Statistics and Data Collection Quiz

Chapter One: Introduction to Statistics

• Statistics is the science of collecting, classifying, presenting, and interpreting data.
• All facets of modern life are affected by statistics.
• Statistics is essential in many academic disciplines, including the sciences, engineering, business, political science, economics, psychology, sociology, education, medicine, nursing, and other health-related areas.
• Accurate description, making decisions and estimations are enabled via using statistical methods correctly, hence the "art" of statistics is important.

Introduction of Basic Terms

• Population: A collection of all individuals or objects of interest. Populations can be finite or infinite.
  • Example of a finite population: All students enrolled in year two at a university.
  • Example of an infinite population: The stars in the sky.
• Sample: A subset of a population.
  • Example of a sample: A collection of cars from a specific faculty car park.
• Variable: A characteristic of interest about each element in a population or sample.
  • Example of variables: Matriculation number, year, department.

Data

• Data: Raw facts and unprocessed information.
• Types of Data:
  • Quantitative/Numerical: Representing measurable values (e.g., weight, height).
    • Discrete: Values that can only take on certain specific values (e.g., number of students in a class).
    • Continuous: Values able to assume any value within a range(e.g., weight of weight lifters).
  • Qualitative/Non-Numerical: Representing categories or attributes (e.g., hair color).
    • Ordinal: Categories with a natural order (e.g., age group).
    • Categorical: Categories without a natural order (e.g., gender, country)

Data Collection

• Primary data: Collected directly from respondents or from the source.
• Secondary data: Collected from existing data banks.
• Data Collection Steps:
  • Defining the survey objectives
  • Defining the target population
  • Defining the strategy and method to be used for data collection and measurement.
  • Ascertaining data analysis methods to use (descriptive or inferential)

Chapter Two: Summary and Display of Data

• Large data sets are difficult to interpret.
• Frequency distributions condense data into manageable classes
• Relative frequency distribution shows each class's proportion of the total data.
• Cumulative frequency distribution presents cumulative frequencies for each class.
• Various graphical representations: Dot plots, bar charts, histograms, and pie charts.

Chapter Three: Descriptive Analysis

• Measures of Location (Central Tendency):
  • Mean: The arithmetic average of all values.
  • Median: The middle value when ranked in order.
  • Mode: The value that appears most frequently.
• Measures of Dispersion (Spread):
  • Range: Difference between maximum and minimum values.
  • Variance: Measures the average squared difference from the mean.
  • Standard Deviation: The square root of the variance.
• Sigma (Σ) notation: Used for summation in mathematics

Chapter Four: Introduction to Probability

• Experiment: Any process that yields a result or observation.
• Outcome: A particular result of an experiment.
• Sample Space: The set of all possible outcomes of an experiment.
• Sample point: An individual outcome in the sample space
• Event: A subset (or combination) of the sample space.
• Probability(P(A)): The likelihood of an event A occurring. Calculated as the number of favorable outcomes divided by the total possible outcomes.

Chapter Five: Discrete Probability Distribution

• Random variable (x): A variable whose value is determined by a probability experiment.

• Probability distribution: A listing of possible outcomes of a probability experiment along with their probabilities.

  • Mean of a probability distribution: Average of all possible outcomes. (mu)=∑(xP(x))
  • Variance of a probability distribution: Measures how spread out the probability distribution is. σ2 = ∑(x−μ)²P(x)
  • Standard deviation of a probability distribution

• Binomial distribution: The probability that an event will occur a specific number of times in a fixed number of independent trials.

• Bernoulli distribution: Represents outcomes (success or failure) of a single experiment, or binary outcome such as toss of a coin, or any Bernoulli trial.

• Poisson distribution: Represents the number of events happening in a fixed interval of time or space. These occurrences are discrete.

Chapter Six: The Normal Distribution

• The normal distribution is a continuous probability distribution that is symmetric and bell-shaped.
• Standard Normal Distribution: Used when the population mean and standard deviation are known.
• Z-scores: Standard scores that measure how many standard deviations a data point lies from the mean.
• Normal Approximation to Binomial: When 'n' is large, the binomial distribution can be approximated by the normal distribution. The approximation is accurate if np ≥ 5 and n(1-p) ≥ 5.

