AP Statistics Flashcards: Setting Up Tests For Population Mean

What this deck covers

This deck focuses on Setting Up Tests For Population Mean, giving you a quick way to review the definitions, rules, and examples that matter most for AP Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Identify the correct conclusion template when $p \le \alpha$ in a test of $H_0: \mu=\mu_0$.

Answer: Reject $H_0$; evidence supports $H_a$. Small $p$-value indicates data unlikely under $H_0$.

Flashcard 2: Compute the degrees of freedom for a one-sample $t$ test when $n = 18$.

Answer: $df = 17$. Apply $df = n - 1 = 18 - 1 = 17$.

Flashcard 3: Which alternative hypothesis matches a two-sided claim that $\mu$ differs from $\mu_0$?

Answer: $H_a: \mu \ne \mu_0$. Two-sided test uses $\ne$ for any difference from $\mu_0$.

Flashcard 4: What is the correct alternative hypothesis for a two-sided test about a mean?

Answer: $H_a: \mu \ne \mu_0$. Two-sided tests check for any difference from the hypothesized value.

Flashcard 5: Which tail is used for $H_a: \mu > \mu_0$ when finding a $p$-value from a $t$ distribution?

Answer: Right-tailed ($P(T \ge t_{obs})$). Upper tail probability for testing mean exceeds claimed value.

Flashcard 6: Identify the correct conclusion template when $p > \alpha$ in a test of $H_0: \mu=\mu_0$.

Answer: Fail to reject $H_0$; insufficient evidence for $H_a$. Large $p$-value means data consistent with $H_0$.

Flashcard 7: What parameter is tested in a one-sample $t$ test for a population mean?

Answer: The population mean $\mu$. The test examines whether the population mean equals a hypothesized value.

Flashcard 8: Find and correct the error: using $z = \frac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}}$ for a mean test with unknown $\sigma$.

Answer: Use $t = \frac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}}$ with $df=n-1$. Must use $t$ distribution when $\sigma$ is unknown, not $z$.

Flashcard 9: Compute the standard error when $s = 10$ and $n = 25$ for a one-sample $t$ test.

Answer: $SE = \frac{10}{\sqrt{25}} = 2$. Substitute values into $SE = \frac{s}{\sqrt{n}}$.

Flashcard 10: What is the correct null hypothesis form for testing a population mean against a specific value?

Answer: $H_0: \mu = \mu_0$. Null hypothesis always states equality to the hypothesized value $\mu_0$.

Flashcard 11: Which option correctly matches a claim of 'mean is higher than $\mu_0$' to $H_a$?

Answer: $H_a: \mu > \mu_0$. "Higher than" indicates a one-sided test in the positive direction.

Flashcard 12: Which feature in the data is most concerning for a one-sample $t$ test when $n$ is small?

Answer: Strong skewness or outliers. These violate normality assumption when sample size is small.

Flashcard 13: What is the correct alternative hypothesis for a left-tailed test about a mean?

Answer: $H_a: \mu < \mu_0$. Left-tailed tests check if the mean is below the hypothesized value.

Flashcard 14: Identify the condition that justifies using a one-sample $t$ test: how must the sample be selected?

Answer: Random sample or randomized experiment. Random selection ensures the sample represents the population.

Flashcard 15: State the one-sample $t$ test statistic for a population mean using $\bar{x}$, $\mu_0$, $s$, and $n$.

Answer: $t = \frac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}}$. Standardizes sample mean using standard error $\frac{s}{\sqrt{n}}$.

Flashcard 16: Which tail is used for $H_a: \mu < \mu_0$ when finding a $p$-value from a $t$ distribution?

Answer: Left-tailed ($P(T \le t_{obs})$). Lower tail probability for testing mean below claimed value.

Flashcard 17: Which test is appropriate when $\sigma$ is unknown and you test a single population mean?

Answer: One-sample $t$ test for $\mu$. Uses $t$ distribution when population SD is estimated from sample.

Flashcard 18: State the test statistic for a one-sample $t$ test for a population mean.

Answer: $t = \frac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}}$. Standardizes the difference between sample and hypothesized means.

Flashcard 19: What condition checks randomness for a one-sample mean test using a random sample?

Answer: Random sample (or random assignment) stated. Random sampling ensures unbiased representation of population.

Flashcard 20: If the population distribution is clearly normal, what sample size condition is sufficient for a $t$ test?

Answer: No minimum $n$ (normal population is sufficient). Normal population makes $\bar{x}$ normal for any sample size.

