Probability, Normal Curves, Sampling Distributions, and Confidence Intervals

Probability Foundations and Set Operations

• Definition of Probability:

  • Probability represents the likelihood of an event occurring.

  • Mathematical bounds: $0 \le P(E) \le 1$ (or $0\%$ to $100\%$), where $0$ signifies no chance of occurrence and $1$ (or $100\%$) signifies absolute certainty.

  • Notation: $P(E)$ denotes the probability that event $E$ occurs.

• Classical Probability Formula:

  • $P(E) = \frac{N(E)}{N(S)}$

  • $N(E)$ is the number of ways event $E$ can occur.

  • $N(S)$ is the total number of possible outcomes in the sample space $S$.

• Complement Rule:

  • The complement of event $E$ (denoted as $\text{not } E$ or $E^c$) consists of all outcomes in the sample space where $E$ does not happen.

  • $P(\text{not } E) = 1 - P(E)$

  • The sum of an event and its complement always equals certainty: $P(E) + P(\text{not } E) = 1$.

  • Example: If the probability of snow is $P(\text{snow}) = 0.49$, then $P(\text{not snow}) = 1 - 0.49 = 0.51$.

• Basic Probability Example:

  • In a class consisting of $32$ students, $20$ are coffee drinkers and $12$ are non-coffee drinkers.

  • Probability of randomly selecting a coffee drinker:     $P(\text{coffee}) = \frac{20}{32} = 0.625 = 62.5\%$

• Two-Way Contingency Tables:

  • Used to categorize individuals across two categorical variables simultaneously.

  • Reference distribution of $32$ students classified by academic standing (Year $1$ through $5$) and coffee consumption:

  • Year $1$: $7$ coffee drinkers, $6$ non-coffee drinkers (Total = $13$ students).

  • Year $2$: $9$ coffee drinkers, $5$ non-coffee drinkers (Total = $14$ students).

  • Year $3$: $2$ coffee drinkers, $0$ non-coffee drinkers (Total = $2$ students).

  • Year $4$: $1$ coffee drinker, $1$ non-coffee drinker (Total = $2$ students).

  • Year $5$: $1$ coffee drinker, $0$ non-coffee drinkers (Total = $1$ student).

  • Column Totals: Total coffee drinkers = $20$; Total non-coffee drinkers = $12$.

  • Grand Total: $N(S) = 32$.

  • Marginal Probability Calculation:     $P(\text{2nd year}) = \frac{9 + 5}{32} = \frac{14}{32} = 0.4375$

• Addition Rules for Union ($A \cup B$ / $A \text{ OR } B$):

  • General Addition Rule (accounts for overlapping outcomes to avoid double-counting):     $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

  • Special Addition Rule (for mutually exclusive / disjoint events where $P(A \cap B) = 0$):     $P(A \cup B) = P(A) + P(B)$

  • Tabular Example: Probability of choosing a $2\text{nd}$ year student OR a coffee drinker:

  • Method 1 (summing individual mutually exclusive cells): $P(\text{2nd } \cup \text{ coffee}) = \frac{7 + 9 + 2 + 1 + 1 + 5}{32} = \frac{25}{32} = 0.78125$

  • Method 2 (using general formula): $P(\text{2nd}) + P(\text{coffee}) - P(\text{2nd } \cap \text{ coffee}) = \frac{14}{32} + \frac{20}{32} - \frac{9}{32} = \frac{25}{32} = 0.78125$

• Joint Probability ($A \cap B$ / $A \text{ AND } B$):

  • Represents the probability that both event $A$ and event $B$ occur simultaneously (the intersection/overlap).

  • Tabular Example: Probability of choosing a $4\text{th}$ year student and a coffee drinker:     $P(\text{4th } \cap \text{ coffee}) = \frac{1}{32} = 0.03125$

• Multiplication Rule for Independent Events:

  • Special Multiplication Rule: If events $A$ and $B$ are independent, the occurrence of one does not affect the occurrence of the other:     $P(A \cap B) = P(A) \times P(B)$

  • Multi-event independence example: If the probability of snow on any given day is $P(\text{snow}) = 0.50$ ($50\%$) independently over $3$ consecutive days:     $P(S_1 \cap S_2 \cap S_3) = P(S) \times P(S) \times P(S) = (0.50)^3 = 0.125 = 12.5\%$

Conditional Probability and Independence

• Definition of Conditional Probability:

  • The probability of an event occurring given that another event has already occurred.

