Setting Up Tests for Population Mean - AP Statistics
Card 1 of 30
State the test statistic for a one-sample $t$ test for a population mean.
State the test statistic for a one-sample $t$ test for a population mean.
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$t = \frac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}}$. Standardizes the difference between sample and hypothesized means.
$t = \frac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}}$. Standardizes the difference between sample and hypothesized means.
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Identify the hypotheses for testing whether the mean is less than $12$ (state $H_0$ and $H_a$).
Identify the hypotheses for testing whether the mean is less than $12$ (state $H_0$ and $H_a$).
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$H_0: \mu = 12$; $H_a: \mu < 12$. "Less than" indicates a one-sided test in the negative direction.
$H_0: \mu = 12$; $H_a: \mu < 12$. "Less than" indicates a one-sided test in the negative direction.
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Which tail is used for $H_a: \mu < \mu_0$ when finding a $p$-value from a $t$ distribution?
Which tail is used for $H_a: \mu < \mu_0$ when finding a $p$-value from a $t$ distribution?
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Left-tailed ($P(T \le t_{obs})$). Lower tail probability for testing mean below claimed value.
Left-tailed ($P(T \le t_{obs})$). Lower tail probability for testing mean below claimed value.
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Identify the correct hypotheses for testing a claim that $\mu$ is at most $50$.
Identify the correct hypotheses for testing a claim that $\mu$ is at most $50$.
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$H_0: \mu = 50$, $H_a: \mu < 50$. "At most 50" means $\mu \le 50$; test opposite in $H_a$.
$H_0: \mu = 50$, $H_a: \mu < 50$. "At most 50" means $\mu \le 50$; test opposite in $H_a$.
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Which tail is used for $H_a: \mu > \mu_0$ when finding a $p$-value from a $t$ distribution?
Which tail is used for $H_a: \mu > \mu_0$ when finding a $p$-value from a $t$ distribution?
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Right-tailed ($P(T \ge t_{obs})$). Upper tail probability for testing mean exceeds claimed value.
Right-tailed ($P(T \ge t_{obs})$). Upper tail probability for testing mean exceeds claimed value.
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Identify the correct hypotheses for testing a claim that $\mu$ is at least $12$.
Identify the correct hypotheses for testing a claim that $\mu$ is at least $12$.
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$H_0: \mu = 12$, $H_a: \mu > 12$. "At least 12" means $\mu \ge 12$; test opposite in $H_a$.
$H_0: \mu = 12$, $H_a: \mu > 12$. "At least 12" means $\mu \ge 12$; test opposite in $H_a$.
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Compute the test statistic for $\bar{x}=52$, $\mu_0=50$, $s=8$, $n=16$ using a one-sample $t$ test.
Compute the test statistic for $\bar{x}=52$, $\mu_0=50$, $s=8$, $n=16$ using a one-sample $t$ test.
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$t = 1$. $t = \frac{52-50}{8/\sqrt{16}} = \frac{2}{2} = 1$.
$t = 1$. $t = \frac{52-50}{8/\sqrt{16}} = \frac{2}{2} = 1$.
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What is the $p$-value expression for a two-sided test $H_a: \mu \ne \mu_0$ using $t_{obs}$?
What is the $p$-value expression for a two-sided test $H_a: \mu \ne \mu_0$ using $t_{obs}$?
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$p = 2P(T \ge |t_{obs}|)$. Two-tailed test doubles one-tail probability for symmetry.
$p = 2P(T \ge |t_{obs}|)$. Two-tailed test doubles one-tail probability for symmetry.
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What are the degrees of freedom for a one-sample $t$ test for a mean with sample size $n$?
What are the degrees of freedom for a one-sample $t$ test for a mean with sample size $n$?
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$df = n-1$. Loses one degree of freedom when estimating $\sigma$ with $s$.
$df = n-1$. Loses one degree of freedom when estimating $\sigma$ with $s$.
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What condition justifies approximate normality of $\bar{x}$ when the population is not known to be normal?
