Study sheet: Statistics for Management

📋 Course Outline

1. Statistical Thinking and Data Context
2. Data Types and Sampling
3. Displaying Qualitative and Quantitative Data
4. Center, Spread, and Distribution Shape
5. Contingency Tables and Association
6. Correlation and Linear Regression
7. Regression Fit and Residuals
8. Probability Rules and Conditioning
9. Discrete Random Variables
10. Binomial and Poisson Models
11. Continuous and Normal Models
12. Sampling Distributions
13. Sampling Distribution of the Mean
14. Sampling Distribution of Proportions
15. Point and Interval Estimation
16. Confidence Intervals for Means
17. Confidence Intervals for Proportions
18. One-Sample Hypothesis Testing

📖 1. Statistical Thinking and Data Context

🔑 Key Concepts & Definitions

• Statistics : both a way of reasoning and a collection of tools and methods designed to analyze data, while statistics in the plural are results of calculations made with data.
• Data : any collection of numbers, characters, images, or other items that provide information about something; they are values that vary in a context.

📝 Essential Points

• 🔄 Doing statistics involves three stages:

  1. Thinking before the analysis
  2. Showing results with mathematics and charts
  3. Telling others the results so they can understand them correctly

• Data require six contextual questions: who the cases are, what variables were measured, why the data were collected, how they were collected, when they were collected, and where they were collected.

📌 Descriptive statistics describe sample data quantitatively, visually, and verbally, whereas inferential statistics use sample data to draw conclusions and make reliable forecasts about a population.

💡 Memory Hook

Think → Show → Tell

📖 2. Data Types and Sampling

🔑 Key Concepts & Definitions

• Simple random sample : selected so that every different sample of size n has an equal chance of selection.
• Identifier variable : a categorical variable, such as a student ID, tax file number, or name, for which there should be only one case in each category.

★ Must-know

📌 Quantitative data are measured by numbers and usually have units, whereas qualitative data identify a group or category.

📌 A numerical variable is quantitative only when calculations with its values make sense, so student ID numbers and classroom numbers are numerical but categorical.

Further detail

• Common sampling techniques include:

  • Systematic sampling
  • Convenience sampling
  • Cluster sampling
  • Stratified sampling

• The four stated sources of data are:

  • Published sources
  • Designed experiments
  • Surveys
  • Observational studies

💡 Memory Hook

Quantitative measures amount; qualitative identifies category

📖 3. Displaying Qualitative and Quantitative Data

🔑 Key Concepts & Definitions

• Histogram : partitions quantitative values into equal-width class intervals and places a bar over each interval whose height represents its frequency or relative frequency.

★ Must-know

📌 Bar charts and pie charts are adapted for qualitative variables, whereas quantitative data are better represented with summaries or displays such as stem-and-leaf plots and histograms.

Further detail

• A frequency table lists each category and its number of observations, and it may also show percentages.

• A Pareto diagram is a bar graph whose categories are arranged by decreasing height from left to right.

• For the European teenage-smoking data, the bin frequencies for [10–20), [20–30), [30–40), and [40–50) were 6, 15, 12, and 3, respectively. — European School Survey Project on Alcohol and Other Drugs, 2007

💡 Memory Hook

Bars have gaps; histograms do not

📖 4. Center, Spread, and Distribution Shape

🔑 Key Concepts & Definitions

• Mean : The sample mean is the sum of all observations divided by the number of observations, written as $\bar{x}=\frac{\sum_{i=1}^{n}x_i}{n}$, and it is affected by extreme values.
• Median : the middle value in an ordered sequence, or the average of the two middle values when the number of observations is even.
• Standard deviation : a measure of dispersion that describes variation around the mean, with larger values indicating more spread-out data.
• Interquartile range : The interquartile range is the difference between the third and first quartiles, $\mathrm{IQR}=Q_3-Q_1$, and describes the spread of the middle 50% of the data.

