Unit 4 Beta

Inference for Quantitative Data: Means: the one-page sheet

4.1 Sampling Distributions for Sample Means

The sample mean x-bar has a sampling distribution centered at μ with standard deviation (σ)/(√n) (10% condition). It is normal for a normal population and approximately normal for n ≥ 30 by the central limit theorem. Standardize with z = (x-bar − μ)/(σ/√n) to find probabilities about a sample mean.

Center: $\mu_{\bar{x}} = \mu$ for every sample size.
Spread: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ , valid when the sample is less than 10% of the population. Four times the sample size halves the spread.
Shape: normal if the population is normal; approximately normal if $n \ge 30$ (central limit theorem); otherwise it keeps some of the population's skew.
The central limit theorem is about the distribution of many sample means, not about the data in one sample.

high and low values partly cancel, so means vary less than single values μ_(x-bar) = μ : the sample mean is unbiased the standard deviation of x-bar is σ_(x-bar) = (σ)/(√n) (when the sample is under 10% of the population) for n ≥ 30 the sampling distribution is approximately normal (the central limit theorem); for a normal population it is normal at any n a z-score (x-bar − μ)/(σ/√n) gives probabilities about sample means

standard deviation of the sample mean: The standard deviation of the sampling distribution of the sample mean: <math class="stat-m"><semantics><mrow><msub><mi>σ</mi><mover accent="true"><mi>x</mi><mo stretchy="true">¯</mo></mover></msub><mo>=</mo><mfrac><mi>σ</mi><msqrt><mi>n</mi></msqrt></mfrac></mrow><annotation encoding="application/x-tex">\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}</annotation></semantics></math>, for random samples of size <math class="stat-m"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> that are less than 10% of the population. It measures how far sample means typically fall from <math class="stat-m"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>.

4.2 Constructing a Confidence Interval for a Population Mean or Population Mean Difference

When σ is unknown, a confidence interval for a mean is x-bar ± t*s/(√n) with t* from the t distribution with n − 1 degrees of freedom. Check random, 10% and normal/large sample. For paired data, compute the differences and build the same interval for the mean difference μ_d .

One-sample t interval: $\bar{x} \pm t^*\frac{s}{\sqrt{n}}$ , with $t^*$ from the t distribution with $\text{df} = n - 1$ .
Conditions: random sample (or random assignment); 10% condition when sampling without replacement; normal/large sample (population normal, or $n \ge 30$ , or a graph of the sample with no strong skew or outliers).
Paired data (two measurements on the same individual or matched pair): work with the differences, one sample, parameter $\mu_d$ .
Interpretation: "We are 95% confident that the interval from ... to ... captures the true mean ..."

estimate it with the sample standard deviation s , so the standard error is s/(√n) (x-bar − μ)/(s/√n) varies more than a z-score: it follows a t distribution with n − 1 degrees of freedom the critical value t* is larger than z* , more so for small samples the one-sample t interval is x-bar ± t*s/(√n) the same interval on the differences, x-bar_d ± t*(s_d)/(√n) , estimates the mean difference μ_d

t distribution: A family of symmetric, bell-shaped density curves centered at 0, with heavier tails than the standard normal. Each is set by its degrees of freedom; as df grows, the t distribution approaches the standard normal. Its critical values are written <math class="stat-m"><semantics><mrow><msup><mi>t</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">t^*</annotation></semantics></math>.
one-sample t interval: A confidence interval for a population mean when <math class="stat-m"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> is unknown: <math class="stat-m"><semantics><mrow><mover accent="true"><mi>x</mi><mo stretchy="true">¯</mo></mover><mo>±</mo><msup><mi>t</mi><mo>∗</mo></msup><mfrac><mi>s</mi><msqrt><mi>n</mi></msqrt></mfrac></mrow><annotation encoding="application/x-tex">\bar{x} \pm t^*\frac{s}{\sqrt{n}}</annotation></semantics></math>, with <math class="stat-m"><semantics><mrow><mtext>df</mtext><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\text{df} = n - 1</annotation></semantics></math>. Conditions: random, 10%, normal/large sample.
paired data: Two measurements that belong together, on the same individual or on two matched individuals (before and after, left and right, twins). Analyze the differences as one sample; the parameter is the true mean difference <math class="stat-m"><semantics><mrow><msub><mi>μ</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">\mu_d</annotation></semantics></math>.
normal/large sample condition: The shape condition for t procedures: the population is normal, or the sample size is at least 30, or a graph of the sample data (dotplot, boxplot, histogram) shows no strong skew and no outliers.