What condition justifies approximate normality of $\bar{x}$ when the population is not known to be normal?
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Large sample condition: $n \ge 30$. CLT ensures $\bar{x}$ is approximately normal for large samples.
Large sample condition: $n \ge 30$. CLT ensures $\bar{x}$ is approximately normal for large samples.
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If the population distribution is clearly normal, what sample size condition is sufficient for a $t$ test?
If the population distribution is clearly normal, what sample size condition is sufficient for a $t$ test?
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No minimum $n$ (normal population is sufficient). Normal population makes $\bar{x}$ normal for any sample size.
No minimum $n$ (normal population is sufficient). Normal population makes $\bar{x}$ normal for any sample size.
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Identify the correct conclusion template when $p \le \alpha$ in a test of $H_0: \mu=\mu_0$.
Identify the correct conclusion template when $p \le \alpha$ in a test of $H_0: \mu=\mu_0$.
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Reject $H_0$; evidence supports $H_a$. Small $p$-value indicates data unlikely under $H_0$.
Reject $H_0$; evidence supports $H_a$. Small $p$-value indicates data unlikely under $H_0$.
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Identify the correct conclusion template when $p > \alpha$ in a test of $H_0: \mu=\mu_0$.
Identify the correct conclusion template when $p > \alpha$ in a test of $H_0: \mu=\mu_0$.
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Fail to reject $H_0$; insufficient evidence for $H_a$. Large $p$-value means data consistent with $H_0$.
Fail to reject $H_0$; insufficient evidence for $H_a$. Large $p$-value means data consistent with $H_0$.
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What parameter is tested in a one-sample test for a population mean?
What parameter is tested in a one-sample test for a population mean?
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$\mu$ (the population mean). Tests whether the true population average differs from a claimed value.
$\mu$ (the population mean). Tests whether the true population average differs from a claimed value.
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What condition checks independence when sampling without replacement from a finite population?
What condition checks independence when sampling without replacement from a finite population?
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$n \le 0.10N$ (the $10%$ condition). Ensures sample is small relative to population for independence.
$n \le 0.10N$ (the $10%$ condition). Ensures sample is small relative to population for independence.
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What is the correct null hypothesis form for a one-sample mean test with claimed value $\mu_0$?
What is the correct null hypothesis form for a one-sample mean test with claimed value $\mu_0$?
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$H_0: \mu = \mu_0$. Null always states equality with the claimed value.
$H_0: \mu = \mu_0$. Null always states equality with the claimed value.
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Which alternative hypothesis matches a two-sided claim that $\mu$ differs from $\mu_0$?
Which alternative hypothesis matches a two-sided claim that $\mu$ differs from $\mu_0$?
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$H_a: \mu \ne \mu_0$. Two-sided test uses $\ne$ for any difference from $\mu_0$.
$H_a: \mu \ne \mu_0$. Two-sided test uses $\ne$ for any difference from $\mu_0$.
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Which alternative hypothesis matches a claim that the mean is greater than $\mu_0$?
Which alternative hypothesis matches a claim that the mean is greater than $\mu_0$?
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$H_a: \mu > \mu_0$. Right-tailed test uses $>$ when claiming mean exceeds $\mu_0$.
$H_a: \mu > \mu_0$. Right-tailed test uses $>$ when claiming mean exceeds $\mu_0$.
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Which alternative hypothesis matches a claim that the mean is less than $\mu_0$?
Which alternative hypothesis matches a claim that the mean is less than $\mu_0$?
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$H_a: \mu < \mu_0$. Left-tailed test uses $<$ when claiming mean is below $\mu_0$.
$H_a: \mu < \mu_0$. Left-tailed test uses $<$ when claiming mean is below $\mu_0$.
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State the one-sample $t$ test statistic for a population mean using $\bar{x}$, $\mu_0$, $s$, and $n$.
State the one-sample $t$ test statistic for a population mean using $\bar{x}$, $\mu_0$, $s$, and $n$.