★ Must-know

📐 Formula — The range equals the largest observation minus the smallest observation: $\mathrm{Range}=x_{\mathrm{largest}}-x_{\mathrm{smallest}}$.

📌 When a distribution is not unimodal or symmetric or contains outliers, gaps, or clusters, the median and IQR are generally preferred to the mean and standard deviation.

📌 For a bell-shaped, unimodal, symmetric distribution, the empirical rule states that approximately 68%, 95%, and 99.7% of observations lie within one, two, and three standard deviations of the mean, respectively.

📐 Formula — A sample z-score measures the number of sample standard deviations a value lies from the sample mean: $z=\frac{x-\bar{x}}{s}$.

Further detail

📐 Formula — The coefficient of variation compares standard deviation with the mean: $\mathrm{CV}=\frac{s}{\bar{x}}$.

💡 Memory Hook

Mean and SD are sensitive; median and IQR are resistant

📖 5. Contingency Tables and Association

🔑 Key Concepts & Definitions

• Contingency table : organizes counts for combinations of two categorical variables, with one variable in rows and the other in columns.
• Independence : when the distribution of one variable is the same for every category of the other variable.
• Cramer’s V : a standardized association coefficient derived from chi-squared that ranges from 0 for no association to 1 for perfect association.
• Simpson’s paradox : occurs when a relationship observed within several groups disappears after the groups are combined according to a third, lurking variable.

📝 Essential Points

📐 Formula — Under independence, the expected count in row i and column j is $E_{ij}=\frac{R_iC_j}{n}$, where Ri is the row total, Cj is the column total, and n is the sample size.

📐 Formula — The chi-squared statistic compares observed and expected counts: $\chi^2=\sum_i\sum_j\frac{(O_{ij}-E_{ij})^2}{E_{ij}}$.

💡 Memory Hook

Condition on a variable → compare distributions → assess independence

📖 6. Correlation and Linear Regression

🔑 Key Concepts & Definitions

• Scatterplot : represents each case as a point (Xi,Yi) to visualize the association between two quantitative variables, with X as the explanatory variable and Y as the response variable.
• Correlation coefficient : measures the direction and strength of a linear association between two quantitative variables and ranges from −1 to 1.

★ Must-know

📌 Correlation does not prove causation because an association may result from coincidence, causality, or a common underlying cause called a lurking variable.

📐 Formula — A linear regression model has the form $\hat{y}=b_0+b_1x$, where b0 is the y-intercept and b1 is the slope.

📐 Formula — A residual is the observed response minus the predicted response: $e=y-\hat{y}$.

📌 Interpolation predicts within the domain used to construct a model, whereas extrapolation predicts outside that domain and requires caution.

• The coefficient of determination is the squared correlation, $r^2$, and represents the percentage of variation in Y accounted for by the linear regression model.

Further detail

📌 The least-squares regression line is the unique line that minimizes the sum of squared residuals.

📌 A useful residual plot should show roughly equal scatter across the range, no bends, and few or no outliers.

💡 Memory Hook

Scatterplot → correlation → regression → residuals

📖 7. Regression Fit and Residuals

🔑 Key Concepts & Definitions

• Standard error of regression : measures how much the y-values vary around the fitted line and is also called the root mean squared error

★ Must-know

📐 Formula — The coefficient of determination is $r^2=\frac{\text{variation accounted for by the model}}{\text{total variation}}$.

• For the manatee-deaths and powerboat-registrations example, $r^2=0.895$, so the model accounts for 89.5% of the variation in manatee deaths and leaves 10.5% unexplained.

• A useful residual plot should have:

  • roughly constant horizontal scatter
  • no bends
  • no or very few outliers

Further detail

• In the manatee example, the standard error is 7.663 manatees killed, so the empirical rule implies that about 95% of deaths lie within 15.3 manatees of the regression line.

