; choose chi-square independence test for one population classified by two categorical variables. The deciding feature is the design and variable structure, not which calculator menu looks familiar. For the selected method, use $$z=({\widehat{p}}_1-{\widehat{p}}_2)/S{E}_0.$$ The neighboring method instead centers on $${\chi }^2=\sum (O-E{)}^2/E.$$ Interpretation: For two-proportion z test, conclude about evidence for a population difference; for the neighboring method, conclude about association rather than causation. Validity: Selected-method check: pool only under the equal-proportions null. Neighboring-method check: check randomization, independent observations, and expected counts. Error to reject: A method can produce a number even when its parameter and data structure are wrong, so calculator output is not validation.

Worked solution P23-Tough-8. Choose one-sample t interval for one quantitative sample with unknown sigma; choose least-squares regression for two quantitative variables. The deciding feature is the design and variable structure, not which calculator menu looks familiar. For the selected method, use $$\overline{x}\pm t^{*}s/\sqrt{n}.$$ The neighboring method instead centers on $$\widehat{y}=a+bx.$$ Interpretation: For one-sample t interval, report the interval in measurement units; for the neighboring method, interpret slope and predictions within the observed x range. Validity: Selected-method check: check randomization, independence, and robustness to shape. Neighboring-method check: inspect form, residual pattern, and influential points. Error to reject: A method can produce a number even when its parameter and data structure are wrong, so calculator output is not validation.

Worked solution P23-Tough-9. Choose paired t test for matched or before-after quantitative measurements; choose randomized experiment for experimental units assigned to treatments. The deciding feature is the design and variable structure, not which calculator menu looks familiar. For the selected method, use $$t=\overline{d}/(s_d/\sqrt{n}).$$ The neighboring method instead centers on $$\text{effect estimate}={\overline{x}}_T-{\overline{x}}_C.$$ Interpretation: For paired t test, conclude about the mean within-pair change; for the neighboring method, support causation while limiting population generalization to the sampling design. Validity: Selected-method check: analyze independent pair differences. Neighboring-method check: use random assignment, comparison, control, and replication. Error to reject: A method can produce a number even when its parameter and data structure are wrong, so calculator output is not validation.

Worked solution P23-Tough-10. Choose chi-square independence test for one population classified by two categorical variables; choose one-variable description for one quantitative variable. The deciding feature is the design and variable structure, not which calculator menu looks familiar. For the selected method, use $${\chi }^2=\sum (O-E{)}^2/E.$$ The neighboring method instead centers on $$\overline{x}=\sum x_i/n.$$ Interpretation: For chi-square independence test, conclude about association rather than causation; for the neighboring method, describe shape, center, variability, and unusual features in context. Validity: Selected-method check: check randomization, independent observations, and expected counts. Neighboring-method check: check shape and unusual values before choosing resistant or nonresistant summaries. Error to reject: A method can produce a number even when its parameter and data structure are wrong, so calculator output is not validation.

Worked solution P23-Tough-11. Choose least-squares regression for two quantitative variables; choose two-way table analysis for two categorical variables. The deciding feature is the design and variable structure, not which calculator menu looks familiar. For the selected method, use $$\widehat{y}=a+bx.$$ The neighboring method instead centers on $$P(A\mid B)=P(A\cap B)/P(B).$$ Interpretation: For least-squares regression, interpret slope and predictions within the observed x range; for the neighboring method, compare conditional distributions without claiming causation. Validity: Selected-method check: inspect form, residual pattern, and influential points. Neighboring-method check: use the conditioning group's total as the denominator. Error to reject: A method can produce a number even when its parameter and data structure are wrong, so calculator output is not validation.

Worked solution P23-Tough-12. Choose randomized experiment for experimental units assigned to treatments; choose normal probability for one quantitative variable modeled as normal. The deciding feature is the design and variable structure, not which calculator menu looks familiar. For the selected method, use $$\text{effect estimate}={\overline{x}}_T-{\overline{x}}_C.$$ The neighboring method instead centers on $$z=(x-\mu )/\sigma .$$ Interpretation: For randomized experiment, support causation while limiting population generalization to the sampling design; for the neighboring method, translate standardized area back to the original context. Validity: Selected-method check: use random assignment, comparison, control, and replication. Neighboring-method check: justify the normal model before using normal areas. Error to reject: A method can produce a number even when its parameter and data structure are wrong, so calculator output is not validation.