Flashcard 21: Which test is appropriate for inference about a single population mean $\mu$ with unknown $\sigma$?

Answer: One-sample $t$ test for $\mu$. Use $t$ when population standard deviation $\sigma$ is unknown.

Flashcard 22: Identify the correct hypotheses for testing a claim that $\mu$ is at least $12$.

Answer: $H_0: \mu = 12$, $H_a: \mu > 12$. "At least 12" means $\mu \ge 12$; test opposite in $H_a$.

Flashcard 23: What condition justifies approximate normality of $\bar{x}$ when the population is not known to be normal?

Answer: Large sample condition: $n \ge 30$. CLT ensures $\bar{x}$ is approximately normal for large samples.

Flashcard 24: Identify the hypotheses for testing whether the mean differs from $50$ (state $H_0$ and $H_a$).

Answer: $H_0: \mu = 50$; $H_a: \mu \ne 50$. Two-sided test uses $\ne$ since no direction is specified.

Flashcard 25: Which option correctly matches a claim of 'mean is different from $\mu_0$' to $H_a$?

Answer: $H_a: \mu \ne \mu_0$. "Different from" indicates a two-sided test.

Flashcard 26: Which alternative hypothesis matches a claim that the mean is greater than $\mu_0$?

Answer: $H_a: \mu > \mu_0$. Right-tailed test uses $>$ when claiming mean exceeds $\mu_0$.

Flashcard 27: What are the degrees of freedom for a one-sample $t$ test for a mean with sample size $n$?

Answer: $df = n-1$. Loses one degree of freedom when estimating $\sigma$ with $s$.

Flashcard 28: What parameter is tested in a one-sample test for a population mean?

Answer: $\mu$ (the population mean). Tests whether the true population average differs from a claimed value.

Flashcard 29: Compute the test statistic for $\bar{x}=52$, $\mu_0=50$, $s=8$, $n=16$ using a one-sample $t$ test.

Answer: $t = 1$. $t = \frac{52-50}{8/\sqrt{16}} = \frac{2}{2} = 1$.

Flashcard 30: What are the degrees of freedom for a one-sample $t$ test with sample size $n$?

Answer: $df = n - 1$. One degree of freedom is lost when estimating $\sigma$ with $s$.

Flashcard 31: Identify the independence condition for a one-sample $t$ test using the $10\%$ rule.

Answer: $n \le 0.10N$ (when sampling without replacement). Ensures observations are approximately independent when sampling.

Flashcard 32: What is the correct null hypothesis form for a one-sample mean test with claimed value $\mu_0$?

Answer: $H_0: \mu = \mu_0$. Null always states equality with the claimed value.

Flashcard 33: Identify the hypotheses for testing whether the mean is less than $12$ (state $H_0$ and $H_a$).

Answer: $H_0: \mu = 12$; $H_a: \mu < 12$. "Less than" indicates a one-sided test in the negative direction.

Flashcard 34: Which alternative hypothesis matches a claim that the mean is less than $\mu_0$?

Answer: $H_a: \mu < \mu_0$. Left-tailed test uses $<$ when claiming mean is below $\mu_0$.

Flashcard 35: What is the standard error used in a one-sample $t$ test for $\mu$?

Answer: $SE = \frac{s}{\sqrt{n}}$. Measures the typical error in $\bar{x}$ as an estimate of $\mu$.

Flashcard 36: What normality condition is sufficient for a one-sample $t$ test when the population is not known normal?

Answer: Large sample, typically $n \ge 30$. Central Limit Theorem ensures $\bar{x}$ is approximately normal.

Flashcard 37: Identify the correct hypotheses for testing a claim that $\mu$ is at most $50$.

Answer: $H_0: \mu = 50$, $H_a: \mu < 50$. "At most 50" means $\mu \le 50$; test opposite in $H_a$.

Flashcard 38: What condition checks independence when sampling without replacement from a finite population?

Answer: $n \le 0.10N$ (the $10\%$ condition). Ensures sample is small relative to population for independence.

Flashcard 39: What is the $p$-value expression for a two-sided test $H_a: \mu \ne \mu_0$ using $t_{obs}$?

Answer: $p = 2P(T \ge |t_{obs}|)$. Two-tailed test doubles one-tail probability for symmetry.

Flashcard 40: What is the correct alternative hypothesis for a right-tailed test about a mean?

Answer: $H_a: \mu > \mu_0$. Right-tailed tests check if the mean exceeds the hypothesized value.