4.3 Justifying a Claim Based on a Confidence Interval for a Population Mean or Population Mean Difference

Use a t interval to judge a claim about a mean: a claimed value outside the interval is not plausible, so the interval gives convincing evidence against it; a value inside is plausible, and nothing is proven. Write it with "Because every value in the interval ..." and name the parameter in context. Interval width grows with confidence and with s , and shrinks like 1/(√n) .

Claim from an interval: "Because every value in the interval is less than 500 (or: 500 is in the interval), the interval gives (does not give) convincing evidence that the true mean ..."
A plausible value is not a proven value: many values inside the interval are equally plausible.
Interval width $= 2t^*\frac{s}{\sqrt{n}}$ : more confidence, wider; more data, narrower; more variable data, wider.
A $C$ % interval matches a two-sided test at $\alpha = 1 - C$ : it excludes exactly the values that test would reject.

a claimed value outside it is not plausible, and one inside it is the interval gives convincing evidence that the true mean is on that side the interval does not give convincing evidence against it, and does not prove it either higher confidence (larger t* ) widens it; a larger sample narrows it, by about half for four times the data

interval width: The distance between the two ends of a confidence interval, twice the margin of error. For a t interval it is <math class="stat-m"><semantics><mrow><mn>2</mn><msup><mi>t</mi><mo>∗</mo></msup><mfrac><mi>s</mi><msqrt><mi>n</mi></msqrt></mfrac></mrow><annotation encoding="application/x-tex">2t^*\frac{s}{\sqrt{n}}</annotation></semantics></math>: it grows with the confidence level and with the variability, and shrinks as the sample size grows.

4.4 Setting Up a Test for a Population Mean or Population Mean Difference

To set up a t test for a mean, name the parameter in context, write H_0:μ = μ_0 and an alternative whose direction comes from the question, and check random, 10% and normal/large sample. For paired data, test the mean difference μ_d with a paired t test on the differences, saying which way you subtracted.

One-sample t test: $H_0: \mu = \mu_0$ against $H_a: \mu > \mu_0$ , $\mu < \mu_0$ or $\mu \ne \mu_0$ .
Paired t test: a one-sample t test on the differences, $H_0: \mu_d = 0$ . The order of subtraction sets the direction of $H_a$ .
Hypotheses use parameters ( $\mu$ , $\mu_d$ ), never statistics ( $\bar{x}$ ), and the direction comes from the question, never from the sample result.
Conditions: random; 10% when sampling without replacement; normal population, $n \ge 30$ , or a graph of the sample (or the differences) without strong skew or outliers.

it gives the parameter, the hypothesized mean μ_0 and the direction of H_a define μ in context: "the true mean ... of all ..." the data are paired: test H_0:μ_d = 0 on the differences, with the order of subtraction stated a one-sample (or paired) t test can be trusted, with df = n − 1