  • Represents a restriction of the sample space to a specific subset.

  • Key indicator phrases signaling a conditional probability include: "out of", "if", "given", and "among".

• Conditional Probability Notation and Anatomy:

  • Notation: $P(A \mid B)$, read as "the probability of event $A$ given event $B$".

  • Event $A$ is the event of interest (represented in the shaded numerator).

  • Event $B$ is the given condition or restriction (represented in the outlined denominator).

• Conditional Probability Formulas:

  • Definition formula:     $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$

  • General Multiplication Rule for any two events:     $P(A \cap B) = P(A) \times P(B \mid A) = P(B) \times P(A \mid B)$

• Mathematical Condition for Independence:

  • Two events $A$ and $B$ are statistically independent if and only if:     $P(B \mid A) = P(B)$     (Equivalently: $P(A \mid B) = P(A)$)

• Worked Contingency Table Applications ($32$-Student Reference Table):

  • Example 1: Probability of selecting a $1\text{st}$ year student out of the coffee drinkers:

  • Restriction (denominator): Coffee drinkers ($20$ total).

  • Event of interest (numerator): $1\text{st}$ year coffee drinkers ($7$ students).

  • Tabular calculation:       $P(\text{1st} \mid \text{coffee}) = \frac{7}{20} = 0.35$

  • Formal formula verification:       $P(\text{1st} \mid \text{coffee}) = \frac{P(\text{1st } \cap \text{ coffee})}{P(\text{coffee})} = \frac{\frac{7}{32}}{\frac{20}{32}} = \frac{7}{20} = 0.35$

  • Example 2: Probability of selecting a non-coffee drinker if they are a $4\text{th}$ year student:

  • Restriction (denominator): $4\text{th}$ year students ($2$ total: $1$ coffee, $1$ non-coffee).

  • Event of interest (numerator): Non-coffee drinkers among $4\text{th}$ years ($1$ student).

  • Tabular calculation:       $P(\text{non-coffee} \mid \text{4th}) = \frac{1}{2} = 0.50$

Normal Probability Distributions and the Empirical Rule

• Core Properties of the Normal Distribution:

  • Continuous probability distribution characterized by a symmetric, unimodal, bell-shaped density curve.

  • Centered at the population mean $\mu$ (or sample mean $\bar{x}$), with spread determined by the standard deviation $\sigma$ (or sample standard deviation $s$).

  • Notation: $X \sim N(\mu, \sigma)$, where $X$ is the continuous random variable.

  • Perfect symmetry: Exactly $50\%$ of the data lies to the left of the mean $\mu$, and $50\%$ lies to the right.

  • Area as probability: The total area under the density curve equals $1$ (or $100\%$ total probability). The area under the curve between any two points equals the probability that the random variable falls within that interval.

• The Empirical Rule ($68$-$95$-$99.7$ Rule for Bell-Shaped Distributions):

  • Within $1$ standard deviation of the mean ($\mu \pm 1\sigma$):

  • Captures approximately $68\%$ of the data.

  • Subdivided into $34\%$ from $\mu - 1\sigma$ to $\mu$, and $34\%$ from $\mu$ to $\mu + 1\sigma$.

  • Within $2$ standard deviations of the mean ($\mu \pm 2\sigma$):

  • Captures approximately $95\%$ of the data.

  • Adds $13.5\%$ in the interval from $\mu - 2\sigma$ to $\mu - 1\sigma$, and $13.5\%$ from $\mu + 1\sigma$ to $\mu + 2\sigma$.

  • Within $3$ standard deviations of the mean ($\mu \pm 3\sigma$):

  • Captures approximately $99.7\%$ of the data.

