Definitions:

Population: the whole group Sample: part of the group Parameter: prediction about the population. Statistic: numerical summary of a sample. Categorical (Qualitative) Data: category or classification. Quantitative Data: number that you can add or subtract and it makes sense. Hint: Categorical is usually words and Quantitative is usually numbers with a few exceptions such as: zip codes cannot be added or subtracted. They are a classification of where you live. Level of Measurement:
  1. Categorical options:
    1. Nominal: has no order.
    2. Ordinal: can be put in order.
  2. Quantitative options:
    1. Interval: the value 0 does exist in context.  Example: 0 degrees is a real temperature.
    2. Ratio:  the value 0 does not exist in context.  Example: baby weight of 0 lbs means there is no baby.
Discrete: you can count and finite Continuous: you can measure and infinite Types of Sampling:
  • Simple Random Sample: equal chance of being selected 
  • Stratified Random Sample: split into groups and randomly select from each group 
  • Cluster Random Sample: split into groups and randomly pick the entire group(s)
  • Systematic Random Sample: example: pick every 3rd person 
  • Convenience Sampling: easily accessible, not reliable. 
Note: can sample with replacement or without replacement

ALEKS Topics:

Topic 1: Differentiating between parameters and statistics

Problem 1: In a study of a town’s population, the average age of all residents is 40 years, but a sample of 100 residents has an average age of 42 years. Identify which value is a parameter and which is a statistic, and explain why. Answer: The average age of all residents (40 years) is a parameter because it describes the entire population. The average age of the sample (42 years) is a statistic because it describes a subset of the population. A parameter is a fixed value for the whole population, while a statistic is calculated from a sample. Problem 2: A company claims the mean salary of all employees is $50,000, while a survey of 200 employees yields a mean salary of $48,500. Determine which is a parameter and which is a statistic, and justify your answer. Answer: The mean salary of all employees ($50,000) is a parameter because it represents the entire employee population. The mean salary of the 200 employees ($48,500) is a statistic because it is derived from a sample. Parameters apply to populations, while statistics apply to samples.

Topic 2: Classifying samples

Problem 1: A researcher surveys 50 students from each grade level in a school to study study habits. Classify this sample as random, stratified, systematic, or convenience, and explain your reasoning. Answer: This is a stratified sample. The researcher divides the population into strata (grade levels) and selects a sample from each stratum (50 students per grade). This ensures representation from each group, unlike a random sample (equal chance for all) or convenience sample (easiest to access). Problem 2: A pollster interviews every 10th person entering a mall to gather opinions on a product. Classify this sample as random, stratified, systematic, or convenience, and provide justification. Answer: This is a systematic sample. The pollster selects every 10th person, following a fixed interval, which distinguishes it from a random sample (no pattern) or convenience sample (non-systematic selection). It is not stratified since no subgroups are defined.

Topic 3: Classification of variables

Problem 1: Classify the variable "number of cars owned by a household" as qualitative or quantitative, and explain why. Answer: The variable is quantitative because it represents a numerical count (e.g., 0, 1, 2 cars) that can be measured or compared mathematically, unlike qualitative variables, which describe categories or qualities (e.g., color). Problem 2: Determine if the variable "favorite music genre" is qualitative or quantitative, and justify your classification. Answer: The variable is qualitative because it describes a category or preference (e.g., rock, jazz) that cannot be quantified numerically, unlike quantitative variables, which involve measurable numerical values.

Topic 4: Discrete versus continuous variables

Problem 1: Is the variable "number of students in a classroom" discrete or continuous? Explain why it fits the chosen category. Answer: The variable is discrete because it takes on specific, countable values (e.g., 20, 25 students) with no intermediate values possible. Continuous variables, like height, can take any value within a range. Problem 2: Classify the variable "height of a tree" as discrete or continuous, and provide reasoning for your answer. Answer: The variable is continuous because tree height can take any value within a range (e.g., 5.2 m, 5.25 m), including fractions, unlike discrete variables, which are limited to distinct, countable values.

Topic 5: Choosing units of measurement and an appropriate method to gather data

Problem 1: To measure the length of fish in a lake, choose an appropriate unit (e.g., centimeters or meters) and suggest a method (e.g., observation, survey, or experiment) to collect the data. Explain your choices. Answer: Unit: Centimeters, as fish lengths are typically small and centimeters provide precise measurements. Method: Observation, by catching and measuring fish directly. This is appropriate because it provides accurate data for length, unlike surveys (subjective) or experiments (manipulating variables). Problem 2: For studying the weight of apples in an orchard, select a suitable unit (e.g., grams or kilograms) and a data collection method. Justify your selections. Answer: Unit: Grams, as apples have relatively small weights, and grams allow for precise measurements. Method: Observation, by weighing apples directly with a scale. This ensures accurate weight data, unlike surveys (less reliable) or experiments (unnecessary manipulation).

Topic 6: Choosing an appropriate method to conduct a survey and making an estimation

Problem 1: To estimate the percentage of city residents who support a new park, choose a survey method (e.g., online, phone, or in-person) and estimate the sample size needed for accuracy. Explain your choices. Answer: Method: Online survey, as it reaches a broad audience efficiently and is cost-effective. Sample size: Approximately 400, based on standard statistical practice for a 95% confidence level with a 5% margin of error for a moderate population size. This ensures a representative sample without excessive cost. Problem 2: For estimating the average time students spend on homework, select a survey method and propose a reasonable sample size. Provide reasoning for your decisions. Answer: Method: In-person survey, as it allows direct interaction with students for accurate responses. Sample size: About 200 students, sufficient for a 95% confidence level with a reasonable margin of error for a school-sized population. This balances accuracy and practicality.

Topic 7: Identifying and reducing statistical bias

Problem 1: A survey on exercise habits is conducted only at a gym. Identify the type of bias (e.g., selection bias) and suggest a way to reduce it, such as expanding the sample to non-gym-goers. Answer: Bias: Selection bias, as the sample only includes gym-goers, who are likely more active, skewing results. Reduction: Expand the survey to include non-gym-goers, such as through online or community surveys, to represent a broader population with varied exercise habits. Problem 2: A poll on a new policy is conducted only during weekday mornings at a business district. Identify the bias and propose a method to reduce it, such as including evening or weekend surveys. Answer: Bias: Selection bias, as the poll only captures business district workers during weekday mornings, excluding others like evening workers or residents. Reduction: Conduct the poll at varied times (evenings, weekends) and locations (e.g., residential areas) to include a more diverse sample.

Activity:

Do you like football?