💡 Memory Hook

Better fit → smaller residual variation → larger r²

📖 8. Probability Rules and Conditioning

🔑 Key Concepts & Definitions

• Law of large numbers : states that, as a random trial is repeated, the proportion of times an event occurs approaches a single long-run value called its empirical probability

📝 Essential Points

📐 Formula — For an event A, the empirical probability is $P(A)=\frac{\#\text{ times A occurs}}{\#\text{ trials}}$.

📐 Formula — For any event A, the complement rule is $P(A^C)=1-P(A)$.

📐 Formula — For disjoint events A and B, the addition rule is $P(A\text{ or }B)=P(A)+P(B)$.

📐 Formula — For any two events A and B, the general addition rule is $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.

📐 Formula — The conditional probability of B given A is $P(B\mid A)=\frac{P(A\cap B)}{P(A)}$, provided that $P(A)\ne0$.

📐 Formula — For any two events A and B, the general multiplication rule is $P(A\cap B)=P(A)P(B\mid A)=P(B)P(A\mid B)$.

📌 Events A and B are independent when $P(B\mid A)=P(B)$, equivalently when $P(A\cap B)=P(A)P(B)$.

💡 Memory Hook

Disjoint events cannot occur together; independent events do not influence one another

📖 9. Discrete Random Variables

🔑 Key Concepts & Definitions

• Random variable : assigns numerical values according to the outcomes of a random event

📝 Essential Points

📌 A discrete random variable has a countable number of distinct outcomes, whereas a continuous random variable can take any numerical value within a range.

📌 A probability model must assign nonnegative probabilities to all values and satisfy $p(x)\ge0$ and $\sum p(x)=1$.

📐 Formula — The expected value of a discrete random variable is $\mu=E(X)=\sum x\,p(x)$.

📐 Formula — The variance and standard deviation of a random variable are $\sigma^2=\sum(x-\mu)^2p(x)$ and $\sigma=\sqrt{\sigma^2}$.

💡 Memory Hook

Model → expected value → variance → standard deviation

📖 10. Binomial and Poisson Models

🔑 Key Concepts & Definitions

• Bernoulli trials : Bernoulli trials have only success and failure outcomes, fixed success probability p and failure probability q=1-p, and independent successive trials.
• Poisson distribution : The Poisson distribution models the number of rare events occurring in a measurement interval such as time, length, or area and is a special case of the binomial model when $n\to\infty$ and $p\to0$.

★ Must-know

📐 Formula — For a binomial random variable with n trials and success probability p, the expected value is $\mu=np$ and the standard deviation is $\sigma=\sqrt{npq}$, where $q=1-p$.

Further detail

📐 Formula — The number of combinations of r successes among n trials is $\binom nr=\frac{n!}{r!(n-r)!}$.

• In the typing-errors example, a transaction of 255 words corresponds to 3.4 minutes and has Poisson mean 0.34 errors per transaction.

💡 Memory Hook

Binomial counts successes in fixed trials; Poisson counts rare events in an interval

📖 11. Continuous and Normal Models

🔑 Key Concepts & Definitions

• Probability density function : represents the probability that a continuous random variable falls within an interval by the area under its curve over that interval
• Normal model : A Normal model describes a continuous distribution that is approximately unimodal and symmetric with mean μ and standard deviation σ, written N(μ,σ).

📝 Essential Points

📌 The 68-95-99.7 rule states that approximately 68%, 95%, and 99.7% of Normal-model values lie within one, two, and three standard deviations of the mean, respectively.

📐 Formula — A value is standardized with the z-score formula $z=\frac{x-\mu}{\sigma}$, converting it to the standard Normal model N(0,1).

📌 The Normal approximation to a binomial model is appropriate when the expected numbers of successes and failures satisfy $np\ge15$ and $nq\ge15$.

💡 Memory Hook

A bell-shaped curve with probability represented by shaded area

📖 12. Sampling Distributions

🔑 Key Concepts & Definitions

• Sampling distribution : is the probability distribution of that statistic across all such samples

★ Must-know

📌 A population parameter describes an entire population, whereas a sample statistic describes a sample and is calculated from its observations.