Chapter Seven: Design and Analysis of Sampling

• Sampling: The process of selecting a subset of a population to learn about the entire population.

• Different Sampling Methods:

  • Simple Random Sampling: Every element in the population has an equal chance of being selected.
  • Systematic Random Sampling: Every kth element in the population is selected after a random start.
  • Stratified Random Sampling: Divides the population into strata and then takes random samples from each stratum.
  • Cluster Random Sampling: The whole group (cluster) selected is used to represent the population.
  • Quota sampling: Selecting items that are judged to be 'typical' of the population.

• Sampling distribution of means: The distribution of all possible sample means that can be drawn from a population.

• Standard error of the mean (se): Indicates how variable the means from the different samples are from the population mean. seX = σ/√n (population std dev/√sample size)

Chapter Eight: Confidence Interval Estimation

• Point estimate: A single value estimate of a population parameter.
• Confidence interval: A range of values within which the population parameter is likely to fall.
• Confidence level (1 − α): The probability that the confidence interval contains the true population parameter value.
• Confidence limits: The upper and lower bounds of the confidence interval.
• Use Z or t statistical tables as well.

Chapter Nine: Confidence Interval Estimation involving Two Populations

• Independent Samples: Samples in which observations from one group are independent of the observations from a second group.
• Dependent Samples: Samples are taken from a paired data set; data values from one group are connected to the corresponding values in another group.

Chapter Ten: Test of Statistical Hypotheses

• Null Hypothesis (Ho): A statement of no effect or no difference.
• Alternative Hypothesis (H1): A statement that there is an effect or a difference.
• Level of significance (α): The probability of rejecting a true null hypothesis.
  • One-tailed test: The alternative hypothesis specifies a direction (e.g., greater than, less than).
  • Two-tailed test: The alternative hypothesis does not specify a direction (e.g., not equal to).
• Critical Value: The borderline value that determines whether to reject or fail to reject Ho; dependent on the level of significance and the type of test (One-tailed or Two-tailed).
• Test statistic: Used to compare the observed sample results to the null hypothesis
• P-value: The probability of observing a test statistic as extreme or more extreme than the calculated value, given that Ho is true.

Chapter Eleven: Correlation and Regression Analysis

• Correlation analysis: Used to determine whether there is a linear relationship between two quantitative variables.
• Correlation coefficient (r): A numerical measure of the strength and direction of a linear relationship between two variables. Ranges from -1 to 1.
• Regression analysis: Determines the nature of relationship and the extent of dependencies that exist between two variables.
• Regression line: A line that represents the best fit to a scatter plot and can be used to predict values of one variable based on the values of the other variable.
• Standard error of estimate (se): Measures the average variability of individual values around the regression line.

Chapter Twelve: Time Series Analysis

• Time series: A sequence of observations measured over a period of time.

• Secular trend: Long-term movement in the time series

  • Trend Line: Linear equation determining the trend.

• Seasonal variation: A component of a time series representing patterns that repeat over periods of time (e.g., days, weeks, months, or years).

• Cyclical variation: Repeating upswings and downswings over a period

  • Cycle chart: Used to illustrate cyclical components

• Irregular variation: Random and irregular components that do not reveal any pattern, e.g. unforeseen occurrences or events.

Chapter Thirteen: Index Numbers

• Index numbers are statistical measures showing the changes in quantities such as prices, wages, production etc., over a period of time.
• Simple price relatives: Indicate the relative change from a base period.
• Simple quantity relatives: Show the relative changes in the quantities associated with a given period.
• Weighted aggregate price index: Method for averaging price relatives to obtain an aggregate price index.
• Paasche's index and Laspeyres' index: Formulas for deriving aggregate price indexes.
• Consumer price index (CPI): An economic indicator to show inflation in the cost of living from one period to another.