Descriptive Statistics & Distribution
A data set is highly right-skewed. Which is greater: the mean or the median? Explain why.
Mean, because it is pulled toward the long tail (higher values)
What does a z-score represent in context?
Number of standard deviations a value is from the mean
Two independent events have probabilities 0.5 and 0.3. What is the probability both occur?
0.15
What is the difference between a population and a sample?
Population = entire group; sample = subset
What is the only continent that has no deserts?
Europe
Two classes have the same mean test score. One has a much larger standard deviation. What does this tell you about the distributions?
One class has more variability/spread; scores are less consistent
A value has a z-score of -2. What does this indicate about its position?
It is 2 standard deviations below the mean
A card is drawn from a standard deck. What is the probability it is a heart or a king?
16/52 (13 hearts + 4 kings − 1 overlap)
Why is random sampling important in studies?
Reduces bias and increases representativeness
Which planet in our solar system has the most moons?
Saturn
A data set has a mean of 80 and a standard deviation of 5. Approximately what percent of data falls between 75 and 85?
About 68%
In a normal distribution, what percent of values lie beyond ±2 standard deviations?
About 5% (roughly 2.5% in each tail)
A game pays $100 with probability 0.2 and $0 otherwise. What is the expected value?
$20
A survey is conducted via social media. What type of bias may occur?
Voluntary response bias
What is the hardest natural substance on Earth?
Diamond
You add an extreme outlier to a data set. Which statistics are most affected: mean, median, IQR?
Mean is most affected
Student A scored 85 (mean = 80, SD = 5). Student B scored 90 (mean = 70, SD = 10). Who performed better relative to their group?
Student B performed better
A game works like this: You flip a fair coin. If it lands heads, you win $20. If tails, you flip again—if the second flip is heads, you win $10; otherwise you win $0. What is the expected value of the game? Should you play if it costs $8?
EV = (0.5 × 20) + (0.5 × 0.5 × 10) + (0.5 × 0.5 × 0)
EV = 10 + 2.5 + 0 = $12.50 → Yes, positive EV (+$4.50 profit)
A 95% confidence interval for a mean is (70, 80). Interpret this correctly.
We are 95% confident the true population mean lies between 70 and 80
Which NFL team has won the most Super Bowl titles?
Pittsburgh Steelers and New England Patriots (tied with 6 each)
Two data sets have identical means and ranges but very different standard deviations. What does this imply about how the data is distributed?
One is more clustered around the mean, the other more spread out
Why are z-scores useful when comparing values from different distributions?
They standardize values, allowing fair comparison across different scales
A student claims a game is “fair” because you win 1 out of every 4 times. The prize is $40 and it costs $15 to play. Is the game actually fair? Justify using expected value.
EV = (0.25 × 40) + (0.75 × 0) = 10
Net EV = 10 − 15 = −$5 → Not fair, you lose money long-term
A 95% confidence interval for the average amount of time students spend on homework is (2.5 hours, 3.5 hours). A student claims, “95% of students spend between 2.5 and 3.5 hours on homework.”
Is this statement correct? If not, explain the correct interpretation.
No, this is incorrect. The correct interpretation is that we are 95% confident the true population mean lies between 2.5 and 3.5 hours—not that 95% of individual students fall in that range.
Which artist became the first to win Album of the Year at the Grammy Awards four times?
Taylor Swift