Worked solution P23-Tough-13. Choose one-variable description for one quantitative variable; choose binomial model for a fixed number of independent binary trials. The deciding feature is the design and variable structure, not which calculator menu looks familiar. For the selected method, use $$\overline{x}=\sum x_i/n.$$ The neighboring method instead centers on $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}.$$ Interpretation: For one-variable description, describe shape, center, variability, and unusual features in context; for the neighboring method, interpret probability over repetitions of the full n-trial process. Validity: Selected-method check: check shape and unusual values before choosing resistant or nonresistant summaries. Neighboring-method check: verify BINS, including constant p. Error to reject: A method can produce a number even when its parameter and data structure are wrong, so calculator output is not validation.

Worked solution P23-Tough-14. Choose two-way table analysis for two categorical variables; choose sampling distribution for a mean for repeated random samples of size n. The deciding feature is the design and variable structure, not which calculator menu looks familiar. For the selected method, use $$P(A\mid B)=P(A\cap B)/P(B).$$ The neighboring method instead centers on $$S{E}_{\overline{X}}=\sigma /\sqrt{n}.$$ Interpretation: For two-way table analysis, compare conditional distributions without claiming causation; for the neighboring method, distinguish variation of sample means from variation of individuals. Validity: Selected-method check: use the conditioning group's total as the denominator. Neighboring-method check: check independence and population shape or CLT support. Error to reject: A method can produce a number even when its parameter and data structure are wrong, so calculator output is not validation.

Worked solution P23-Tough-15. Choose normal probability for one quantitative variable modeled as normal; choose one-proportion z interval for one random sample with a binary response. The deciding feature is the design and variable structure, not which calculator menu looks familiar. For the selected method, use $$z=(x-\mu )/\sigma .$$ The neighboring method instead centers on $$\widehat{p}\pm z^{*}\sqrt{\widehat{p}(1-\widehat{p})/n}.$$ Interpretation: For normal probability, translate standardized area back to the original context; for the neighboring method, state confidence as a property of the interval method. Validity: Selected-method check: justify the normal model before using normal areas. Neighboring-method check: check randomization, independence, and large observed counts. Error to reject: A method can produce a number even when its parameter and data structure are wrong, so calculator output is not validation.

Worked solution P23-Toughest-1. Reject the proposed method and use one-variable description, because the target is a population center or distribution and the observed structure is one quantitative variable. State the setup explicitly: $$\overline{x}=\sum x_i/n.$$ Then substitute only values that correspond to the defined symbols and retain guard digits. Interpretation: The final contextual statement must describe shape, center, variability, and unusual features in context and must respect the study's random sampling or random assignment scope. Validity: The validity audit must check shape and unusual values before choosing resistant or nonresistant summaries; a failed condition requires qualification, an alternative method, or no formal inference. Error to reject: The original choice, binomial model, targets a count of successes rather than a population center or distribution. That parameter mismatch is substantive, not cosmetic.

Worked solution P23-Toughest-2. Reject the proposed method and use two-way table analysis, because the target is conditional proportions and association and the observed structure is two categorical variables. State the setup explicitly: $$P(A\mid B)=P(A\cap B)/P(B).$$ Then substitute only values that correspond to the defined symbols and retain guard digits. Interpretation: The final contextual statement must compare conditional distributions without claiming causation and must respect the study's random sampling or random assignment scope. Validity: The validity audit must use the conditioning group's total as the denominator; a failed condition requires qualification, an alternative method, or no formal inference. Error to reject: The original choice, sampling distribution for a mean, targets the behavior of sample means rather than conditional proportions and association. That parameter mismatch is substantive, not cosmetic.

Worked solution P23-Toughest-3. Reject the proposed method and use normal probability, because the target is a modeled tail or interval probability and the observed structure is one quantitative variable modeled as normal. State the setup explicitly: $$z=(x-\mu )/\sigma .$$ Then substitute only values that correspond to the defined symbols and retain guard digits. Interpretation: The final contextual statement must translate standardized area back to the original context and must respect the study's random sampling or random assignment scope. Validity: The validity audit must justify the normal model before using normal areas; a failed condition requires qualification, an alternative method, or no formal inference. Error to reject: The original choice, one-proportion z interval, targets a population proportion rather than a modeled tail or interval probability. That parameter mismatch is substantive, not cosmetic.

