MATH&146 Lesson 7: Simulating One Categorical Variable

Flashcards covering key concepts from MATH&146 Lesson 7 on simulating one categorical variable, including statistical inference, proportion calculations, random number assignments, StatKey usage, and probability examples.

What is statistical inference?

Statistical inference means drawing conclusions about a population parameter based on sample data.

According to this diagram, what is the full process cycle connecting a population to sample statistics and back?

Sampling is used to obtain sample statistics from a population with parameters, and statistical inference is used to draw conclusions about the population parameters from those sample statistics.

Why is the proportion the primary parameter focused on in Lesson 7?

Proportions are focused on due to their simplicity and ubiquity (appearing everywhere), making them easier to calculate and understand than other parameters.

How is a proportion defined and calculated?

A proportion is a measure of a part of a whole, calculated by dividing the number of items in the part by the total number of items in the whole.

Why are simulations used to estimate probabilities?

Simulations are used when an appropriate mathematical formula for a problem is not known or when no formula exists.

How can integer random numbers from $1$ to $10$ be assigned to simulate a $20\%$ chance of winning a cereal box prize?

Assign numbers $1$ and $2$ to represent winning a prize ($20\%$), and numbers $3, 4, 5, 6, 7, 8, 9, 10$ to represent not winning a prize ($80\%$).

In Mika's cereal box simulation, how many total random numbers and columns were generated in Random.org to run $20$ trials of $6$ boxes each?

$120$ random numbers formatted in $6$ columns (yielding $20$ sets of $6$ digits).

In the cereal box simulation results shown in this table, what was the estimated probability that Mika gets at least one prize?

$\frac{16}{20} = 80\%$ (because $16$ out of the $20$ sets included at least one prize digit of $1$ or $2$).

How can random integers from $1$ to $100$ be assigned to simulate an outcome with a $28\%$ probability of success?

Assign numbers $1$ through $28$ to represent success, and numbers $29$ through $100$ to represent failure.

In StatKey's 'Sampling Distribution for a Proportion' applet, how are the left-tail, right-tail, and two-tail options used?

Use left-tail for probabilities of sample values less than or equal to a testing value; right-tail for probabilities greater than or equal to a testing value; and two-tail for probabilities between two values.

If approximately $70\%$ of statistics students complete their homework on time, how many students in a class of $30$ are expected to complete their homework on time?

$21$ students ($30 \times 0.70 = 21$).

In a sample of $50$ frogs where a genetic trait normally occurs in $1\text{ out of }8$ frogs, what sample proportion corresponds to finding the trait in at most $5$ frogs?

A sample proportion of $\le \frac{5}{50} = 0.10$ (or $10\%$).

If $30\%$ of entering university students drop out, what sample proportion corresponds to at most $500$ students dropping out in an entering class of $1800$?

A sample proportion of $\le \frac{500}{1800} \approx 0.2778$ (or $27.78\%$).

What is the simulated probability of passing a $20$-question True-False exam with at least $70\%$ ($14$ correct answers) by randomly guessing?

$0.0470$ (or $4.70\%$).

What is the simulated probability of passing a $50$-question True-False exam with at least $70\%$ by randomly guessing?

$0.0030$ (or $0.30\%$).

What is the simulated probability of passing a $20$-question multiple-choice exam (with $1$ correct and $3$ incorrect options per question) with at least $70\%$ by randomly guessing?

$0.0000$ (or $0\%$).

Between Shelly ($50$ coin flips) and Diane ($10$ coin flips), who is more likely to get $20\%$ or fewer heads, and why?

Diane is more likely to get $20\%$ or fewer heads (simulated proportion $0.054$ vs. $0.0000$ for Shelly) because smaller sample sizes exhibit greater variability from the expected proportion of $0.50$.