Setting Up Tests for Population Mean - AP Statistics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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