  • Adds $2.35\%$ in the interval from $\mu - 3\sigma$ to $\mu - 2\sigma$, and $2.35\%$ from $\mu + 2\sigma$ to $\mu + 3\sigma$.

  • Extreme Tails (beyond $3$ standard deviations):

  • Exactly $0.15\%$ of data falls below $\mu - 3\sigma$, and $0.15\%$ falls above $\mu + 3\sigma$.

• Cumulative Probability and Interval Operations:

  • Left-tail cumulative probability: Software functions (such as Minitab cumulative distribution function) return the area to the left: $P(X \le a)$.

  • Right-tail probability: Computed via complement: $P(X \ge a) = 1 - P(X \le a)$.

  • Interval probability: Computed via subtraction: $P(a \le X \le b) = P(X \le b) - P(X \le a)$.

• Application Example A — Student Heights ($X \sim N(68, 2.5)$):

  • Distribution parameters: Mean $\mu = 68''$, standard deviation $\sigma = 2.5''$.

  • Empirical Rule scale marks: $60.5''$ ($-3\sigma$), $63.0''$ ($-2\sigma$), $65.5''$ ($-1\sigma$), $68.0''$ ($\mu$), $70.5''$ ($+1\sigma$), $73.0''$ ($+2\sigma$), $75.5''$ ($+3\sigma$).

  • Probability less than or equal to $67''$:     $P(X \le 67) = 0.345$     ($34.5\%$ probability that a randomly chosen student is $67''$ tall or shorter).

  • Probability greater than or equal to $65''$:     $P(X \ge 65) = 1 - P(X \le 65) = 1 - 0.115 = 0.885$     ($88.5\%$ probability that a student is at least $65''$ tall).

  • Probability between $62''$ and $65''$:     $P(62 \le X \le 65) = P(X \le 65) - P(X \le 62) = 0.115 - 0.008 = 0.107$     ($10.7\%$ probability that a student's height is between $62''$ and $65''$).

  • Inverse Normal Calculation ($40\text{th}$ Percentile):

  • Given a left-tail area of $0.40$, find cutoff $k$ such that $P(X \le k) = 0.40$.

  • Output: $k = 67.37''$. Thus, $40\%$ of students are below $67.37''$.

  • Inverse Complement Application:

  • Find the height cutoff where $60\%$ of students are taller.

  • Left-tail equivalent: $100\% - 60\% = 40\%$.

  • Cutoff remains $67.37''$.

• Application Example B — Sample Distribution of Heights ($n = 21$, $\bar{x} = 68$, $s = 4$):

  • Distribution model: $X \sim N(68, 4)$.

  • Scale marks: $56''$ ($-3s$), $60''$ ($-2s$), $64''$ ($-1s$), $68''$ ($\bar{x}$), $72''$ ($+1s$), $76''$ ($+2s$).

  • Left-tail calculation: $P(X \le 67) = 0.4013$ ($40.13\%$).

  • Right-tail calculation: $P(X > 71) = 0.2266$ ($22.66\%$).

  • Middle-interval calculation: $P(64 \le X \le 70) = 0.5328$ ($53.28\%$).

  • $23\text{rd}$ Percentile calculation: Finding $k$ such that $P(X \le k) = 0.23$ yields $k = 65.04''$.

The Standard Normal Distribution and Normality Assessment

• Definition of the Standard Normal Distribution:

  • A specific normal distribution where the center is zero ($\mu = 0$) and spread is one ($\sigma = 1$).

  • Notation: $Z \sim N(0, 1)$.

• Standardization and the $Z$-Score:

  • A $Z$-score transforms any value $X$ from a normal distribution into standard normal units.

  • Formula (population parameters): $Z = \frac{X - \mu}{\sigma}$

  • Formula (sample statistics): $Z = \frac{X - \bar{x}}{s}$

  • Definitional meaning: The $Z$-score represents the exact number of standard deviations an individual raw observation is located away from the mean.

• Probability Calculations for $Z$:

  • Continuous interval probability formula: $P(a \le Z \le b) = \text{normalcdf}(a, b, 0, 1)$.

