Q1. A data set has mean 40. If you subtract 40 from every value and add up the results, you get:

Background

Topic: Properties of the Mean

This question tests your understanding of how the mean relates to the sum of deviations from the mean.

Key Terms and Formulas

• Mean (): The average value of a data set.

• Deviation: for each data value .

• Property: The sum of deviations from the mean is always zero: .

Step-by-Step Guidance

1. Let the data set have values: with mean .

2. Subtract 40 from each value: for each .

3. Add up all these results: .

4. Recall that , so .

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Q2. Which pair of measures are both resistant to extreme values?

Background

Topic: Measures of Center and Spread

This question tests your knowledge of which statistics are resistant (not affected much by outliers).

Key Terms and Formulas

• Resistant measure: A statistic that is not greatly influenced by extreme values (outliers).

• Median: The middle value; resistant.

• Interquartile Range (IQR): ; resistant.

• Mean, standard deviation, range, and variance are not resistant.

Step-by-Step Guidance

1. Recall which measures are resistant: median and IQR are resistant; mean and standard deviation are not.

2. Review each answer choice to see which pair includes only resistant measures.

3. Eliminate choices that include mean, standard deviation, range, or variance.

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Q3. Two populations are measured in the same units. The population with the larger standard deviation:

Background

Topic: Standard Deviation and Dispersion

This question tests your understanding of what standard deviation measures about a data set.

Key Terms and Formulas

• Standard deviation ( or ): Measures the typical distance of data values from the mean.

• Dispersion: The spread of data values.

Step-by-Step Guidance

1. Recall that a larger standard deviation means data are more spread out from the mean.

2. Review the answer choices for the one that describes greater spread or variability.

3. Eliminate choices that refer to the mean or median, which are not directly related to standard deviation.

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Q4. According to the Empirical Rule, about what percentage of data in a bell-shaped distribution lie within 3 standard deviations of the mean?

Background

Topic: Empirical Rule (68-95-99.7 Rule)

This question tests your knowledge of the Empirical Rule for normal (bell-shaped) distributions.

Key Terms and Formulas

• Empirical Rule: For bell-shaped distributions:

  • Within 1 SD: about 68%

  • Within 2 SD: about 95%

  • Within 3 SD: about 99.7%

Step-by-Step Guidance

1. Recall the Empirical Rule percentages for 1, 2, and 3 standard deviations from the mean.

2. Match the correct percentage to the range "within 3 standard deviations".

3. Eliminate answer choices that do not match the Empirical Rule values.

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Q5. An observation has a z-score of 2.3. This means the observation is:

Background

Topic: Z-scores and Standardized Values

This question tests your understanding of what a z-score represents in terms of standard deviations from the mean.

Key Terms and Formulas

• Z-score: (for population) or (for sample)

• Interpretation: The number of standard deviations a value is above or below the mean.

Step-by-Step Guidance

1. Recall that a positive z-score means the value is above the mean.

2. The magnitude of the z-score tells you how many standard deviations away from the mean the value is.

3. Review the answer choices for the one that matches this interpretation.

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Q6. Two variables are positively associated when:

Background

Topic: Association and Correlation

This question tests your understanding of positive association between two variables.

Key Terms and Formulas

• Positive association: As one variable increases, the other tends to increase as well.

• Correlation coefficient (): Measures the strength and direction of a linear relationship.

Step-by-Step Guidance

1. Recall the definition of positive association.

2. Review each answer choice for the one that describes both variables increasing together.

3. Eliminate choices that describe negative association or non-linear relationships.

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Q7. In a boxplot, the median is left of the center of the box and the right whisker is much longer than the left whisker. The distribution is:

Background

Topic: Boxplots and Skewness

This question tests your ability to interpret the shape of a distribution from a boxplot.

Key Terms and Formulas

• Boxplot: Visual summary of the five-number summary.

• Skewed right: Longer right whisker, median left of center.

• Skewed left: Longer left whisker, median right of center.

Step-by-Step Guidance

1. Recall that the direction of skewness matches the longer whisker.

2. If the median is left of center and the right whisker is longer, the distribution is skewed right.

3. Eliminate choices that do not match this description.

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Q8. A distribution's shape is unknown. Which tool can you still use to describe the percentage of data within k standard deviations of the mean?