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$t = \frac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}}$. Standardizes sample mean using standard error $\frac{s}{\sqrt{n}}$.
$t = \frac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}}$. Standardizes sample mean using standard error $\frac{s}{\sqrt{n}}$.
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Which test is appropriate when $\sigma$ is unknown and you test a single population mean?
Which test is appropriate when $\sigma$ is unknown and you test a single population mean?
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One-sample $t$ test for $\mu$. Uses $t$ distribution when population SD is estimated from sample.
One-sample $t$ test for $\mu$. Uses $t$ distribution when population SD is estimated from sample.
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What condition checks randomness for a one-sample mean test using a random sample?
What condition checks randomness for a one-sample mean test using a random sample?
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Random sample (or random assignment) stated. Random sampling ensures unbiased representation of population.
Random sample (or random assignment) stated. Random sampling ensures unbiased representation of population.
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Compute the standard error when $s = 10$ and $n = 25$ for a one-sample $t$ test.
Compute the standard error when $s = 10$ and $n = 25$ for a one-sample $t$ test.
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$SE = \frac{10}{\sqrt{25}} = 2$. Substitute values into $SE = \frac{s}{\sqrt{n}}$.
$SE = \frac{10}{\sqrt{25}} = 2$. Substitute values into $SE = \frac{s}{\sqrt{n}}$.
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Compute the degrees of freedom for a one-sample $t$ test when $n = 18$.
Compute the degrees of freedom for a one-sample $t$ test when $n = 18$.
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$df = 17$. Apply $df = n - 1 = 18 - 1 = 17$.
$df = 17$. Apply $df = n - 1 = 18 - 1 = 17$.
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Identify the hypotheses for testing whether the mean differs from $50$ (state $H_0$ and $H_a$).
Identify the hypotheses for testing whether the mean differs from $50$ (state $H_0$ and $H_a$).
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$H_0: \mu = 50$; $H_a: \mu \ne 50$. Two-sided test uses $\ne$ since no direction is specified.
$H_0: \mu = 50$; $H_a: \mu \ne 50$. Two-sided test uses $\ne$ since no direction is specified.
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What is the standard error used in a one-sample $t$ test for $\mu$?
What is the standard error used in a one-sample $t$ test for $\mu$?
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$SE = \frac{s}{\sqrt{n}}$. Measures the typical error in $\bar{x}$ as an estimate of $\mu$.
$SE = \frac{s}{\sqrt{n}}$. Measures the typical error in $\bar{x}$ as an estimate of $\mu$.
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Which test is appropriate for inference about a single population mean $\mu$ with unknown $\sigma$?
Which test is appropriate for inference about a single population mean $\mu$ with unknown $\sigma$?
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One-sample $t$ test for $\mu$. Use $t$ when population standard deviation $\sigma$ is unknown.
One-sample $t$ test for $\mu$. Use $t$ when population standard deviation $\sigma$ is unknown.
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What parameter is tested in a one-sample $t$ test for a population mean?
What parameter is tested in a one-sample $t$ test for a population mean?
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The population mean $\mu$. The test examines whether the population mean equals a hypothesized value.
The population mean $\mu$. The test examines whether the population mean equals a hypothesized value.
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What is the correct null hypothesis form for testing a population mean against a specific value?
What is the correct null hypothesis form for testing a population mean against a specific value?
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$H_0: \mu = \mu_0$. Null hypothesis always states equality to the hypothesized value $\mu_0$.
$H_0: \mu = \mu_0$. Null hypothesis always states equality to the hypothesized value $\mu_0$.
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What is the correct alternative hypothesis for a right-tailed test about a mean?
What is the correct alternative hypothesis for a right-tailed test about a mean?
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$H_a: \mu > \mu_0$. Right-tailed tests check if the mean exceeds the hypothesized value.
$H_a: \mu > \mu_0$. Right-tailed tests check if the mean exceeds the hypothesized value.
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