  • Tail probability examples:

  • $P(Z < -2.20) = 0.014$

  • $P(Z \le -1.5) = 0.0668$

• Percentiles of $Z$ using Inverse Normal Technology:

  • Calculator/software syntax: invnorm(area to left, 0, 1).

  • $99\text{th}$ Percentile determination:

  • Area to the left = $0.99$; area to the right = $0.01$.

  • Output: $Z = 2.326$.

• Criteria for Assessing Normality of Data:

  • Given information: Explicitly told in the problem statement that the population is normally distributed.

  • Graphical shape: A histogram of sample data is symmetric and bell-shaped.

  • Normal Probability Plot (NPP):

  • A scatter plot graphing observed data values against corresponding expected normal values.

  • Condition for normality: The data is deemed approximately normally distributed if the NPP is fairly linear.

  • In formal statistical tests of the NPP, normality is supported when the associated $p$-value is greater than $0.05$ ($p > 0.05$).

Sampling Distributions for Proportions and Means

• Nature of a Sampling Distribution:

  • A sample statistic varies from sample to sample, making it a random variable with its own probability distribution.

  • A sampling distribution represents the distribution of values taken by a statistic across all possible distinct random samples of a fixed size $n$ drawn from the same population.

• Sampling Distribution of the Sample Proportion ($\hat{p}$) — Categorical Data:

  • Context: Categorical data measuring success ($1$) versus failure ($0$).

  • Point estimate statistic: $\hat{p} = \frac{x}{n}$, where $x$ is the number of individuals possessing the characteristic, and $n$ is sample size.

  • Center (Mean of sample proportions):     $\mu_{\hat{p}} = p$

  • Spread (Standard error of sample proportions):     $\sigma_{\hat{p}} = \text{SE}(\hat{p}) = \sqrt{\frac{p(1 - p)}{n}}$     where $p$ is the probability of success and $1 - p$ is the probability of failure.

  • Shape and Normality Condition:

  • The distribution of $\hat{p}$ is approximately normal if:       $np(1 - p) \ge 10$

  • Complete model notation: $\hat{p} \sim N\left(p, \sqrt{\frac{p(1 - p)}{n}}\right)$.

  • Worked Example — High School Statistics Enrollment:

  • Background: The National Center for Education Statistics reports that the proportion of US high school students taking high school statistics is $p = 0.11$ ($11\%$). A random sample of $n = 150$ students is taken.

  • Normality check: $np(1 - p) = 150 \times 0.11 \times (1 - 0.11) = 150 \times 0.11 \times 0.89 = 14.685$. Because $14.685 \ge 10$, the shape is approximately normal.

  • Center: $\mu_{\hat{p}} = p = 0.11$.

  • Spread: $\text{SE}(\hat{p}) = \sqrt{\frac{0.11(1 - 0.11)}{150}} = \sqrt{\frac{0.0979}{150}} \approx 0.0255$.

  • Probability calculation for $P(\hat{p} > 0.15)$:       $P(0.15 < \hat{p} < \infty) = \text{normalcdf}(0.15, 10000, 0.11, 0.0255) = 0.0584 = 5.84\%$

• Sampling Distribution of the Sample Mean ($\bar{x}$) — Numerical Data:

  • The Central Limit Theorem (CLT):

  • Regardless of the underlying population shape, the sampling distribution of the sample mean $\bar{x}$ becomes approximately normal as the sample size $n$ increases.

  • Operational threshold: $n \ge 30$ is sufficiently large to assume approximate normality.

  • Conditions to Assume Normality for $\bar{x}$:

  1. Sample size $n \ge 30$, OR

  2. Directly told that the underlying population is normal ($X \sim N(\mu, \sigma)$), OR

  3. The Normal Probability Plot (NPP) of the raw data is linear.

  • Center (Mean of sample means):     $\mu_{\bar{x}} = \mu$

  • Spread (Standard error of sample means):     $\sigma_{\bar{x}} = \text{SE}(\bar{x}) = \frac{\sigma}{\sqrt{n}}$     As sample size $n$ increases, the standard error spread decreases.

  • Complete model notation: $\bar{x} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)$.