📌 The Central Limit Theorem states that the sampling distribution of a random sample mean becomes approximately Normal as sample size increases, usually with n≥30 being sufficient depending on the original distribution.

Further detail

• For a sample of 200 college women with low BMI, the reported mean weight is 140 pounds, while the population parameters for all 18-year-old women are μ=143.74 pounds and σ=51.54 pounds.

💡 Memory Hook

Larger samples → smaller standard error → more nearly normal sample means

📖 13. Sampling Distribution of the Mean

★ Must-know

📐 Formula — For a quantitative variable with population mean μ and standard deviation σ, the sampling distribution of the mean has mean $E(\bar{x})=\mu$ and standard deviation $SD(\bar{x})=\frac{\sigma}{\sqrt{n}}$, called the standard error.

📌 If the population distribution is normal, the sampling distribution of the mean is normal; even when the population is not normal, it tends toward normality as n increases.

📌 The Central Limit Theorem states that the mean of a random sample of size n can be approximated by a normal model, with the approximation improving as n increases and usually being adequate when n≥30.

Further detail

• For normally distributed heights with mean 172 cm and standard deviation 10 cm, the sampling distribution of means from samples of size 50 has standard deviation $10/\sqrt{50}\approx1.41$ cm, and the middle 95% of sample means is approximately 169.2 to 174.8 cm.

💡 Memory Hook

Larger samples → smaller standard error → more precise sample means

📖 14. Sampling Distribution of Proportions

🔑 Key Concepts & Definitions

• Sampling distribution for the proportion : the distribution of the proportions obtained from all samples of size n

★ Must-know

📐 Formula — For a qualitative variable with population proportion p, the sampling distribution of the proportion has mean $E(\hat{p})=p$ and standard deviation $SD(\hat{p})=\sqrt{\frac{p(1-p)}{n}}$, called the standard error.

📌 The sampling distribution of a proportion is approximately normal when $np\ge15$ and $n(1-p)\ge15$.

Further detail

• When 52% of voters support a budget and n=400, the sampling distribution of the sample proportion is approximately normal with mean 0.52 and standard deviation 0.025.

💡 Memory Hook

Means describe quantitative variables, whereas proportions describe qualitative properties

📖 15. Point and Interval Estimation

🔑 Key Concepts & Definitions

• Target parameter : the unknown population parameter, such as a mean or proportion, that we want to estimate
• Point estimator : a rule or formula that uses sample data to calculate a single number as an estimate of a population parameter
• Interval estimator : or confidence interval, uses sample data to calculate a range of values estimating the target parameter while accounting for sample-to-sample variation

📝 Essential Points

📐 Formula — The general confidence-interval formula is $\text{Point Estimate}\pm(\text{Critical Value})(\text{Standard Error})$.

📌 A 95% confidence level means that 95% of confidence intervals constructed from all possible random samples by the same method would contain the true population parameter.

💡 Memory Hook

Point estimates give one value, whereas confidence intervals give a range with uncertainty

📖 16. Confidence Intervals for Means

★ Must-know

📐 Formula — When σ is known, a confidence interval for a population mean is $\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$.

📌 A z-interval for a population mean requires a random sample, known population standard deviation σ, and a large sample size such as n≥30 to support an approximately normal sampling distribution.

📐 Formula — When σ is unknown, a confidence interval for a population mean is $\bar{x}\pm t_{\alpha/2}\frac{s}{\sqrt{n}}$, with degrees of freedom $df=n-1$.

Further detail

📌 The Student t distribution is symmetric and bell-shaped like the normal distribution but has heavier tails, and it approaches the normal distribution as sample size increases.

💡 Memory Hook

Known σ uses z, whereas unknown σ uses the Student t distribution

📖 17. Confidence Intervals for Proportions

🔑 Key Concepts & Definitions

• Margin of error : the amount added to and subtracted from the point estimate to form a confidence interval, representing estimation imprecision

📝 Essential Points

📐 Formula — A confidence interval for a population proportion is $\hat{p}\pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.

