STA2023 Practice Midterm 1: Step-by-Step Statistics Guidance

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 possible).

2. Determine which group was actually measured or observed in the study (this is the sample).

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

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

Try solving on your own before revealing the answer!

Q2. Classifying Variables: Type, Discrete/Continuous, and Level of Measurement

Background

Topic: Types of Variables and Levels of Measurement

This question tests your ability to classify variables as qualitative or quantitative, further distinguish quantitative variables as discrete or continuous, and 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: Categories with no order.

  • Ordinal: Categories with a meaningful order.

  • Interval: Ordered, equal intervals, no true zero.

  • Ratio: Ordered, equal intervals, true zero.

Step-by-Step Guidance

1. For each variable, ask: Is it describing a quality (category) or a quantity (number)?

2. If quantitative, determine if it can take on any value (continuous) or only specific values (discrete).

3. Decide the level of measurement by considering if the data has order, equal intervals, and a true zero.

Try classifying each variable before checking the answer!

Q3. Observational Study vs. Experiment, Explanatory and Response Variables, and 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 and observe effects.

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

• Response Variable: The outcome measured in the study.

• Causation: Whether a cause-and-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 changes (explanatory), and which is the outcome (response).

3. Consider whether the study design allows for causal conclusions.

Try identifying each part 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 affect survey results: 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. For each scenario, consider how the data collection process could lead to unrepresentative or inaccurate results.

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

Try matching each scenario to the correct bias 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 interpret categorical data.

Key Formulas:

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

• Degree Measure (for pie chart): $\text{Degree Measure} = \text{Relative Frequency} \times 360^\circ$

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.

4. Use the table to answer questions about the most popular activity and percentages.

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

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

Background

Topic: Frequency Distributions and Histograms

This question tests your ability to interpret a frequency table, identify class limits, calculate class width, and determine percentages and frequencies.

Key Terms and Formulas:

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

• Class Width: $\text{Class Width} = \text{Upper Class Limit} - \text{Lower Class Limit} + 1$ (or, for equal-width classes, $\text{Class Width} = \text{Lower Limit of Next Class} - \text{Lower Limit of Current Class}$)

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

Step-by-Step Guidance

1. Identify the lower and upper class limits for the specified class by looking at the table.

2. Calculate the class width using the formula above.

3. Sum the frequencies for the relevant classes to answer questions about percentages and counts.

4. Identify the class with the highest frequency by comparing the values.

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 (Normal): 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 and what the data represent.

2. Think about whether the data are likely to be symmetric, have a long tail, or be uniform.

3. Match the scenario to the appropriate 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, and Shape

Background

Topic: Descriptive Statistics (Measures of Center and Spread)

This question tests your ability to compute the mean, variance, standard deviation, median, and mode for a population, and to interpret 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 are ordered.

• Mode: Most frequent value(s).

Step-by-Step Guidance

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

2. Subtract the mean from each data point, square the result, sum these squares, and divide by the population size to find the variance.

3. Take the square root of the variance to find the standard deviation.

4. Order the data and find the median (middle value).

5. Identify the mode by looking for repeated values.

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

Try calculating each measure before checking the answer!

Q9. Sample Mean and Sample Variance (First 5 Commute Times)

Background

Topic: Sample Statistics

This question tests your ability to compute the mean and variance for a sample (not the whole population).

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. Add up the first five commute times and divide by 5 to find the sample mean.

2. Subtract the sample mean from each value, square the differences, sum them, and divide by (n-1) to find the sample variance.

Try calculating the sample 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 minimum, first quartile (Q1), median, third quartile (Q3), maximum, calculate the interquartile range (IQR), and use fences to identify outliers.

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, median, Q3, and maximum.

2. Calculate the IQR using the formula above.

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

4. Compare each data point to the fences to identify any 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: Normal Distribution and Empirical Rule

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:

• Empirical Rule: For a normal distribution:

  • About 68% of data within 1 standard deviation of the mean

  • About 95% within 2 standard deviations

  • About 99.7% within 3 standard deviations

• Interval Calculation: $\text{Interval} = \mu \pm k\sigma$ (where $k$ is 1, 2, or 3)

Step-by-Step Guidance

1. Use the mean and standard deviation from earlier to calculate the intervals for 68%, 95%, and 99.7% of the data.

2. For specific intervals, determine how many standard deviations from the mean the endpoints are.

3. Use the Empirical Rule to estimate the percentage of data within those intervals.

Try applying the Empirical Rule before checking the answer!