Q1. What would it take to convince you the fortune teller has a real, special ability, rather than just guessing?

Background

Topic: Statistical Evidence and Probability

This question is testing your understanding of how to distinguish between random chance and evidence of a special ability using statistical reasoning.

Key Terms:

• Statistical evidence: Information that supports a claim based on data and probability.

• Random guessing: Making predictions with no special knowledge, where each outcome is equally likely.

• Convincing evidence: Results that are unlikely to occur by random chance alone.

Step-by-Step Guidance

1. Consider what would be expected if the fortune teller was just guessing. For a fair coin, each guess has a 50% chance of being correct.

2. Think about how many correct guesses would be surprising compared to what is expected by chance.

3. Reflect on whether a result is so unusual that it suggests something more than random guessing is happening.

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Q2. How many correct guesses would you expect the fortune teller to get if they were just randomly guessing heads or tails (no real ability)? Explain.

Background

Topic: Expected Value in Probability

This question is testing your ability to calculate the expected number of successes in a binomial experiment.

Key Terms and Formula:

• Expected value: The average outcome you would expect over many trials.

• Binomial probability: Used when there are two possible outcomes (success/failure).

• = number of trials (coin tosses)

• = probability of a correct guess (for a fair coin, )

Step-by-Step Guidance

1. Identify the number of coin tosses: .

2. Determine the probability of a correct guess: .

3. Set up the expected value formula: .

4. Multiply the values to find the expected number of correct guesses.

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Q3. Based on the results, do we have some evidence that something more than random guessing is going on?

Background

Topic: Statistical Significance and Probability

This question is testing your ability to interpret whether observed results are unusual compared to what is expected by chance.

Key Terms:

• Statistical significance: When results are unlikely to occur by random chance.

• Probability: The likelihood of an event occurring.

Step-by-Step Guidance

1. Compare the observed number of correct guesses (14 out of 17) to the expected value (about 8.5).

2. Consider how likely it is to get 14 or more correct guesses by random chance.

3. Think about whether this result is rare enough to suggest something other than random guessing.

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Q4. How many correct guesses out of 17 would the fortune teller have to get to convince you that something more than random guessing is going on?

Background

Topic: Statistical Thresholds and Probability

This question is testing your understanding of how to set a threshold for statistical significance.

Key Terms:

• Threshold: The cutoff point where results are considered unusual.

• Statistical significance: When results are unlikely to occur by random chance.

Step-by-Step Guidance

1. Recall the expected number of correct guesses (about 8.5).

2. Think about how many correct guesses would be so rare that it would be unlikely to occur by chance.

3. Consider using a probability calculation (such as the binomial distribution) to find the cutoff.

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Q5. What does each dot represent in the dotplot of correct guesses?

Background

Topic: Data Visualization – Dotplots

This question is testing your understanding of how dotplots represent data.

Key Terms:

• Dotplot: A graphical display of data using dots.

• Each dot: Represents one observation or result.

Step-by-Step Guidance

1. Look at the dotplot and identify what is being measured (number of correct guesses).

2. Understand that each dot corresponds to a single trial or participant's result.

3. Think about how the dots show the distribution of results across the class.

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Q6. How is the dotplot from the random sample (#5) different than the dotplot for the convenience sample (#4)? Which do you think is a better estimator of the true average word length?

Background

Topic: Sampling Methods and Estimation

This question is testing your understanding of how sampling methods affect the accuracy of estimates.

Key Terms:

• Convenience sample: A sample chosen based on ease of access, not randomness.

• Random sample: A sample chosen so every item has an equal chance of being selected.

• Estimator: A statistic used to estimate a population parameter.

Step-by-Step Guidance

1. Compare the spread and clustering of dots in both dotplots.

2. Consider whether the random sample is more likely to represent the true average word length.

3. Think about why random sampling reduces bias compared to convenience sampling.

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Q7. What is the difference between voluntary response and nonresponse?

Background

Topic: Survey Bias

This question is testing your understanding of different types of bias in survey sampling.

Key Terms:

• Voluntary response: When participants choose to respond, often leading to bias.

• Nonresponse: When selected participants do not respond, potentially leading to bias.

Step-by-Step Guidance

1. Define voluntary response and nonresponse.

2. Think about how each type of bias affects the results of a survey.

3. Consider examples of each type from the scenarios provided.

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