Inference for Quantitative Data: Means · ⁨الاستدلال للبيانات الكمية: المتوسطات⁩

Enduring Understanding (VAR-1): Given that variation may be random or not, conclusions are uncertain.

Learning Objective VAR-1.I: Identify questions suggested by probabilities of errors in statistical inference. [Skill 1.A]

• VAR-1.I.1 Random variation may result in errors in statistical inference.

Enduring Understanding (VAR-7): The $t$-distribution may be used to model variation.

Learning Objective VAR-7.A: Describe $t$-distributions. [Skill 3.C]

• VAR-7.A.1 When $s$ is used instead of $\sigma$ to calculate a test statistic, the corresponding distribution, known as the $t$-distribution, varies from the normal distribution in shape, in that more of the area is allocated to the tails of the density curve than in a normal distribution.
• VAR-7.A.2 As the degrees of freedom increase, the area in the tails of a $t$-distribution decreases.

Enduring Understanding (UNC-4): An interval of values should be used to estimate parameters, in order to account for uncertainty.

Learning Objective UNC-4.O: Identify an appropriate confidence interval procedure for a population mean, including the mean difference between values in matched pairs. [Skill 1.D]

• UNC-4.O.1 Because $\sigma$ is typically not known for distributions of quantitative variables, the appropriate confidence interval procedure for estimating the population mean of one quantitative variable for one sample is a one-sample $t$-interval for a mean.
• UNC-4.O.2 For one quantitative variable, $X$, that is normally distributed, the distribution of $t = \dfrac{(\overline{x} - \mu)}{\frac{s}{\sqrt{n}}}$ is a $t$-distribution with $n-1$ degrees of freedom.
• UNC-4.O.3 Matched pairs can be thought of as one sample of pairs. Once differences between pairs of values are found, inference for confidence intervals proceeds as for a population mean.

Learning Objective UNC-4.P: Verify the conditions for calculating confidence intervals for a population mean, including the mean difference between values in matched pairs. [Skill 4.C]

• UNC-4.P.1 In order to calculate confidence intervals to estimate a population mean, we must check for independence and that the sampling distribution is approximately normal:
  • a. To check for independence:
    • i. Data should be collected using a random sample or a randomized experiment.
    • ii. When sampling without replacement, check that $n \leq 10\%N$, where $N$ is the size of the population.
  • b. To check that the sampling distribution of $\overline{x}$ is approximately normal (shape):
    • i. If the observed distribution is skewed, $n$ should be greater than 30.
    • ii. If the sample size is less than 30, the distribution of the sample data should be free from strong skewness and outliers.

Learning Objective UNC-4.Q: Determine the margin of error for a given sample size for a one-sample $t$-interval. [Skill 3.D]

• UNC-4.Q.1 The critical value $t^*$ with $n-1$ degrees of freedom can be found using a table or computer-generated output.
• UNC-4.Q.2 The standard error for a sample mean is given by $SE = \dfrac{s}{\sqrt{n}}$, where $s$ is the sample standard deviation.
• UNC-4.Q.3 For a one-sample $t$-interval for a mean, the margin of error is the critical value ($t^*$) times the standard error ($SE$), which equals $t^*\left(\dfrac{s}{\sqrt{n}}\right)$.

Learning Objective UNC-4.R: Calculate an appropriate confidence interval for a population mean, including the mean difference between values in matched pairs. [Skill 3.D]

• UNC-4.R.1 The point estimate for a population mean is the sample mean, $\overline{x}$.
• UNC-4.R.2 For the population mean for one sample with unknown population standard deviation, the confidence interval is $\overline{x} \pm t^* \dfrac{s}{\sqrt{n}}$.

Boundary statement: Formulas for interval estimates do not appear explicitly on the AP Statistics Formula Sheet provided with the AP Statistics Exam. However, these formulas do not need to be memorized, as they can be constructed based on the general test statistic formula and the relevant standard error formulas that are provided on the formula sheet.

Enduring Understanding (UNC-4): An interval of values should be used to estimate parameters, in order to account for uncertainty.

Learning Objective UNC-4.S: Interpret a confidence interval for a population mean, including the mean difference between values in matched pairs. [Skill 4.B]

• UNC-4.S.1 A confidence interval for a population mean either contains the population mean or it does not, because each interval is based on data from a random sample, which varies from sample to sample.
• UNC-4.S.2 We are C% confident that the confidence interval for a population mean captures the population mean.
• UNC-4.S.3 An interpretation of a confidence interval for a population mean includes a reference to the sample taken and details about the population it represents.
  • Illustrative examples for UNC-4.S.3: For interpreting a 96% confidence interval for mean foot length for all footprints found in a cave based on a particular randomly selected sample of footprints in the cave: "We are 96% confident that the mean foot length for all footprints found in the cave falls within the confidence interval" (based on 2000 FRQ 2).