  • Worked Example — Pre-Calculus Final Exam Scores:

  • Background: Population mean score is $\mu = 75$ with standard deviation $\sigma = 9$. A random sample of $n = 41$ students is selected.

  • Normality check: $n = 41 \ge 30$, satisfying the CLT requirement for normality.

  • Center: $\mu_{\bar{x}} = \mu = 75$.

  • Spread: $\text{SE}(\bar{x}) = \frac{\sigma}{\sqrt{n}} = \frac{9}{\sqrt{41}} \approx 1.4059$.

  • Probability calculation for $P(\bar{x} > 79)$:       $P(\bar{x} > 79) = \text{normalcdf}(79, 10^{99}, 75, 1.4059) = 0.0022 = 0.22\%$

Confidence Intervals for Population Proportions

• Theoretical Principles:

  • Point Estimate: A single statistic that serves as the best estimate of a parameter. For categorical proportions, the sample proportion $\hat{p} = \frac{x}{n}$ serves as the point estimate for $p$.

  • Interval Estimation Rationale: Because $\hat{p}$ varies from sample to sample, a single point estimate cannot be expected to provide a complete picture of the population parameter. A confidence interval builds an interval of plausible values around the point estimate.

  • Confidence Level ($1 - \alpha$): The theoretical proportion of intervals that will contain the true population parameter $p$ if repeated samples of size $n$ are drawn indefinitely.

  • Precision vs. Confidence Trade-off: Increasing the level of confidence requires a larger critical value, resulting in a wider interval.

• Mathematical Structure of Confidence Intervals for $p$:

  • General format:     $\text{Point Estimate} \pm \text{Margin of Error}$

  • Specific formula:     $\hat{p} \pm Z_{\frac{\alpha}{2}} \times \text{SE}(\hat{p}) = \hat{p} \pm Z_{\frac{\alpha}{2}} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

  • Margin of error component: $E = Z_{\frac{\alpha}{2}} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$.

  • Critical value $Z_{\frac{\alpha}{2}}$: Represents the number of standard errors away from center; depends exclusively on the chosen confidence level and corresponds to an area of $\frac{\alpha}{2}$ in each tail ($\alpha$ is total tail area).

• Common Critical Values ($Z_{\frac{\alpha}{2}}$) via invnorm:

  • $90\%$ Confidence: Tail area $\frac{\alpha}{2} = 0.05 \rightarrow Z_{\frac{\alpha}{2}} = \pm 1.645$

  • $95\%$ Confidence: Tail area $\frac{\alpha}{2} = 0.025 \rightarrow Z_{\frac{\alpha}{2}} = \pm 1.96$

  • $99\%$ Confidence: Tail area $\frac{\alpha}{2} = 0.005 \rightarrow Z_{\frac{\alpha}{2}} = \pm 2.576$

• Normality Verification Check:

  • Condition required to apply the normal distribution to the interval:     $n\hat{p}(1 - \hat{p}) \ge 10$

• Fully Worked Example — Pass/Fail Grade Option:

  • Context: In a random sample of $n = 150$ RIT students, $x = 93$ support a pass/fail grading option. Construct a $95\%$ confidence interval for the population proportion $p$.

  • Point estimate:     $\hat{p} = \frac{x}{n} = \frac{93}{150} = 0.62$

  • Normality check:     $n\hat{p}(1 - \hat{p}) = 150(0.62)(0.38) = 35.34 \ge 10$

  • Standard error calculation:     $\text{SE}(\hat{p}) = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.62(1 - 0.62)}{150}} = \sqrt{\frac{0.2356}{150}} \approx 0.0396$

  • Margin of error ($95\%$ confidence, $Z_{\frac{\alpha}{2}} = 1.96$):     $E = 1.96 \times 0.0396 = 0.077616$

  • Interval boundaries:

  • Upper bound: $0.62 + 0.077616 = 0.697616 \approx 0.6977$

  • Lower bound: $0.62 - 0.077616 = 0.542384 \approx 0.5423$

  • Confidence Interval: $(0.5423, 0.6977)$

• Standard Interpretation Template:

  • Formulation: "We are $[\text{confidence level}]\%$ confident that the population proportion of $[\text{context/variable}]$ is between $[\text{lower bound}]$ and $[\text{upper bound}]$."

