Unit 2 practice test: Probability, Random Variables, and Probability Distributions

Each value is a statistic: the proportion of supporters in one random sample of 100 voters.

• The opinion of one voter, support or not, in a single sample: One voter gives a yes or no; a sample of 100 gives a proportion.
• The proportion of supporters among the city's voters, 0.40: That is the parameter, one fixed number.
• The number of samples that had 40 supporters out of 100: The bar heights count samples; each value is a proportion.
• Correct: The proportion of supporters in one random sample of 100 voters: Right: one sample, one $\hat{p}$.

The values are centered near 0.40 (their mean is about 0.401), the population proportion: the sampling distribution of $\hat{p}$ is centered at p.

• Correct: Near 0.40, the proportion of all the city's voters who support the tax: Right: $\mu_{\hat{p}} = p$.
• Near 0.50, since voters either support the tax or they do not: Two possible answers do not make each one half as likely.
• Near 100, the number of voters in each sample: The values are proportions between 0 and 1.
• Near 0.60, the proportion of voters who do not support it: 0.60 is the other group; the histogram shows supporters.

Same center, 0.40, but less spread: four times the sample size cuts the spread of $\hat{p}$ in half, from about 0.049 to about 0.024. Larger random samples give statistics closer to the parameter.

• Same center, but four times as spread out: More voters per sample make $\hat{p}$ less variable, not more.
• Correct: Same center, but about half as spread out: Right: spread shrinks like $\frac{1}{\sqrt{n}}$, and $\sqrt{4} = 2$.
• The center would move from 0.40 to 0.10: The center stays at the population proportion, 0.40.
• It would not change, since the city has the same voters: The population is the same, but the sample size changes the spread.

A parameter describes the population (0.40 of all the city's voters). A statistic describes a sample ($\hat{p}$ for each sample of 100), and it varies from sample to sample.

• 0.40 is a statistic; each $\hat{p}$ is a parameter: That is reversed: the 0.40 describes every voter in the city.
• Both are statistics, since both are proportions: Being a proportion does not decide it; population or sample does.
• Correct: 0.40 is a parameter; each $\hat{p}$ is a statistic: Right: population number, then sample numbers.
• Both are parameters, since they describe the same voters: Each $\hat{p}$ describes only its own sample of 100.

V is 2, 4, 6 and F is 5: no outcome is in both. V and G share 4 and 6; F and G share 5.

• V and G: They share 4 and 6, so they can happen together.
• F and G: A 5 is greater than 3, so F and G happen together on a 5.
• G and "the roll is 6": A 6 is in G, so they overlap.
• Correct: V and F: Right: an even roll can never be a 5.

V and F are mutually exclusive, so the probabilities add: $\frac{3}{6} + \frac{1}{6} = \frac{4}{6}$ (the rolls 2, 4, 5, 6).

• $\frac{3}{6}$: That is the even rolls alone, without the 5.
• Correct: $\frac{4}{6}$: Right: 2, 4, 5 and 6.
• $\frac{1}{6}$: That is the 5 alone.
• $\frac{3}{36}$: That multiplies the probabilities, which does not give the chance of one or the other.

V and G are not mutually exclusive: 4 and 6 are in both, so adding counts them twice. Listing the outcomes, 2, 4, 5, 6, gives $\frac{4}{6}$.

• Nothing went wrong, since each event has probability one half: A probability of 1 would mean a 1 or a 3 could never be rolled, which is false.
• The student should have multiplied the probabilities to get one fourth: Multiplying does not give the chance of one event or the other.
• Correct: The events overlap at 4 and 6, so those rolls were counted twice: Right: adding works only for mutually exclusive events.
• The die has 6 faces, so the answer should have been divided by 6: The fractions already use 6 as the denominator.