one-sample t test: A significance test of <math class="stat-m"><semantics><mrow><msub><mi>H</mi><mn>0</mn></msub><mo>:</mo><mi>μ</mi><mo>=</mo><msub><mi>μ</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">H_0: \mu = \mu_0</annotation></semantics></math> for a population mean when <math class="stat-m"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> is unknown, using the t distribution with <math class="stat-m"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n - 1</annotation></semantics></math> degrees of freedom. A paired t test is a one-sample t test on the differences of paired data, usually with <math class="stat-m"><semantics><mrow><msub><mi>H</mi><mn>0</mn></msub><mo>:</mo><msub><mi>μ</mi><mi>d</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">H_0: \mu_d = 0</annotation></semantics></math>.
hypothesized mean: The hypothesized mean <math class="stat-m"><semantics><mrow><msub><mi>μ</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mu_0</annotation></semantics></math>: the value of the population mean stated by the null hypothesis <math class="stat-m"><semantics><mrow><msub><mi>H</mi><mn>0</mn></msub><mo>:</mo><mi>μ</mi><mo>=</mo><msub><mi>μ</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">H_0: \mu = \mu_0</annotation></semantics></math>. It comes from the question (a claim, a standard, a past value), never from the sample.

4.5 Carrying Out a Test for a Population Mean or Population Mean Difference

Carry out a one-sample (or paired) t test by computing t = (x-bar − μ_0)/(s/√n) with n − 1 df, finding the p-value in the direction of H_a , and comparing it with α . Conclude in context with evidence language about the true mean, and interpret the p-value as a probability computed assuming H_0 .

t statistic: $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$ (paired: $t$ $= \frac{\bar{x}_d - 0}{s_d/\sqrt{n}}$ ), $\text{df} = n - 1$ .
p-value: right tail for $>$ , left tail for $<$ , twice the smaller tail for $\ne$ . Table B brackets it; technology gives it exactly.
Conclusion: "Because the p-value of 0.0112 is less than 0.05, we reject $H_0$ . There is convincing evidence that the true mean ... is greater than ..."
A large p-value is "not convincing evidence", never "accept $H_0$ " or "the mean is $\mu_0$ ".

sample means vary around μ_0 with standard error s/(√n) t = (x-bar − μ_0)/(s/√n) , with df = n − 1 that area is the p-value (double the tail for ≠ ) reject H_0 if p < α , fail to reject otherwise, and state the evidence for H_a in context

t statistic: The standardized test statistic of a one-sample t test: <math class="stat-m"><semantics><mrow><mi>t</mi><mo>=</mo><mfrac><mrow><mover accent="true"><mi>x</mi><mo stretchy="true">¯</mo></mover><mo>−</mo><msub><mi>μ</mi><mn>0</mn></msub></mrow><mrow><mi>s</mi><mo>/</mo><msqrt><mi>n</mi></msqrt></mrow></mfrac></mrow><annotation encoding="application/x-tex">t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}</annotation></semantics></math>, the number of standard errors between the sample mean and the hypothesized mean. Its p-value comes from the t distribution with <math class="stat-m"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n - 1</annotation></semantics></math> degrees of freedom.

4.6 Sampling Distributions for the Difference Between Two Sample Means

For independent random samples (or randomly assigned groups), x-bar_1 − x-bar_2 has mean μ_1 − μ_2 and standard deviation √((σ_1²)/(n_1) + (σ_2²)/(n_2)) , and it is approximately normal when each sample mean is. Use z = ((x-bar_1 − x-bar_2) − (μ_1 − μ_2))/(σ_(x-bar_1 − x-bar_2)) for probabilities.

Center: $\mu_{\bar{x}_1 - \bar{x}_2}$ $= \mu_1 - \mu_2$ .
Spread: $\sigma_{\bar{x}_1 - \bar{x}_2}$ $= \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$ for independent samples (or groups formed by random assignment), each under 10% of its population when sampling.
Variances add, never standard deviations, and never subtract.
Shape: normal if both populations are normal; approximately normal if both $n_1 \ge 30$ and $n_2 \ge 30$ .