  • Contextual statement: "We are $95\%$ confident that the population proportion of RIT students who support the pass/fail grade option is between $0.5423$ and $0.6977$."

• Rules for Evaluating Claims from Confidence Intervals:

  • "Less than" claim: Supported only when the entire confidence interval lies strictly below the claimed value.

  • "More than" claim: Supported only when the entire confidence interval lies strictly above the claimed value.

  • "Equal to" claim: Supported/plausible if the value of interest falls inside the confidence interval boundaries.

Confidence Intervals for Population Means

• Parameter and Point Estimate:

  • Parameter: Population mean $\mu$ (applied to numerical data, continuous measurements, means).

  • Point Estimate: Sample mean $\bar{x}$ (the midpoint of the confidence interval).

• The Student's $t$-Distribution:

  • Need for a new distribution: In practice, the population standard deviation $\sigma$ is unknown and must be estimated using the sample standard deviation $s$.

  • Extra variability: Estimating both $\mu$ with $\bar{x}$ and $\sigma$ with $s$ from the same sample introduces additional variability. The standard normal curve ($Z$) does not account for this extra variability; intervals constructed using $Z$ would fail to capture $\mu$ at the specified rate (e.g., $90\%$ intervals using $Z$ would contain $\mu$ in fewer than $90$ out of $100$ samples).

  • Defining Characteristics of $t$:

  • Centered at $0$ and symmetric.

  • Similar in shape to the standard normal curve, but shorter at the peak and more spread out in the tails to absorb the extra variability.

  • Shape changes dynamically with sample size based on degrees of freedom ($df$):       $df = n - 1$

  • As degrees of freedom ($n - 1$) increase, the $t$-distribution curve approaches the standard normal distribution.

  • Critical value notation: Uses $t_{\frac{\alpha}{2}}$ instead of $Z_{\frac{\alpha}{2}}$, determined via a $t$-table or invt(area to left, df).

• Mathematical Structure of Confidence Intervals for $\mu$:

  • General formula:     $\bar{x} \pm t_{\frac{\alpha}{2}} \times \text{SE}(\bar{x}) = \bar{x} \pm t_{\frac{\alpha}{2}} \left(\frac{s}{\sqrt{n}}\right)$

  • Midpoint: $\bar{x}$.

  • Margin of error component: $E = t_{\frac{\alpha}{2}} \left(\frac{s}{\sqrt{n}}\right)$.

  • Normality condition: $n \ge 30$ OR the Normal Probability Plot (NPP) is linear.

• Fully Worked Example — Caloric Content in Frozen Dinners:

  • Given sample statistics: $n = 41$, $\bar{x} = 260.5\,\text{cal}$, $s = 12.1\,\text{cal}$.

  • Objective: Construct a $95\%$ confidence interval for the population mean calories $\mu$.

  • Normality check: $n = 41 \ge 30$, fulfilling the CLT normality condition.

  • Degrees of freedom: $df = n - 1 = 41 - 1 = 40$.

  • Critical value: For $95\%$ confidence, tail area is $\frac{1 - 0.95}{2} = 0.025$.

  • Calculator syntax: invt(0.025, 40) yields critical value $t_{\frac{\alpha}{2}} = 2.021$.

  • Standard error calculation:     $\text{SE}(\bar{x}) = \frac{s}{\sqrt{n}} = \frac{12.1}{\sqrt{41}} \approx 1.8897\,\text{cal}$

  • Margin of error calculation:     $E = 2.021 \times 1.8897 = 3.81908 \approx 3.8191\,\text{cal}$

  • Interval boundaries:

  • Upper bound: $260.5 + 3.8191 = 264.3191 \approx 264.32\,\text{cal}$

  • Lower bound: $260.5 - 3.8191 = 256.6809 \approx 256.68\,\text{cal}$

  • Confidence Interval: $(256.68, 264.32)$

• Conclusion and Contextual Interpretation:

  • "We are $95\%$ confident that the population mean calories in this brand of frozen dinners is between $256.68$ and $264.32$."

