STA2023 Practice Midterm 1: Step-by-Step Statistics Study Guide

Q1. Nutritionist Diet Study: Identifying Population, Sample, Individuals, Parameter, and Statistic

Background

Topic: Populations, Samples, Parameters, and Statistics

This question tests your understanding of basic statistical terminology and the ability to distinguish between the population, sample, individuals, parameter, and statistic in a study context.

Key Terms:

• Population: The entire group of individuals or items of interest.

• Sample: A subset of the population, selected for study.

• Individuals: The objects or people described by the data.

• Parameter: A numerical summary describing a characteristic of the population.

• Statistic: A numerical summary describing a characteristic of the sample.

Step-by-Step Guidance

1. Read the scenario carefully and identify who or what the study is about (the broadest group = population).

2. Determine which subset of the population was actually measured (this is the sample).

3. Identify the individual units (people, objects) that data was collected from.

4. Find the value that describes the population (parameter) and the value that describes the sample (statistic).

Try solving on your own before revealing the answer!

Q2. Classifying Variables: Qualitative/Quantitative, Discrete/Continuous, Level of Measurement

Background

Topic: Types of Variables and Levels of Measurement

This question tests your ability to classify variables as qualitative or quantitative, and if quantitative, as discrete or continuous, and to identify the level of measurement (nominal, ordinal, interval, ratio).

Key Terms:

• Qualitative (Categorical): Describes qualities or categories.

• Quantitative: Numerical values; can be discrete (countable) or continuous (measurable).

• Levels of Measurement: Nominal, Ordinal, Interval, Ratio.

Step-by-Step Guidance

1. For each variable, ask: Is it a category or a number? (Qualitative vs. Quantitative)

2. If quantitative, can it take on any value (continuous) or only certain values (discrete)?

3. Determine the level of measurement:

   • Nominal: Categories only, no order.

   • Ordinal: Categories with order.

   • Interval: Numerical, no true zero.

   • Ratio: Numerical, true zero exists.

Try classifying each variable before checking the answer!

Q3. Observational Study vs. Experiment; Explanatory and Response Variables; Causation

Background

Topic: Types of Studies and Variables

This question tests your understanding of the difference between observational studies and experiments, and your ability to identify explanatory and response variables, as well as whether causation can be inferred.

Key Terms:

• Observational Study: Researchers observe without intervention.

• Experiment: Researchers apply a treatment/intervention.

• Explanatory Variable: The variable that is manipulated or categorized to see its effect.

• Response Variable: The outcome measured.

• Causation: Whether a cause-effect relationship can be claimed.

Step-by-Step Guidance

1. Determine if the researchers assigned treatments or just observed existing conditions.

2. Identify which variable is being used to explain or predict (explanatory) and which is the outcome (response).

3. Recall that causation can only be claimed in well-designed experiments, not observational studies.

Try identifying the variables and study type before checking the answer!

Q4. Identifying Types of Bias in Surveys

Background

Topic: Types of Bias in Sampling

This question tests your ability to recognize different types of bias that can occur in survey sampling: sampling bias, nonresponse bias, and response bias.

Key Terms:

• Sampling Bias: When the sample is not representative of the population.

• Nonresponse Bias: When a significant portion of the sample does not respond.

• Response Bias: When respondents give inaccurate or untruthful answers.

Step-by-Step Guidance

1. Read each scenario and ask: Is the issue with who was sampled, who responded, or how they responded?

2. Match the scenario to the correct type of bias based on the definitions above.

Try matching each scenario to the bias type before checking the answer!

Q5. Frequency Table: Relative Frequencies and Degree Measures

Background

Topic: Frequency Distributions and Pie Charts

This question tests your ability to calculate relative frequencies and degree measures for a frequency table, and to interpret categorical data.

Key Formulas:

• Relative Frequency: $\text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total}}$

• Degree Measure: $\text{Degree} = \text{Relative Frequency} \times 360^\circ$

Step-by-Step Guidance

1. Find the total number of households by summing all frequencies.

2. For each activity, calculate the relative frequency using the formula above.

3. For each activity, calculate the degree measure for a pie chart using the formula above.

4. Interpret the table to answer which activity is most popular and what percent prefer Reading.

5. For the True/False question, add the frequencies for Sports and Movies and compare to half the total.

Try filling in the missing values and answering the questions before checking the answer!

Q6. Frequency Distribution: Class Limits, Class Width, Percentages, and Mode

Background

Topic: Grouped Frequency Distributions

This question tests your ability to interpret a frequency distribution table, including identifying class limits, class width, calculating percentages, and finding the modal class.

Key Terms and Formulas:

• Class Limits: The smallest and largest data values that can belong to a class.

