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 ($\bar{x}$): The average value of a data set.

• Deviation: $x_i - \bar{x}$ for each data value $x_i$.

• Property: $\sum (x_i - \bar{x}) = 0$ for any data set.

Step-by-Step Guidance

1. Recall that the mean is the value such that the sum of all deviations from the mean is zero.

2. When you subtract the mean from each value, you are calculating each deviation: $x_i - 40$.

3. Add up all these deviations: $\sum (x_i - 40)$.

4. Think about what this sum represents and what property of the mean it illustrates.

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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:

• Resistant: A statistic is resistant if it is not greatly influenced by extreme values (outliers).

• Median: The middle value; resistant.

• IQR (Interquartile Range): The range of the middle 50% of data; resistant.

• Mean, Standard Deviation, Range, Variance: Not resistant.

Step-by-Step Guidance

1. Review which statistics are affected by outliers (mean, standard deviation, range, variance).

2. Recall which statistics are not affected by outliers (median, IQR).

3. Look for the answer choice that includes only resistant measures.

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

Background

Topic: Measures of Spread

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

Key Terms:

• Standard Deviation ($\sigma$ or $s$): Measures the average 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 values are more spread out from the mean.

2. Compare the answer choices to see which one describes greater spread or variability.

3. Be careful not to confuse standard deviation with mean or median.

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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 distributions.

Key Terms and Formulas:

• Empirical Rule: For bell-shaped (normal) distributions:

  • Within 1 SD: about 68%

  • Within 2 SD: about 95%

  • Within 3 SD: about 99.7%

Step-by-Step Guidance

1. Recall the percentages for 1, 2, and 3 standard deviations from the mean in a normal distribution.

2. Identify which percentage corresponds to within 3 standard deviations.

3. Match this value to the correct answer choice.

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

Background

Topic: Z-scores (Standard Scores)

This question tests your understanding of what a z-score represents.

Key Terms and Formulas:

• Z-score: $z = \frac{x - \mu}{\sigma}$

• It tells how many standard deviations a value is from the mean.

Step-by-Step Guidance

1. Recall the definition of a z-score and what its sign and magnitude mean.

2. Interpret a positive z-score (above the mean) and the value (2.3 standard deviations).

3. Match this interpretation to the correct answer choice.

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Q6. Two variables are positively associated when:

Background

Topic: Correlation and Association

This question tests your understanding of positive association between variables.

Key Terms:

• Positive Association: As one variable increases, the other tends to increase.

• Correlation Coefficient ($r$): Measures the strength and direction of a linear relationship.

Step-by-Step Guidance

1. Recall what it means for two variables to be positively associated.

2. Eliminate answer choices that describe negative or non-linear relationships.

3. Choose the option that best describes positive association.

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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:

• Median: The line inside the box.

• Whiskers: Lines extending from the box to the minimum and maximum (excluding outliers).

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

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

Step-by-Step Guidance

1. Recall how the position of the median and the length of the whiskers indicate skewness.

2. Match the description (median left, right whisker longer) to the type of skewness.

3. Eliminate options that do not fit the boxplot 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 rules apply to all distributions versus only normal distributions.

Key Terms:

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

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

Step-by-Step Guidance

1. Recall the conditions for using the Empirical Rule and Chebyshev's Inequality.

2. Identify which rule can be used when the distribution shape is unknown.

3. Choose the answer that correctly identifies this rule.

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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:

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

Step-by-Step Guidance

1. Consider each pair and think about whether an increase in one variable would likely cause a decrease in the other.

2. Eliminate pairs that are likely to be positively correlated or unrelated.

3. Choose the pair that best fits a negative correlation.

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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 $r$.

Key Terms:

• Correlation Coefficient ($r$): Must be between -1 and 1, inclusive.

Step-by-Step Guidance

1. Recall the range of possible values for $r$.

2. Check each answer choice to see if it falls within this range.

3. Identify the value that is not possible for $r$.

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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 $r$ relates to the strength of association, regardless of sign.

Key Terms:

• Strength: Determined by the absolute value of $r$.

• Weakest: $r$ closest to 0; Strongest: $r$ closest to -1 or 1.

Step-by-Step Guidance

1. Order the values by their absolute value, from smallest to largest.

2. Ignore the sign when considering strength; focus on how close the value is to 0 or 1.

3. Match the correct order to the answer choices.

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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: Interpreting Scatterplots and Correlation

This question tests your understanding of the difference between linear and nonlinear relationships and what $r$ measures.

Key Terms:

• Correlation Coefficient ($r$): Measures linear association only.

• Nonlinear Relationship: $r$ may be near zero even if a strong nonlinear pattern exists.

Step-by-Step Guidance

1. Recall that $r$ measures only linear relationships.

2. Consider what a U-shaped pattern means for linear correlation.

3. Choose the answer that correctly interprets a low $r$ with a clear nonlinear pattern.

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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 $|r|$ to the critical value.

• If $|r| >$ critical value, the correlation is significant.

Step-by-Step Guidance

1. Take the absolute value of $r$ and compare it to the critical value.

2. If $|r|$ is greater, a significant linear relation exists (regardless of sign).

3. Choose the answer that matches this conclusion.

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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 versus conditional distributions in two-way tables.

Key Terms:

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

• Conditional Distribution: Distribution for one variable, given a specific value of the other.

Step-by-Step Guidance

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

2. Identify which answer choice describes a marginal distribution.

3. Eliminate options that refer to conditional distributions or other concepts.

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