Confidence Intervals, Hypothesis Testing for Proportions, and Sample Size Determination

Exam 1 Performance and Exam 2 Preparation

• Exam 1 Statistical Distribution and Class Performance:

  • The exam results reflect a large population of approximately $2000$ students.

  • The mean score across the population is $77$.

  • The highest score achieved in the class was a $92$, scored by Isaac.

  • A normal distribution model is used to describe performance intervals: adding $14.9984$ to the mean yields the upper benchmark, while subtracting $14.99$ establishes the lower boundary.

  • Students scoring outside this standard interval are encouraged to engage in self-reflection regarding their study habits and course engagement.

• Exam 2 Difficulty and Common Pitfalls:

  • Exam 2 historically exhibits the lowest student performance of the semester.

  • The difficulty does not stem from inherently complex mathematical theory, but rather from students' inability to identify which fundamental calculator functions apply to specific problem types.

  • A frequent point of failure is understanding when and how to implement the inverse normal distribution function ($\text{invNorm}$).

  • While Exam 2 is scheduled well in advance (around November 7th), mastering the distinction between problem frameworks cannot be accomplished in a 48-hour cramming window.

  • Success on Exam 2 requires reading the problem text carefully to identify the underlying distribution and parameter structure.

Foundations and Decomposition of Confidence Intervals

• General Architecture of a Confidence Interval:

  • A confidence interval for an unknown population parameter is constructed around a point estimate by adding and subtracting a margin of error:     $\text{Confidence Interval} = \text{Point Estimate} \pm \text{Margin of Error}$

  • For a population proportion, the point estimate is the sample proportion ($\hat{p}$).

  • Collecting a simple random sample of size $n$ and counting the number of successes ($x$) allows the calculation of the sample proportion:     $\hat{p} = \frac{x}{n}$

  • The level of confidence dictates the critical value ($z^*$), which acts as a multiplier scale factor.

  • The margin of error ($ME$) is the product of the critical value ($z^*$) and the standard error ($\text{SE}$):     $ME = z^* \times \text{SE}$

  • The complete analytical model for a confidence interval for a proportion is:     $\hat{p} \pm z^* \times \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

• Deconstructing Confidence Intervals into Core Components:

  • When provided with a finalized confidence interval, algebraic techniques allow the recovery of the original point estimate ($\hat{p}$), margin of error ($ME$), standard error ($\text{SE}$), and sample size ($n$).

  • If an arbitrary interval is specified as $[0.2, 0.4]$, the point estimate lies at the exact center, while the margin of error represents the distance from the center to either bound.

Point Estimate and Margin of Error Formulas and Computation

• Derivation and Formula for the Point Estimate:

  • The sample proportion ($\hat{p}$) is the midpoint of the confidence interval.

  • It is calculated by summing the lower bound and the upper bound and dividing by $2$:     $\hat{p} = \frac{\text{Upper Bound} + \text{Lower Bound}}{2}$

• Derivation and Formula for the Margin of Error:

  • The margin of error ($ME$) corresponds to half the total width of the confidence interval.

  • It is calculated by subtracting the lower bound from the upper bound and dividing by $2$:     $ME = \frac{\text{Upper Bound} - \text{Lower Bound}}{2}$

  • This shortcut replaces systems of linear equations solved via substitution or elimination, yielding the exact distance from $\hat{p}$ to either boundary.

• Comprehensive Example: Audiobook Listening Study:

  • Study Context: A survey investigates the proportion of United States adults who listened to an audiobook in the preceding 12 months.

  • A $90\%$ confidence interval is established with a lower bound of $0.183$ and an upper bound of $0.217$:     $\text{Interval} = [0.183, 0.217]$

  • Calculation of the point estimate ($\hat{p}$):     $\hat{p} = \frac{0.217 + 0.183}{2} = \frac{0.400}{2} = 0.20$

  • Calculation of the margin of error ($ME$):     $ME = \frac{0.217 - 0.183}{2} = \frac{0.034}{2} = 0.017$

• Real-World Application: Political Polling:

  • Margin of error concepts appear routinely in electoral reporting.

  • For instance, if candidate polling shows $47\%$ versus $45\%$ with a reported margin of error of $2\%$ (or $2.2\%$):

  • Adding $2\%$ to $47\%$ gives an upper bound of $49\%$.

  • Subtracting $2\%$ from $47\%$ gives a lower bound of $45\%$.

  • The resulting confidence interval of $[45\%, 49\%]$ illustrates potential sampling variability in voter share.