Learning Objective UNC-4.T: Justify a claim based on a confidence interval for a population mean, including the mean difference between values in matched pairs. [Skill 4.D]

• UNC-4.T.1 A confidence interval for a population mean provides an interval of values that may provide sufficient evidence to support a particular claim in context.

Learning Objective UNC-4.U: Identify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population mean. [Skill 4.A]

• UNC-4.U.1 When all other things remain the same, the width of a confidence interval for a population mean tends to decrease as the sample size increases.
• UNC-4.U.2 For a single mean, the width of the interval is proportional to $\dfrac{1}{\sqrt{n}}$.
• UNC-4.U.3 For a given sample, the width of the confidence interval for a population mean increases as the confidence level increases.

Enduring Understanding (VAR-7): The $t$-distribution may be used to model variation.

Learning Objective VAR-7.B: Identify an appropriate testing method for a population mean with unknown $\sigma$, including the mean difference between values in matched pairs. [Skill 1.E]

• VAR-7.B.1 The appropriate test for a population mean with unknown $\sigma$ is a one-sample $t$-test for a population mean.
• VAR-7.B.2 Matched pairs can be thought of as one sample of pairs. Once differences between pairs of values are found, inference for significance testing proceeds as for a population mean.

Learning Objective VAR-7.C: Identify the null and alternative hypotheses for a population mean with unknown $\sigma$, including the mean difference between values in matched pairs. [Skill 1.F]

• VAR-7.C.1 The null hypothesis for a one-sample $t$-test for a population mean is $H_0 : \mu = \mu_0$, where $\mu_0$ is the hypothesized value. Depending upon the situation, the alternative hypothesis is $H_a : \mu < \mu_0$, or $H_a : \mu > \mu_0$, or $H_a : \mu \neq \mu_0$.
• VAR-7.C.2 When finding the mean difference, $\mu_d$, between values in a matched pair, it is important to define the order of subtraction.

Learning Objective VAR-7.D: Verify the conditions for the test for a population mean, including the mean difference between values in matched pairs. [Skill 4.C]

• VAR-7.D.1 In order to make statistical inferences when testing a population mean, we must check for independence and that the sampling distribution is approximately normal:
  • a. To check for independence:
    • i. Data should be collected using a random sample or a randomized experiment.
    • ii. When sampling without replacement, check that $n \leq 10\%N$.
  • b. To check that the sampling distribution of $\overline{x}$ is approximately normal (shape):
    • i. If the observed distribution is skewed, $n$ should be greater than 30.
    • ii. If the sample size is less than 30, the distribution of the sample data should be free from strong skewness and outliers.

Enduring Understanding (VAR-7): The $t$-distribution may be used to model variation.

Learning Objective VAR-7.E: Calculate an appropriate test statistic for a population mean, including the mean difference between values in matched pairs. [Skill 3.E]

• VAR-7.E.1 For a single quantitative variable when random sampling with replacement from a population that can be modeled with a normal distribution with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of $t = \dfrac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}$ has a $t$-distribution with $n - 1$ degrees of freedom.

Boundary statement: The formulas for test statistics do not appear explicitly on the AP Statistics Formula Sheet provided with the AP Statistics Exam. However, these formulas do not need to be memorized, as they can be constructed based on the general test statistic formula and the relevant standard error formulas that are provided on the formula sheet.

Enduring Understanding (DAT-3): Significance testing allows us to make decisions about hypotheses within a particular context.

Learning Objective DAT-3.E: Interpret the $p$-value of a significance test for a population mean, including the mean difference between values in matched pairs. [Skill 4.B]

• DAT-3.E.1 An interpretation of the $p$-value of a significance test for a population mean should recognize that the $p$-value is computed by assuming that the null hypothesis is true, i.e., by assuming that the true population mean is equal to the particular value stated in the null hypothesis.

Learning Objective DAT-3.F: Justify a claim about the population based on the results of a significance test for a population mean. [Skill 4.E]

• DAT-3.F.1 A formal decision explicitly compares the $p$-value to the significance $\alpha$. If the $p$-value $\leq \alpha$, then reject the null hypothesis, $H_0 : \mu = \mu_0$. If the $p$-value $> \alpha$, then fail to reject the null hypothesis.
• DAT-3.F.2 The results of a significance test for a population mean can serve as the statistical reasoning to support the answer to a research question about the population that was sampled.