Background

Topic: Chebyshev's Inequality vs. Empirical Rule

This question tests your understanding of which rule applies to all distributions, regardless of shape.

Key Terms and Formulas

• Empirical Rule: Applies only to bell-shaped (normal) distributions.

• Chebyshev's Inequality: Applies to any distribution, regardless of shape.

• Chebyshev's formula: At least of data within standard deviations of the mean, for .

Step-by-Step Guidance

1. Recall that the Empirical Rule requires a bell-shaped distribution.

2. Chebyshev's Inequality works for any distribution shape.

3. Identify which answer choice refers to Chebyshev's Inequality.

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Q9. Which pair of variables would most likely have a negative correlation?

Background

Topic: Correlation Direction

This question tests your ability to identify pairs of variables that move in opposite directions.

Key Terms and Formulas

• Negative correlation: As one variable increases, the other decreases.

• Examples: Temperature and heating bill (as temperature rises, heating bill falls).

Step-by-Step Guidance

1. Review each pair and consider whether an increase in one variable is associated with a decrease in the other.

2. Eliminate pairs that are positively correlated or unrelated.

3. Identify the pair where the relationship is most likely negative.

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Q10. Which value cannot be a linear correlation coefficient?

Background

Topic: Correlation Coefficient Range

This question tests your knowledge of the possible values for the correlation coefficient .

Key Terms and Formulas

• Correlation coefficient (): Measures linear association; must be between -1 and 1.

• Possible values: .

Step-by-Step Guidance

1. Recall the allowed range for .

2. Check each answer choice to see if it falls within to $1$.

3. Identify the value that is outside this range.

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Q11. Which list is ordered from weakest to strongest linear association?

Background

Topic: Interpreting Correlation Coefficient Magnitude

This question tests your understanding of how the magnitude of relates to the strength of association, regardless of sign.

Key Terms and Formulas

• Strength of association: Determined by (absolute value of ).

• Weaker association: closer to 0; stronger: $|r|$ closer to 1.

Step-by-Step Guidance

1. Take the absolute value of each in the list.

2. Order the values from smallest (weakest) to largest $|r|$ (strongest).

3. Match the answer choice that follows this order.

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Q12. A scatter diagram shows points forming a clear U-shape, and r = 0.02. Which conclusion is correct?

Background

Topic: Linear vs. Nonlinear Relationships

This question tests your understanding of what the correlation coefficient measures and its limitations.

Key Terms and Formulas

• Correlation coefficient (): Measures strength of linear relationship only.

• Nonlinear relationship: may be near zero even if a strong nonlinear pattern exists.

Step-by-Step Guidance

1. Recall that only measures linear relationships.

2. A U-shape is a strong nonlinear pattern, but will be close to zero.

3. Identify the answer that recognizes a nonlinear relationship despite being near zero.

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Q13. For a sample of n = 6, r = −0.92 and the critical value is 0.811. What should you conclude?

Background

Topic: Testing Significance of Correlation

This question tests your ability to compare the correlation coefficient to a critical value to determine significance.

Key Terms and Formulas

• Critical value: Threshold for significance; compare to the critical value.

• If critical value, the correlation is significant.

• Sign of indicates direction (negative or positive).

Step-by-Step Guidance

1. Calculate .

2. Compare to the critical value (0.811).

3. If is greater, a significant linear relation exists; the sign tells you the direction.

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Q14. In a contingency table, a marginal distribution is:

Background

Topic: Contingency Tables and Marginal Distributions

This question tests your understanding of marginal vs. conditional distributions in two-way tables.

Key Terms and Formulas

• Marginal distribution: Distribution of totals for one variable, ignoring the other variable.

• Conditional distribution: Distribution of one variable for a fixed value of the other variable.

Step-by-Step Guidance

1. Recall that marginal distributions are found in the margins (totals) of the table.

2. Conditional distributions are for a specific value of the other variable.

3. Identify the answer that describes a frequency or relative frequency for a row or column variable alone.

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