the difference is centered at μ_1 − μ_2 for independent samples the variances add, so the difference varies more than either mean σ_(x-bar_1 − x-bar_2) = √((σ_1²)/(n_1) + (σ_2²)/(n_2)) the difference is approximately normal, and z-scores give probabilities

difference in sample means: The statistic <math class="stat-m"><semantics><mrow><msub><mover accent="true"><mi>x</mi><mo stretchy="true">¯</mo></mover><mn>1</mn></msub><mo>−</mo><msub><mover accent="true"><mi>x</mi><mo stretchy="true">¯</mo></mover><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\bar{x}_1 - \bar{x}_2</annotation></semantics></math> from two independent samples or groups. Its sampling distribution has mean <math class="stat-m"><semantics><mrow><msub><mi>μ</mi><mn>1</mn></msub><mo>−</mo><msub><mi>μ</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mu_1 - \mu_2</annotation></semantics></math> and standard deviation <math class="stat-m"><semantics><mrow><msqrt><mrow><mfrac><msubsup><mi>σ</mi><mn>1</mn><mn>2</mn></msubsup><msub><mi>n</mi><mn>1</mn></msub></mfrac><mo>+</mo><mfrac><msubsup><mi>σ</mi><mn>2</mn><mn>2</mn></msubsup><msub><mi>n</mi><mn>2</mn></msub></mfrac></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}</annotation></semantics></math>, and is approximately normal when each sample mean is.

4.7 Constructing a Confidence Interval for the Difference Between Two Population Means

To estimate μ_1 − μ_2 from two independent samples or randomized groups, use (x-bar_1 − x-bar_2) ±t*√((s_1²)/(n_1) + (s_2²)/(n_2)) with technology df or the conservative df. Check random, 10% and normal/large sample for each group, and interpret the interval as capturing the true difference in means, in the stated order.

Two-sample t interval: $(\bar{x}_1 - \bar{x}_2)$ $\pm t^*\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$ .
Degrees of freedom: technology (Welch) gives a non-integer df; the conservative df is the smaller of $n_1 - 1$ and $n_2 - 1$ , which gives a slightly wider interval. Both are accepted.
Conditions for each group: random (two independent random samples or random assignment), 10% (when sampling), normal/large sample.
Two independent groups use this interval; two linked measurements on each unit are paired and use a one-sample t interval on the differences.

its point estimate is x-bar_1 − x-bar_2 the standard error is √((s_1²)/(n_1) + (s_2²)/(n_2)) use t* , with technology df (Welch) or the conservative df, the smaller of n_1 − 1 and n_2 − 1 build (x-bar_1 − x-bar_2) ±t*√((s_1²)/(n_1) + (s_2²)/(n_2)) and interpret it in context, order of subtraction stated

two-sample t interval: A confidence interval for the difference of two population means, <math class="stat-m"><semantics><mrow><msub><mi>μ</mi><mn>1</mn></msub><mo>−</mo><msub><mi>μ</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mu_1 - \mu_2</annotation></semantics></math>, from two independent samples or randomized groups: <math class="stat-m"><semantics><mrow><mo stretchy="false">(</mo><msub><mover accent="true"><mi>x</mi><mo stretchy="true">¯</mo></mover><mn>1</mn></msub><mo>−</mo><msub><mover accent="true"><mi>x</mi><mo stretchy="true">¯</mo></mover><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\bar{x}_1 - \bar{x}_2)</annotation></semantics></math><wbr><math class="stat-m"><semantics><mrow><mo form="infix">±</mo><msup><mi>t</mi><mo>∗</mo></msup><msqrt><mrow><mfrac><msubsup><mi>s</mi><mn>1</mn><mn>2</mn></msubsup><msub><mi>n</mi><mn>1</mn></msub></mfrac><mo>+</mo><mfrac><msubsup><mi>s</mi><mn>2</mn><mn>2</mn></msubsup><msub><mi>n</mi><mn>2</mn></msub></mfrac></mrow></msqrt></mrow><annotation encoding="application/x-tex">\pm t^*\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}</annotation></semantics></math>, with technology df or the conservative df.
conservative degrees of freedom: For two-sample t procedures, the smaller of <math class="stat-m"><semantics><mrow><msub><mi>n</mi><mn>1</mn></msub><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_1 - 1</annotation></semantics></math> and <math class="stat-m"><semantics><mrow><msub><mi>n</mi><mn>2</mn></msub><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_2 - 1</annotation></semantics></math>. It never exceeds the technology (Welch) degrees of freedom, so it gives a slightly larger <math class="stat-m"><semantics><mrow><msup><mi>t</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">t^*</annotation></semantics></math>: a wider interval and a larger p-value. Both choices are accepted.