Enduring Understanding (UNC-4): An interval of values should be used to estimate parameters, in order to account for uncertainty.

Learning Objective UNC-4.V: Identify an appropriate confidence interval procedure for a difference of two population means. [Skill 1.D]

• UNC-4.V.1 Consider a simple random sample from population 1 of size $n_1$, mean $\mu_1$, and standard deviation $\sigma_1$ and a second simple random sample from population 2 of size $n_2$, mean $\mu_2$, and standard deviation $\sigma_2$. If the distributions of populations 1 and 2 are normal or if both $n_1$ and $n_2$ are greater than 30, then the sampling distribution of the difference of means, $\overline{x}_1 - \overline{x}_2$ is also normal. The mean for the sampling distribution of $\overline{x}_1 - \overline{x}_2$ is $\mu_1 - \mu_2$. The standard deviation of $\overline{x}_1 - \overline{x}_2$ is $\sqrt{\dfrac{(\sigma_1)^2}{n_1} + \dfrac{(\sigma_2)^2}{n_2}}$.
• UNC-4.V.2 The appropriate confidence interval procedure for one quantitative variable for two independent samples is a two-sample $t$-interval for a difference between population means.

Learning Objective UNC-4.W: Verify the conditions to calculate confidence intervals for the difference of two population means. [Skill 4.C]

• UNC-4.W.1 In order to calculate confidence intervals to estimate a difference of population means, we must check for independence and that the sampling distribution is approximately normal:
  • a. To check for independence:
    • i. Data should be collected using two independent, random samples or a randomized experiment.
    • ii. When sampling without replacement, check that $n_1 \leq 10\%N_1$ and $n_2 \leq 10\%N_2$.
  • b. To check that the sampling distribution of $(\overline{x}_1 - \overline{x}_2)$ should be approximately normal (shape):
    • i. If the observed distributions are skewed, both $n_1$ and $n_2$ should be greater than 30.

Learning Objective UNC-4.X: Determine the margin of error for the difference of two population means. [Skill 3.D]

• UNC-4.X.1 For the difference of two sample means, the margin of error is the critical value ($t^*$) times the standard error ($SE$) of the difference of two means.
• UNC-4.X.2 The standard error for the difference in two sample means with sample standard deviations, $s_1$ and $s_2$, is $\sqrt{\dfrac{(s_1)^2}{n_1} + \dfrac{(s_2)^2}{n_2}}$.

Learning Objective UNC-4.Y: Calculate an appropriate confidence interval for a difference of two population means. [Skill 3.D]

• UNC-4.Y.1 The point estimate for the difference of two population means is the difference in sample means, $\overline{x}_1 - \overline{x}_2$.
• UNC-4.Y.2 For a difference of two population means where the population standard deviations are not known, the confidence interval is $(\overline{x}_1 - \overline{x}_2) \pm t^* \sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}$ where $\pm t^*$ are the critical values for the central C% of a $t$-distribution with appropriate degrees of freedom that can be found using technology.

Boundary statement: Formulas for interval estimates do not appear explicitly on the AP Statistics Formula Sheet provided with the AP Statistics Exam. However, these formulas do not need to be memorized, as they can be constructed based on the general test statistic formula and the relevant standard error formulas that are provided on the formula sheet.

Enduring Understanding (UNC-4): An interval of values should be used to estimate parameters, in order to account for uncertainty.

Learning Objective UNC-4.Z: Interpret a confidence interval for a difference of population means. [Skill 4.B]

• UNC-4.Z.1 In repeated random sampling with the same sample size, approximately C% of confidence intervals created will capture the difference of population means.
• UNC-4.Z.2 An interpretation for a confidence interval for the difference of two population means should include a reference to the samples taken and details about the populations they represent.
  • Illustrative examples for UNC-4.Z.2: For interpreting a confidence interval for a difference between mean response times for two fire stations (northern - southern): "Based on these samples, one can be 95 percent confident that the difference in the population mean response times (northern - southern) is between -2.37 minutes and 0.37 minutes" (2009 FRQ 4).

Learning Objective UNC-4.AA: Justify a claim based on a confidence interval for a difference of population means. [Skill 4.D]

• UNC-4.AA.1 A confidence interval for a difference of population means provides an interval of values that may provide sufficient evidence to support a particular claim in context.

Learning Objective UNC-4.AB: Identify the effects of sample size on the width of a confidence interval for the difference of two means. [Skill 4.A]

• UNC-4.AB.1 When all other things remain the same, the width of the confidence interval for the difference of two means tends to decrease as the sample sizes increase.

Enduring Understanding (VAR-7): The $t$-distribution may be used to model variation.