📌 A confidence interval for a population proportion requires a random or representative sample and sufficiently large counts satisfying $n\hat{p}\ge15$ and $n(1-\hat{p})\ge15$.

📌 Increasing the confidence level requires a larger interval, which increases certainty but decreases precision.

💡 Memory Hook

Higher confidence → larger margin of error → lower precision

📖 18. One-Sample Hypothesis Testing

🔑 Key Concepts & Definitions

• Null hypothesis : is the population-parameter claim tested in a hypothesis test, usually representing a status quo or equality to a hypothesized value
• Alternative hypothesis : is the population-parameter claim opposite to the null hypothesis and is supported only when the sample provides convincing evidence against H₀
• Chi-squared test of independence : tests whether two categorical variables are independent in the population from which a sample is drawn by comparing observed and expected contingency-table counts

📝 Essential Points

📌 Rejecting H₀ provides evidence supporting Hₐ, whereas failing to reject H₀ means only that there is insufficient evidence to support Hₐ.

• A Type I error occurs when a true null hypothesis is rejected, and its probability is the significance level α; a Type II error occurs when a false null hypothesis is not rejected, and its probability is β.

• 🔄 A hypothesis test proceeds by: stating H₀ and Hₐ, checking conditions, calculating a test statistic, finding critical values or a p-value, concluding by rejecting or failing to reject H₀ in context

📌 The p-value is the conditional probability of obtaining a test statistic at least as extreme as the observed one, assuming H₀ is true; reject H₀ when p-value<α and fail to reject H₀ when p-value>α.

📌 Valid large-sample inference for the difference between two population means requires two independent random samples and sample sizes of at least 30 in both populations.

📐 Formula — The large-sample confidence interval for the difference between two population means is $\bar{x}_1-\bar{x}_2\pm z_{\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}$.

📐 Formula — The large-sample test statistic for two population means is approximately $z=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1-\mu_2)}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$.

📌 Valid small-sample inference for the difference between two population means requires independent random samples, approximately normal populations, and equal population variances.

📐 Formula — The pooled variance for two independent small samples is $s_p^2=\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}$.

📐 Formula — The small-sample confidence interval for two population means is $\bar{x}_1-\bar{x}_2\pm t_{\alpha/2}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}$ with $n_1+n_2-2$ degrees of freedom.

📌 Valid large-sample inference for the difference between two population proportions requires independent random samples and, for each sample, at least 15 expected successes and 15 expected failures based on the sample proportion.

📐 Formula — The large-sample confidence interval for two population proportions is $\hat{p}_1-\hat{p}_2\pm z_{\alpha/2}\sqrt{\frac{\hat{p}_1\hat{q}_1}{n_1}+\frac{\hat{p}_2\hat{q}_2}{n_2}}$.

📐 Formula — For testing equality of two population proportions, the pooled proportion is $\hat{p}=\frac{x_1+x_2}{n_1+n_2}$ and the test statistic is $z=\frac{\hat{p}_1-\hat{p}_2-(p_1-p_2)}{\sqrt{\hat{p}\hat{q}\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}$, where $\hat{q}=1-\hat{p}$.

💡 Memory Hook

Hypotheses → conditions → statistic → critical value or p-value → conclusion

📊 Synthesis Tables

Measures of Center and Spread

Measure	What it describes	Sensitivity to outliers
Mean	Balance point of the data	Sensitive
Median	Middle of ordered data	Resistant
Range	Maximum minus minimum	Sensitive
Standard deviation	Spread around the mean	Sensitive
Interquartile range	Spread of the middle 50%	Resistant

Discrete and continuous random variables

Feature	Discrete	Continuous
Possible values	Countable distinct outcomes	Any numerical value in an interval
Probability focus	Probability of individual values	Probability over intervals
Typical examples	Number of successes or errors	Height, distance, or time