Worked solution P23-Toughest-4. Reject the proposed method and use binomial model, because the target is a count of successes and the observed structure is a fixed number of independent binary trials. State the setup explicitly: $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}.$$ Then substitute only values that correspond to the defined symbols and retain guard digits. Interpretation: The final contextual statement must interpret probability over repetitions of the full n-trial process and must respect the study's random sampling or random assignment scope. Validity: The validity audit must verify BINS, including constant p; a failed condition requires qualification, an alternative method, or no formal inference. Error to reject: The original choice, two-proportion z test, targets equality of two population proportions rather than a count of successes. That parameter mismatch is substantive, not cosmetic.

Worked solution P23-Toughest-5. Reject the proposed method and use sampling distribution for a mean, because the target is the behavior of sample means and the observed structure is repeated random samples of size n. State the setup explicitly: $$S{E}_{\overline{X}}=\sigma /\sqrt{n}.$$ Then substitute only values that correspond to the defined symbols and retain guard digits. Interpretation: The final contextual statement must distinguish variation of sample means from variation of individuals and must respect the study's random sampling or random assignment scope. Validity: The validity audit must check independence and population shape or CLT support; a failed condition requires qualification, an alternative method, or no formal inference. Error to reject: The original choice, one-sample t interval, targets a population mean rather than the behavior of sample means. That parameter mismatch is substantive, not cosmetic.

Worked solution P23-Toughest-6. Reject the proposed method and use one-proportion z interval, because the target is a population proportion and the observed structure is one random sample with a binary response. State the setup explicitly: $$\widehat{p}\pm z^{*}\sqrt{\widehat{p}(1-\widehat{p})/n}.$$ Then substitute only values that correspond to the defined symbols and retain guard digits. Interpretation: The final contextual statement must state confidence as a property of the interval method and must respect the study's random sampling or random assignment scope. Validity: The validity audit must check randomization, independence, and large observed counts; a failed condition requires qualification, an alternative method, or no formal inference. Error to reject: The original choice, paired t test, targets a population mean difference rather than a population proportion. That parameter mismatch is substantive, not cosmetic.

Worked solution P23-Toughest-7. Reject the proposed method and use two-proportion z test, because the target is equality of two population proportions and the observed structure is two independent groups with binary responses. State the setup explicitly: $$z=({\widehat{p}}_1-{\widehat{p}}_2)/S{E}_0.$$ Then substitute only values that correspond to the defined symbols and retain guard digits. Interpretation: The final contextual statement must conclude about evidence for a population difference and must respect the study's random sampling or random assignment scope. Validity: The validity audit must pool only under the equal-proportions null; a failed condition requires qualification, an alternative method, or no formal inference. Error to reject: The original choice, chi-square independence test, targets association between the variables rather than equality of two population proportions. That parameter mismatch is substantive, not cosmetic.

Worked solution P23-Toughest-8. Reject the proposed method and use one-sample t interval, because the target is a population mean and the observed structure is one quantitative sample with unknown sigma. State the setup explicitly: $$\overline{x}\pm t^{*}s/\sqrt{n}.$$ Then substitute only values that correspond to the defined symbols and retain guard digits. Interpretation: The final contextual statement must report the interval in measurement units and must respect the study's random sampling or random assignment scope. Validity: The validity audit must check randomization, independence, and robustness to shape; a failed condition requires qualification, an alternative method, or no formal inference. Error to reject: The original choice, least-squares regression, targets a linear predictive association rather than a population mean. That parameter mismatch is substantive, not cosmetic.

Worked solution P23-Toughest-9. Reject the proposed method and use paired t test, because the target is a population mean difference and the observed structure is matched or before-after quantitative measurements. State the setup explicitly: $$t=\overline{d}/(s_d/\sqrt{n}).$$ Then substitute only values that correspond to the defined symbols and retain guard digits. Interpretation: The final contextual statement must conclude about the mean within-pair change and must respect the study's random sampling or random assignment scope. Validity: The validity audit must analyze independent pair differences; a failed condition requires qualification, an alternative method, or no formal inference. Error to reject: The original choice, randomized experiment, targets a treatment effect rather than a population mean difference. That parameter mismatch is substantive, not cosmetic.