• Follow-Up Claim Analysis Based on the $95\%$ CI $(256.68, 264.32)$:;

  • Claim: "Can the company claim that the dinners have less than $260$ calories?"

  • Evaluation: No. The claim requires the entire interval to be below $260$, but the interval extends up to $264.32$.

  • Claim: "Can we conclude that the dinners have more than $250$ calories?"

  • Evaluation: Yes. The entire interval lies strictly above $250$ ($256.68 > 250$).

  • Claim: "Can we state that the dinners have (are equal to) $258$ calories?"

  • Evaluation: Yes. The value $258$ falls within the interval boundaries of $256.68$ and $264.32$.

Summary of Center, Spread, Normality Checks, and Software Procedures

• Normal Probability Distribution ($X$):

  • Data type: Numerical observations.

  • Center: Mean $\mu$ (or $\bar{x}$).

  • Spread: Standard deviation $\sigma$ (or $s$).

  • Normality verification: Explicitly stated/told in problem setup, histogram is symmetric and bell-shaped, or NPP is fairly linear ($p > 0.05$).

  • Notation: $X \sim N(\mu, \sigma)$.

• Standard Normal Distribution ($Z$):

  • Data type: Standardized numerical units.

  • Center: Mean = $0$.

  • Spread: Standard deviation = $1$.

  • Normality verification: Explicitly stated/told.

  • Notation: $Z \sim N(0, 1)$.

• Sampling Distribution of Proportions ($\hat{p}$):

  • Data type: Categorical data (proportions, success rates).

  • Center: Mean = $p$.

  • Spread: Standard error $\text{SE}(\hat{p}) = \sqrt{\frac{p(1 - p)}{n}}$.

  • Normality verification: $np(1 - p) \ge 10$ (or explicitly told).

  • Notation: $\hat{p} \sim N\left(p, \sqrt{\frac{p(1 - p)}{n}}\right)$.

• Sampling Distribution of Means ($\bar{x}$):

  • Data type: Numerical data (sample averages).

  • Center: Mean = $\mu$.

  • Spread: Standard error $\text{SE}(\bar{x}) = \frac{\sigma}{\sqrt{n}}$.

  • Normality verification: Sample size $n \ge 30$, population explicitly stated as normal, or NPP is linear.

  • Notation: $\bar{x} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)$.

• Confidence Intervals for Proportions ($p$):

  • Data type: Categorical data (percentages, proportions of a whole, yes/no queries).

  • Center (Point estimate): $\hat{p} = \frac{x}{n}$.

  • Spread (Standard error): $\text{SE}(\hat{p}) = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$.

  • Normality verification: $n\hat{p}(1 - \hat{p}) \ge 10$.

  • Calculation formula:     $\hat{p} \pm Z_{\frac{\alpha}{2}} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$

  • Minitab menu path: Stat > Basic Statistics > 1 Proportion.

• Confidence Intervals for Means ($\mu$):

  • Data type: Numerical data (means, continuous measurements).

  • Center (Point estimate): $\bar{x}$.

  • Spread (Standard error): $\text{SE}(\bar{x}) = \frac{s}{\sqrt{n}}$.

  • Normality verification: Sample size $n \ge 30$ or NPP of raw data is linear.

  • Degrees of freedom: $df = n - 1$.

  • Calculation formula:     $\bar{x} \pm t_{\frac{\alpha}{2}} \left(\frac{s}{\sqrt{n}}\right)$

  • Minitab menu path: Stat > Basic Statistics > 1 Sample t.

• Calculator and Software Technology Syntax Summary:

  • Normal cumulative probability: normalcdf(lower_bound, upper_bound, center, spread).

  • Standard normal cumulative probability: normalcdf(lower_bound, upper_bound, 0, 1).

  • Normal cutoff values: invnorm(area_to_left, center, spread).

  • Standard normal critical values: invnorm(area_to_left, 0, 1).

  • Student's $t$ critical values: invt(area_to_left, df).