A researcher divides a city into neighborhoods, randomly selects several neighborhoods, and surveys everyone living in them. This is which sampling method?
A) Stratified
B) Systematic
C) Cluster
D) Simple random
E) Convenience
Solution:
Whole pre-existing groups (neighborhoods) are randomly chosen, then fully sampled.
Sampling all members of selected groups is cluster sampling.
\(\boxed{\text{Cluster}}\)
Problem 14
What is the primary purpose of random assignment of treatments in an experiment?
A) Increase the sample size
B) Eliminate all variability
C) Create bias on purpose
D) Reduce the effect of confounding variables
E) Guarantee a normal distribution
Solution:
Random assignment balances lurking variables across treatment groups.
This lets us attribute differences to the treatment, not to confounders.
\(\boxed{\text{Reduce the effect of confounding variables}}\)
Problem 15
A website posts a poll and uses only the responses of people who chose to answer. This is most likely to produce which kind of bias?
A) Undercoverage bias
B) Response bias
C) Nonresponse bias
D) Question-wording bias
E) Voluntary response bias
Solution:
People who self-select into a poll tend to have stronger opinions.
A self-selected sample produces voluntary response bias.
\(\boxed{\text{Voluntary response bias}}\)
Problem 16
Events \(A\) and \(B\) are mutually exclusive with \(P(A) = 0.3\) and \(P(B) = 0.4\). Find \(P(A \text{ or } B)\).
A) 0.7
B) 0.12
C) 0.1
D) 1.2
E) 0.5
Solution:
Mutually exclusive means \(P(A \text{ and } B) = 0\).
\(P(A \text{ or } B) = P(A) + P(B) = 0.3 + 0.4\).
\(= \boxed{0.7}\)
Problem 17
Given \(P(A \text{ and } B) = 0.2\) and \(P(B) = 0.5\), find the conditional probability \(P(A \mid B)\).
A) 0.1
B) 0.4
C) 0.7
D) 0.25
E) 2.5
Solution:
\(P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)}\).
\(= \dfrac{0.2}{0.5}\).
\(= \boxed{0.4}\)
Problem 18
A random variable \(X\) takes the values \(0, 1, 2\) with probabilities \(0.5, 0.3, 0.2\). Find \(E(X)\).
A) 1.0
B) 1.5
C) 0.7
D) 0.5
E) 0.3
Solution:
\(E(X) = \sum x \cdot P(x)\).
\(= 0(0.5) + 1(0.3) + 2(0.2) = 0 + 0.3 + 0.4\).
\(= \boxed{0.7}\)
Problem 19
For a binomial distribution with \(n = 10\) and \(p = 0.5\), find the mean.
A) 10
B) 2.5
C) 25
D) 5
E) 1
Solution:
Binomial mean \(= np\).
\(= 10 \times 0.5\).
\(= \boxed{5}\)
Problem 20
For a binomial distribution with \(n = 100\) and \(p = 0.5\), find the standard deviation.
A) 25
B) 50
C) 2.5
D) 10
E) 5
Solution:
Binomial SD \(= \sqrt{np(1-p)}\).
\(= \sqrt{100 \cdot 0.5 \cdot 0.5} = \sqrt{25}\).
\(= \boxed{5}\)
Problem 21
A fair coin is flipped 3 times. Find the probability of getting exactly 2 heads.
A) 0.375
B) 0.125
C) 0.25
D) 0.5
E) 0.75
Solution:
\(P(X = 2) = \binom{3}{2}(0.5)^2(0.5)^1\).
\(= 3 \cdot 0.25 \cdot 0.5 = \dfrac{3}{8}\).
\(= \boxed{0.375}\)
Problem 22
Independent events have \(P(A) = 0.6\) and \(P(B) = 0.5\). Find \(P(A \text{ and } B)\).
A) 1.1
B) 0.3
C) 0.6
D) 0.1
E) 0.5
Solution:
For independent events, \(P(A \text{ and } B) = P(A)\cdot P(B)\).
\(= 0.6 \times 0.5\).
\(= \boxed{0.3}\)
Problem 23
Independent random variables have \(\text{Var}(X) = 4\) and \(\text{Var}(Y) = 9\). Find \(\text{Var}(X + Y)\).
A) 5
B) 25
C) 13
D) 6.5
E) \(\sqrt{13}\)
Solution:
For independent variables, variances add: \(\text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y)\).
\(= 4 + 9\).
\(= \boxed{13}\)
Problem 24
In a geometric setting, each trial succeeds with probability \(p = 0.25\). What is the expected number of trials until the first success?
A) 0.25
B) 2
C) 16
D) 4
E) 1
Solution:
For a geometric distribution, the mean is \(\dfrac{1}{p}\).
\(= \dfrac{1}{0.25}\).
\(= \boxed{4}\)
Problem 25
A population has standard deviation \(\sigma = 10\). For samples of size \(n = 25\), find the standard deviation of the sampling distribution of the sample mean (the standard error).
A) 10
B) 0.4
C) 50
D) 5
E) 2
Solution:
Standard error \(= \dfrac{\sigma}{\sqrt{n}}\).
\(= \dfrac{10}{\sqrt{25}} = \dfrac{10}{5}\).
\(= \boxed{2}\)
Problem 26
A population proportion is \(p = 0.4\). What is the mean of the sampling distribution of the sample proportion \(\hat{p}\)?
A) 0.4
B) 0.6
C) 0.2
D) 0.16
E) 0.5
Solution:
The sampling distribution of \(\hat{p}\) is centered at the true proportion.
\(\mu_{\hat{p}} = p = 0.4\).
\(= \boxed{0.4}\)
Problem 27
By the Central Limit Theorem, for a large sample size the sampling distribution of the sample mean is approximately what shape, regardless of the population's shape?
A) Skewed right
B) Approximately normal
C) Uniform
D) Bimodal
E) Skewed left
Solution:
The Central Limit Theorem says that as \(n\) grows, \(\bar{x}\) becomes approximately normal.
This holds even when the population distribution is not normal.
\(\boxed{\text{Approximately normal}}\)
Problem 28
What is the critical value \(z^*\) used for a 95% confidence interval for a proportion?
A) 1.645
B) 2.576
C) 1.96
D) 1.0
E) 2.0
Solution:
A 95% confidence level leaves 2.5% in each tail of the standard normal curve.
The \(z\)-value with 0.025 in the upper tail is \(1.96\).
\(z^* = \boxed{1.96}\)
Problem 29
Test scores are normally distributed with mean 100 and standard deviation 15. Using the empirical rule, about what percent of scores fall between 70 and 130?
A) 68%
B) 99.7%
C) 50%
D) 95%
E) 34%
Solution:
\(70 = 100 - 2(15)\) and \(130 = 100 + 2(15)\), so this is \(\mu \pm 2\sigma\).
By the empirical rule, about 95% of data lies within 2 standard deviations.
\(= \boxed{95\%}\)
Problem 30
A hypothesis test yields a \(p\)-value of 0.03. At significance level \(\alpha = 0.05\), what is the correct decision?
A) Fail to reject \(H_0\)
B) Accept \(H_0\)
C) The result is insignificant
D) Increase \(\alpha\)
E) Reject \(H_0\)
Solution:
Compare the \(p\)-value to \(\alpha\): \(0.03 < 0.05\).
When the \(p\)-value is less than \(\alpha\), there is significant evidence against \(H_0\).