Worked solution P23-Toughest-10. Reject the proposed method and use chi-square independence test, because the target is association between the variables and the observed structure is one population classified by two categorical variables. State the setup explicitly: $${\chi }^2=\sum (O-E{)}^2/E.$$ Then substitute only values that correspond to the defined symbols and retain guard digits. Interpretation: The final contextual statement must conclude about association rather than causation and must respect the study's random sampling or random assignment scope. Validity: The validity audit must check randomization, independent observations, and expected counts; a failed condition requires qualification, an alternative method, or no formal inference. Error to reject: The original choice, one-variable description, targets a population center or distribution rather than association between the variables. That parameter mismatch is substantive, not cosmetic.

Worked solution P23-Toughest-11. Reject the proposed method and use least-squares regression, because the target is a linear predictive association and the observed structure is two quantitative variables. State the setup explicitly: $$\widehat{y}=a+bx.$$ Then substitute only values that correspond to the defined symbols and retain guard digits. Interpretation: The final contextual statement must interpret slope and predictions within the observed x range and must respect the study's random sampling or random assignment scope. Validity: The validity audit must inspect form, residual pattern, and influential points; a failed condition requires qualification, an alternative method, or no formal inference. Error to reject: The original choice, two-way table analysis, targets conditional proportions and association rather than a linear predictive association. That parameter mismatch is substantive, not cosmetic.

Worked solution P23-Toughest-12. Reject the proposed method and use randomized experiment, because the target is a treatment effect and the observed structure is experimental units assigned to treatments. State the setup explicitly: $$\text{effect estimate}={\overline{x}}_T-{\overline{x}}_C.$$ Then substitute only values that correspond to the defined symbols and retain guard digits. Interpretation: The final contextual statement must support causation while limiting population generalization to the sampling design and must respect the study's random sampling or random assignment scope. Validity: The validity audit must use random assignment, comparison, control, and replication; a failed condition requires qualification, an alternative method, or no formal inference. Error to reject: The original choice, normal probability, targets a modeled tail or interval probability rather than a treatment effect. That parameter mismatch is substantive, not cosmetic.

Worked solution P23-Toughest-13. Reject the proposed method and use one-variable description, because the target is a population center or distribution and the observed structure is one quantitative variable. State the setup explicitly: $$\overline{x}=\sum x_i/n.$$ Then substitute only values that correspond to the defined symbols and retain guard digits. Interpretation: The final contextual statement must describe shape, center, variability, and unusual features in context and must respect the study's random sampling or random assignment scope. Validity: The validity audit must check shape and unusual values before choosing resistant or nonresistant summaries; a failed condition requires qualification, an alternative method, or no formal inference. Error to reject: The original choice, binomial model, targets a count of successes rather than a population center or distribution. That parameter mismatch is substantive, not cosmetic.

Worked solution P23-Toughest-14. Reject the proposed method and use two-way table analysis, because the target is conditional proportions and association and the observed structure is two categorical variables. State the setup explicitly: $$P(A\mid B)=P(A\cap B)/P(B).$$ Then substitute only values that correspond to the defined symbols and retain guard digits. Interpretation: The final contextual statement must compare conditional distributions without claiming causation and must respect the study's random sampling or random assignment scope. Validity: The validity audit must use the conditioning group's total as the denominator; a failed condition requires qualification, an alternative method, or no formal inference. Error to reject: The original choice, sampling distribution for a mean, targets the behavior of sample means rather than conditional proportions and association. That parameter mismatch is substantive, not cosmetic.

Worked solution P23-Toughest-15. Reject the proposed method and use normal probability, because the target is a modeled tail or interval probability and the observed structure is one quantitative variable modeled as normal. State the setup explicitly: $$z=(x-\mu )/\sigma .$$ Then substitute only values that correspond to the defined symbols and retain guard digits. Interpretation: The final contextual statement must translate standardized area back to the original context and must respect the study's random sampling or random assignment scope. Validity: The validity audit must justify the normal model before using normal areas; a failed condition requires qualification, an alternative method, or no formal inference. Error to reject: The original choice, one-proportion z interval, targets a population proportion rather than a modeled tail or interval probability. That parameter mismatch is substantive, not cosmetic.