Learning Objective VAR-7.F: Identify an appropriate selection of a testing method for a difference of two population means. [Skill 1.E]

• VAR-7.F.1 For a quantitative variable, the appropriate test for a difference of two population means is a two-sample $t$-test for a difference of two population means.

Learning Objective VAR-7.G: Identify the null and alternative hypotheses for a difference of two population means. [Skill 1.F]

• VAR-7.G.1 The null hypothesis for a two-sample $t$-test for a difference of two population means, $\mu_1$ and $\mu_2$, is: $H_0 : \mu_1 - \mu_2 = 0$, or $H_0 : \mu_1 = \mu_2$. The alternative hypothesis is $H_a : \mu_1 - \mu_2 < 0$, or $H_a : \mu_1 - \mu_2 > 0$, or $H_a : \mu_1 - \mu_2 \neq 0$, or $H_a : \mu_1 > \mu_2$, or $H_a : \mu_1 < \mu_2$, or $H_a : \mu_1 \neq \mu_2$.

Learning Objective VAR-7.H: Verify the conditions for the significance test for the difference of two population means. [Skill 4.C]

• VAR-7.H.1 In order to make statistical inferences when testing a difference between population means, we must check for independence and that the sampling distribution is approximately normal:
  • a. Individual observations should be independent:
    • i. Data should be collected using simple random samples or a randomized experiment.
    • ii. When sampling without replacement, check that $n_1 \leq 10\%N_1$ and $n_2 \leq 10\%N_2$.
  • b. The sampling distribution of $\overline{x}_1 - \overline{x}_2$ should be approximately normal (shape).
    • i. If the observed distribution is skewed, both $n_1$ and $n_2$ should be greater than 30.
    • ii. If the sample size is less than 30, the distribution of the sample data should be free from strong skewness and outliers. This should be checked for BOTH samples.

Enduring Understanding (VAR-7): The $t$-distribution may be used to model variation.

Learning Objective VAR-7.I: Calculate an appropriate test statistic for a difference of two means. [Skill 3.E]

• VAR-7.I.1 For a single quantitative variable, data collected using independent random samples or a randomized experiment from two populations, each of which can be modeled with a normal distribution, the sampling distribution of $t = \dfrac{(\overline{x}_1 - \overline{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}}$ is an approximate $t$-distribution with degrees of freedom that can be found using technology. The degrees of freedom fall between the smaller of $n_1 - 1$ and $n_2 - 1$ and $n_1 + n_2 - 2$.
  • Illustrative examples for VAR-7.I.1: In a study comparing mean recovery times for two surgical procedures to repair a torn anterior cruciate ligament (ACL), the group receiving one procedure had a sample size of 110, while the group receiving the other procedure had a sample size of 100. The degrees of freedom fall between 100 (the smaller of 110 and 100) and 208 (110 + 100 - 2). The degrees of freedom may be determined using technology. If the test statistic for this study is $t \approx 7.13$, then the $p$-value is the area greater than 7.13 for a $t$-distribution with $df = 207.18$ (2018 FRQ 4).

Boundary statement: The formulas for test statistics do not appear explicitly on the AP Statistics Formula Sheet provided with the AP Statistics Exam. However, these formulas do not need to be memorized, as they can be constructed based on the general test statistic formula and the standard error formulas for each of the relevant test statistics that are provided on the formula sheet.

Enduring Understanding (DAT-3): Significance testing allows us to make decisions about hypotheses within a particular context.

Learning Objective DAT-3.G: Interpret the $p$-value of a significance test for a difference of population means. [Skill 4.B]

• DAT-3.G.1 An interpretation of the $p$-value of a significance test for a two-sample difference of population means should recognize that the $p$-value is computed by assuming that the null hypothesis is true, i.e., by assuming that the true population means are equal to each other.

Learning Objective DAT-3.H: Justify a claim about the population based on the results of a significance test for a difference of two population means in context. [Skill 4.E]

• DAT-3.H.1 A formal decision explicitly compares the $p$-value to the significance $\alpha$. If the $p$-value $\leq \alpha$, then reject the null hypothesis, $H_0 : \mu_1 - \mu_2 = 0$, or $H_0 : \mu_1 = \mu_2$. If the $p$-value $> \alpha$, then fail to reject the null hypothesis.
• DAT-3.H.2 The results of a significance test for a two-sample test for a difference between two population means can serve as the statistical reasoning to support the answer to a research question about the populations that were sampled.

This topic is intended to focus on the skill of selecting an appropriate inference procedure, now that students have a range of options. Students should be given opportunities to practice when and how to apply all learning objectives relating to inference involving proportions or means.