• Class Width: $\text{Class Width} = \text{Upper Limit of a Class} - \text{Lower Limit of the Same Class} + 1$ (or difference between lower limits of consecutive classes).

• Percentage: $\text{Percent} = \frac{\text{Frequency}}{\text{Total}} \times 100\%$

• Modal Class: The class with the highest frequency.

Step-by-Step Guidance

1. Identify the lower and upper class limits for the specified class.

2. Calculate the class width using the formula above.

3. To find the percent scoring above a certain value, sum the frequencies for those classes and divide by the total.

4. To find how many scored below a certain value, sum the frequencies for those classes.

5. Identify the class with the highest frequency.

Try working through each part before checking the answer!

Q7. Identifying Distribution Shapes

Background

Topic: Shapes of Distributions

This question tests your ability to recognize the likely shape of a distribution (bell-shaped, uniform, skewed left, skewed right) based on context.

Key Terms:

• Bell-shaped: Symmetrical, most data near the center.

• Uniform: All outcomes equally likely.

• Skewed Left: Tail on the left (few low values).

• Skewed Right: Tail on the right (few high values).

Step-by-Step Guidance

1. Consider the context: Are there a few extreme high or low values?

2. Recall typical distributions for common variables (e.g., salaries, heights, dice rolls, waiting times).

3. Match the scenario to the most likely distribution shape.

Try matching each scenario to a distribution shape before checking the answer!

Q8. Daily Commute Times: Population Mean, Variance, Standard Deviation, Median, Mode, Shape

Background

Topic: Descriptive Statistics for Populations

This question tests your ability to compute the population mean, variance, standard deviation, median, mode, and to describe the shape of the distribution.

Key Formulas:

• Population Mean: $\mu = \frac{\sum x_i}{N}$

• Population Variance: $\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$

• Population Standard Deviation: $\sigma = \sqrt{\sigma^2}$

• Median: Middle value when data is ordered.

• Mode: Most frequent value.

Step-by-Step Guidance

1. Sum all commute times and divide by the number of employees to find the mean.

2. Subtract the mean from each value, square the result, sum these squares, and divide by the population size for variance.

3. Take the square root of the variance for standard deviation.

4. Order the data and find the median (middle value or average of two middle values).

5. Identify the mode (if any value repeats).

6. Compare mean and median to describe the shape (e.g., symmetric, skewed).

Try calculating each statistic before checking the answer!

Q9. Sample Mean and Sample Variance for First 5 Commute Times

Background

Topic: Descriptive Statistics for Samples

This question tests your ability to compute the sample mean and sample variance for a subset of data.

Key Formulas:

• Sample Mean: $\bar{x} = \frac{\sum x_i}{n}$

• Sample Variance: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$

Step-by-Step Guidance

1. Sum the sample values and divide by the sample size to find the mean.

2. Subtract the sample mean from each value, square the result, sum these squares, and divide by $n-1$ for the variance.

Try calculating the mean and variance before checking the answer!

Q10. Five-Number Summary, IQR, and Outliers for Commute Times

Background

Topic: Five-Number Summary and Outlier Detection

This question tests your ability to find the five-number summary, calculate the interquartile range (IQR), and determine outliers using fences.

Key Formulas:

• Five-number summary: Minimum, Q1, Median, Q3, Maximum

• IQR: $\text{IQR} = Q_3 - Q_1$

• Lower Fence: $Q_1 - 1.5 \times \text{IQR}$

• Upper Fence: $Q_3 + 1.5 \times \text{IQR}$

Step-by-Step Guidance

1. Order the data and find the minimum, Q1 (25th percentile), median, Q3 (75th percentile), and maximum.

2. Calculate the IQR using the formula above.

3. Compute the lower and upper fences to check for outliers.

4. Identify any data points outside the fences as outliers.

Try finding the five-number summary and outliers before checking the answer!

Q11. Empirical Rule (68–95–99.7 Rule) for Commute Times

Background

Topic: Empirical Rule for Normal Distributions

This question tests your ability to apply the Empirical Rule to interpret the spread of data in a normal distribution using the mean and standard deviation.

Key Formulas:

• 68% of data within $\mu \pm 1\sigma$

• 95% of data within $\mu \pm 2\sigma$

• 99.7% of data within $\mu \pm 3\sigma$

Step-by-Step Guidance

1. Use the mean and standard deviation from Q8.

2. Calculate $\mu \pm 2\sigma$ for the 95% interval.

3. For specific intervals, determine how many standard deviations from the mean the endpoints are, and use the Empirical Rule to estimate the percentage.

Try applying the Empirical Rule before checking the answer!