Basic Stats Exam 2

What is a random variable?

A variable that assigns a numerical value to each outcome of a chance process.

What is a discrete random variable?

A random variable with a finite or countable number of possible values.

What is a continuous random variable?

A random variable with infinitely many possible measured values.

What is a probability distribution?

A distribution that gives probabilities for the values of a random variable.

What conditions must a discrete probability distribution satisfy?

Each probability is between 0 and 1, and all probabilities sum to 1.

What is the mean of a probability distribution?

μ = E(X) = Σ[xP(x)]; the long-run average of X.

What is the variance of a discrete random variable?

σ² = Σ[(x − μ)²P(x)].

What is the standard deviation of a probability distribution?

σ = √σ².

What is the range rule of thumb?

Unusually low: x < μ − 2σ; unusually high: x > μ + 2σ.

What is the rare event rule?

If an observed result is unlikely under an assumption, the assumption may be questioned.

What are the conditions for a binomial setting?

Fixed n, independent trials, 2 outcomes, and constant probability of success.

What does n represent in a binomial distribution?

The fixed number of trials.

What does x represent in a binomial distribution?

The number of successes in the n trials.

What does p represent in a binomial distribution?

The probability of success on one trial.

What does q represent in a binomial distribution?

The probability of failure; q = 1 − p.

What is the binomial probability formula?

P(x) = [n!/((n-x)!x!)]*pˣqⁿ⁻ˣ.

What is the mean of a binomial distribution?

μ = np.

What is the variance of a binomial distribution?

σ² = npq.

What is the standard deviation of a binomial distribution?

σ = √npq.

What is a Poisson distribution used for?

Counting occurrences within a fixed interval.

What are Poisson events like?

Random, independent, and uniformly distributed within the interval.

What is the Poisson probability formula?

P(x) = e⁻ᵘμˣ/x!.

What are the mean and variance of a Poisson distribution?

The mean μ and variance σ² are both equal to μ.

What is a normal distribution?

A continuous, bell-shaped, symmetric distribution.

What are the mean and standard deviation of the standard normal distribution?

μ = 0 and σ = 1.

What is a z-score formula?

z = (x − μ)/σ.

What does a z-score tell you?

How many standard deviations a value is from the mean.

What does the area under a normal curve represent?

Probability.

What does a positive z-score mean?

The value is above the mean.

What does a negative z-score mean?

The value is below the mean.

What does a z-table usually give?

Cumulative area to the left of a z-score.

How do you find P(Z > z) using a z-table?

Subtract the left-tail area from 1.

How do you find P(a < Z < b)?

Area left of b minus area left of a.

How do you find a z-score from a given area?

Find the closest area in the z-table and read the corresponding z-score.

What is a critical value?

A z-score that marks the boundary of a specified tail area.

What is the empirical rule?

About 68%, 95%, and 99.7% lie within 1, 2, and 3 standard deviations.

What is standardization?

Converting an original value into a z-score.

How do you convert a z-score back to a raw value?

x = μ + zσ.

What is a sampling distribution?

The distribution of a statistic from all possible samples of the same size.

What is sampling variability?

The natural variation among statistics from different random samples.

What is the sampling distribution of x̄?

The distribution of sample means from all possible samples.

What happens to spread as sample size increases?

The sampling distribution becomes less spread out.

What does p represent?

The true population proportion.

What does p̂ represent?

The proportion observed in a random sample.

What is the center of the sampling distribution of p̂?

μp̂ = p.

What is an estimator?

A statistic used to estimate a population parameter.

What is an unbiased estimator?

An estimator whose sampling distribution is centered at the true parameter.

What is a biased estimator?

An estimator whose sampling distribution is not centered at the true parameter.

What does x̄ estimate?

The population mean μ.

What does p̂ estimate?

The population proportion p.

What does s estimate?

The population standard deviation σ.

What does s² estimate?

The population variance σ².

What does the Central Limit Theorem say?

For large n, the sampling distribution of x̄ becomes approximately normal.

What happens to the standard error as sample size increases?

It decreases.

What is the standard error of x̄?

SE = σ/√n, or s/√n when σ is unknown.

What is the standard error of p̂?

SE = √[p(1 − p)/n].

What happens to p̂ as sample size increases?

Its sampling distribution becomes more nearly normal.

When can the CLT be used for x̄?

When the population is normal or n is sufficiently large.

How do you assess normality?

Look for reasonable normality using histograms or Q-Q plots.

Does data need to be perfectly normal?

No, it only needs reasonable evidence of normality.

What should you do with outliers?

Investigate them before removing them.

What is a point estimate?

A single value used to estimate a population parameter.

What is a confidence interval?

A range of values used to estimate a population parameter.

What is the general confidence interval form?

Point estimate ± margin of error.

What is a confidence level?

The long-run success rate of the interval-building procedure.

What are common confidence levels?

90%, 95%, and 99%.

What is α for a confidence level?

α = 1 − confidence level.

What is α for a 95% confidence level?

α = 0.05.

Where is α placed for a two-sided confidence interval?

It is split equally between the two tails.

What happens to interval width when confidence level increases?

The interval becomes wider.

What happens to interval width when sample size increases?

The interval becomes narrower.

What happens to interval width when variability increases?

The interval becomes wider.

What is margin of error?

The maximum likely amount of error in estimating a parameter.

What is the margin of error for a population proportion?

E = z*√[p̂(1 − p̂)/n].

What are requirements for a confidence interval for p?

Random sample, independence, and at least 5 successes and 5 failures.

What is the population mean?

μ, the true mean of the entire population.

What is the sample mean?

x̄, the mean calculated from a sample.

What is the point estimate for μ?

The sample mean x̄.

What is the margin of error for μ when σ is known?

E = z*(σ/√n).

What is the margin of error for μ when σ is unknown?

E = t*(s/√n).

When do you use a t-distribution?

When the population standard deviation σ is unknown.

What are degrees of freedom for a one-sample t procedure?

df = n − 1.

Why does the t-distribution have heavier tails?

It accounts for uncertainty from estimating σ with s.

What happens to the t-distribution as df increases?

It approaches the standard normal distribution.

What are requirements for a confidence interval for μ?

Random sample, independence, and normal population or sufficiently large n.

What is the confidence interval for μ with known σ?

x̄ ± z*(σ/√n).

What is the confidence interval for μ with unknown σ?

x̄ ± t*(s/√n).

What is the sample size formula for estimating a proportion?

n = p̂(1 − p̂)(z*/E)².

What if no prior estimate of p is available?

Use p̂ = 0.50 for the largest required sample size.

How should a required sample size be rounded?

Always round up to the next whole number.

What information is needed to plan a sample size?

Confidence level, acceptable margin of error, and prior estimate of p.

What can be used for a prior estimate of p?

A previous comparable study or a pilot study.

Why should sample-size calculations be rounded up?

Rounding down could make the margin of error too large.

What is the main idea of inferential statistics?

Use sample information to make conclusions about a population.

What does 95% confidence mean?

The method captures the true parameter in about 95% of repeated samples.

What does a narrower confidence interval indicate?

Less uncertainty and a more precise estimate.

What does a wider confidence interval indicate?

More uncertainty and a less precise estimate.

What is the difference between a parameter and a statistic?

A parameter describes a population; a statistic describes a sample.

What does sampling with replacement help ensure?

Independence between observations.