Standard Error and Constructing New Confidence Intervals

• Definition and Properties of Standard Error:

  • Standard error ($\text{SE}$) represents the standard deviation of the sampling distribution of a statistic.

  • In sampling distributions, the term standard error is preferred over standard deviation to indicate that it quantifies variability across sample estimates.

  • Mathematically, the margin of error equation can be solved for standard error:     $ME = z^* \times \text{SE} \implies \text{SE} = \frac{ME}{z^*}$

• Calculating Standard Error for the Audiobook Data:

  • For a $90\%$ confidence level, the critical value obtained from standard statistical tables is:     $z^* = 1.645$

  • Using the calculated margin of error ($ME = 0.017$):     $\text{SE} = \frac{0.017}{1.645} \approx 0.01033$

• Constructing a 95% Confidence Interval from Existing Data:

  • The building blocks extracted from the $90\%$ interval ($\hat{p} = 0.20$ and $\text{SE} \approx 0.01033$) can be repurposed to create a $95\%$ confidence interval for the same population.

  • For a $95\%$ confidence level, the critical value is:     $z^* = 1.96$

  • Calculation of the lower bound:     $\text{Lower Bound} = \hat{p} - (z^* \times \text{SE}) = 0.20 - (1.96 \times 0.01033) \approx 0.20 - 0.02025 = 0.17975 \approx 0.179$

  • Calculation of the upper bound:     $\text{Upper Bound} = \hat{p} + (z^* \times \text{SE}) = 0.20 + (1.96 \times 0.01033) \approx 0.20 + 0.02025 = 0.22025 \approx 0.226$

  • The new interval spans from approximately $17.9\%$ to $22.6\%$ ($[0.179, 0.226]$).

• The Inverse Trade-Off Between Precision and Confidence Level:

  • Comparing the two intervals demonstrates the relationship between confidence and width:

  • $90\%$ Confidence Interval: $[0.183, 0.217]$ (width = $0.034$)

  • $95\%$ Confidence Interval: $[0.179, 0.226]$ (width = $0.047$)

  • Higher confidence levels require a wider confidence interval.

  • Conceptual Analogy: To have greater certainty of catching the true parameter, one must cast a wider net.

Sample Size Determination for Desired Margin of Error

• Mathematical Formulation for Sample Size:

  • To achieve a targeted margin of error ($ME$), the required sample size ($n$) is derived by solving the margin of error equation:     $ME = z^* \times \sqrt{\frac{p(1 - p)}{n}}$     $n = p(1 - p)\left(\frac{z^*}{ME}\right)^2$

• Default Planning Value for Proportion:

  • When no preliminary estimate or historical data exists for $p$, always adopt the conservative planning value:     $p = 0.5$

  • Setting $p = 0.5$ maximizes the variance product $p(1 - p) = (0.5)(0.5) = 0.25$, guaranteeing that the calculated sample size is sufficiently large regardless of the true proportion.

• Step-by-Step Problem: Michelle's Presidential Campaign:

  • Context: Michelle is running for student council president. A preliminary simple random sample is gathered to project voter support.

  • Preliminary Data:

  • Sample size: $n = 150$

  • Intending to vote for Michelle: $x = 87$

  • Estimated sample proportion:       $\hat{p} = \frac{87}{150} = 0.58$

  • Complement of the proportion:       $1 - \hat{p} = 1 - 0.58 = 0.42$

  • Design Constraints:

  • Desired margin of error: $ME \le 0.05$

  • Required confidence level: $95\%$

  • Critical value ($z^*$) for $95\%$ confidence: $1.96$

  • Calculation Execution:     $n = (0.58)(0.42)\left(\frac{1.96}{0.05}\right)^2$     $n = (0.2436)\left(39.2\right)^2$     $n = (0.2436)(1536.64) = 374.325504$

• Mandatory Rounding Rule for Sample Sizes:

  • Sample sizes denote discrete observational units (e.g., individual persons) and cannot be fractional.

  • In sample size calculations, always round up to the next integer, regardless of the decimal value.

  • Even an answer such as $374.00000444$ must round up to $375$.

  • Michelle's campaign manager must sample at least $375$ students.

  • Academic Grading Impact: Failing to round up results in an automatic deduction of $2$ points on a standard $6$-point exam problem.

• Handling Percentages in Margin of Error Calculations:

  • When a problem specifies a margin of error as a percentage, it must be converted to a decimal before calculating.

  • For example, a margin of error given as $3\%$ must be entered as $0.03$, never as the integer $3$.

Hypothesis Testing for a Population Proportion: ESRC Case Study

• Research Scenario: Economic and Social Research Council (ESRC):

  • The ESRC is a non-developmental public body funded by the UK government that provides research funding and training support in the social sciences.

  • Each year, the ESRC compiles reports detailing the proportion of grant applications funded.

  • Historical Baseline: In 2016, the grant success rate was $21\%$ ($p = 0.21$).

  • Research Question: A researcher named Martina investigates whether the current year's grant success rate has increased beyond the 2016 benchmark.

• Sample Observational Data:

  • Total grant requests sampled: $n = 573$

  • Number of grants successfully funded: $x = 132$

• Formulation of Statistical Hypotheses:

  • Hypotheses must strictly be stated in terms of the population parameter ($p$), never sample statistics ($\hat{p}$).

  • Null Hypothesis ($H_0$): Asserts that the success rate remains unchanged from the historical baseline:     $H_0: p = 0.21$

  • Alternative Hypothesis ($H_a$): Reflects the researcher's directional conjecture of an increase:     $H_a: p > 0.21$

  • Test Classification: Because the alternative hypothesis uses the greater-than sign ($>$), this is a one-tailed (right-tailed) hypothesis test, not a two-tailed ($\neq$) test.

Assumptions and Conditions: Hypothesis Tests vs. Confidence Intervals

• Fundamental Assumptions:

  • The data must be drawn via a simple random sample from the target population.

  • Individual observations within the sample must be independent of one another.

• Success/Failure Condition Distinction:

  • Both inference procedures require at least $10$ expected successes and $10$ expected failures to satisfy the normality criteria of the sampling distribution.

  • For Hypothesis Testing:

  • Normality conditions must be verified using the hypothesized null proportion ($p_0$), assuming the null hypothesis is true:       $n p_0 \ge 10 \quad \text{and} \quad n(1 - p_0) \ge 10$

  • Evaluation for ESRC problem:       $n p_0 = 573 \times 0.21 = 120.33 \ge 10$       $n(1 - p_0) = 573 \times (1 - 0.21) = 573 \times 0.79 = 452.67 \ge 10$

  • For Confidence Intervals:

  • The true parameter is unknown, so normality conditions are evaluated using the sample proportion ($\hat{p}$):       $n \hat{p} \ge 10 \quad \text{and} \quad n(1 - \hat{p}) \ge 10$

Technology Implementation: 1-PropZTest on the Graphing Calculator

• Selection of Test Statistic and Function:

  • When conducting a hypothesis test for a single population proportion, the test statistic is a standard normal $z$-score.

  • The graphing calculator tool used for this test is 1-PropZTest.

• Calculator Input Parameters:

  • Null parameter ($p_0$): enter $0.21$

  • Number of successes ($x$): enter $132$

  • Sample size ($n$): enter $573$

  • Direction of alternative hypothesis: select > p_0 to specify a right-tailed test, then navigate to Calculate.

• Output Results and Interpretation:

  • Sample proportion ($\hat{p}$):     $\hat{p} = \frac{132}{573} \approx 0.23037$

  • Test statistic ($z$):     $z \approx 1.149209$

  • Calculated $p$-value:     $p\text{-value} \approx 0.0329821$

  • Summary: The single calculator function generates both the test statistic ($z$) and the corresponding probability ($p$-value), providing the necessary metrics to draw a formal statistical conclusion.

Questions & Discussion

• Student Discussion on Exam Scores:

  • Student Observation: Inquiry regarding distance from the mean exam score of $77$.

  • Response: Individual scores are private, known only to the individual student and the instructor. Isaac received the highest score in the class with a $92$.

• Student Chatter on Sports and Fantasy Football Drafts:

  • Purdue Athletics: Brief observation regarding Purdue's performance challenges.

  • NFL Roster and Trade Moves: Commentary regarding the Minnesota Vikings' trade interest, potential trade impacts on the Detroit Lions, and Lions' defensive assets.

  • Player Injuries and Team Management: Critique regarding player injury management, noting that an athlete sustained injuries because team medical staff permitted premature play.

  • Fantasy Draft Strategy and Player Selections:

  • Deliberation between drafting Dalton Schultz versus David Montgomery, concluding running backs are generally easier to acquire, leading to the selection of Schultz.

  • Matthew Stafford's protection by the offensive line.

  • Selection of Kenneth Walker III (K9), characterized as a major draft steal.

  • Additional player mentions including Jaxon Smith-Njigba (JSN), Drake London, and Treyvon Walker.