4.8 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Means

Use an interval for μ_1 − μ_2 to judge claims: if 0 is outside, there is convincing evidence of a difference in the direction of the interval's sign; if 0 is inside, there is not, and no difference is proven. Then judge practical significance by comparing the plausible values with what matters in context, and judge cause and generalization from the design.

0 outside the interval: "Because every value in the interval is greater than 0, the interval gives convincing evidence that the true mean ... is greater for group 1 than for group 2."
0 inside the interval: "Because 0 is in the interval, the interval does not give convincing evidence that the true means differ." Never "the means are equal".
Practical significance: a difference can be statistically significant yet too small to matter; compare the plausible values with a threshold from the context.
Reversing the order of subtraction flips the interval's signs, not the conclusion.

outside: convincing evidence of a difference (its sign gives the direction); inside: no convincing evidence that judges practical significance, which a p-value or "0 is outside" cannot random assignment supports cause and effect; random sampling supports generalizing is never proof of no difference: the plausible values still range from one direction to the other

practical significance: Whether a difference is large enough to matter in context. A result can be statistically significant (0 outside the interval, or a small p-value) but practically unimportant, especially with very large samples. Judge it by comparing the plausible values in the interval with a meaningful threshold.

4.9 Setting Up a Test for the Difference Between Two Population Means

To set up a test for two means, decide from the design whether the data are paired or two independent groups. For two groups, define μ_1 and μ_2 in context, write H_0:μ_1 − μ_2 = 0 with an alternative from the question, and check random, 10% and normal/large sample for each group. This is a two-sample t test.

Two-sample t test: $H_0: \mu_1 - \mu_2 = 0$ (or $\mu_1 = \mu_2$ ) against $H_a: \mu_1 - \mu_2 > 0$ , $< 0$ or $\ne 0$ .
Define both parameters in context: " $\mu_X$ = the true mean battery life of all new brand X phones".
Conditions for each group: random (independent random samples or random assignment), 10% (when sampling), normal/large sample.
Paired or two-sample is decided by the design: same units twice or matched units, paired; separate groups, two-sample.

the parameter is μ_1 − μ_2 and the test is a two-sample t test define μ_1 and μ_2 in context, with the order of subtraction the two-sample t test can be trusted the data are paired, so use a paired t test on the differences instead; pairing removes unit-to-unit variation

two-sample t test: A significance test of <math class="stat-m"><semantics><mrow><msub><mi>H</mi><mn>0</mn></msub><mo>:</mo><msub><mi>μ</mi><mn>1</mn></msub><mo>−</mo><msub><mi>μ</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">H_0: \mu_1 - \mu_2 = 0</annotation></semantics></math> for two independent random samples or two randomly assigned groups, with <math class="stat-m"><semantics><mrow><msub><mi>σ</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_1</annotation></semantics></math> and <math class="stat-m"><semantics><mrow><msub><mi>σ</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_2</annotation></semantics></math> unknown. It uses <math class="stat-m"><semantics><mrow><msub><mi>s</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">s_1</annotation></semantics></math> and <math class="stat-m"><semantics><mrow><msub><mi>s</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">s_2</annotation></semantics></math> separately (no combined standard deviation) and the t distribution with technology or conservative df.

4.10 Carrying Out a Test for the Difference Between Two Population Means

A two-sample t test computes t = (x-bar_1 − x-bar_2)/(√(s_1²/n_1 + s_2²/n_2)) , finds the p-value from the t distribution with technology or conservative df in the direction of H_a , and concludes about μ_1 − μ_2 in context. Random assignment allows a causal conclusion; random samples allow generalizing.

Two-sample t statistic: $t$ $= \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$ .
p-value: left tail for $<$ , right tail for $>$ , twice the smaller tail for $\ne$ ; technology df, or the conservative df (also accepted).
Conclusion: "Because the p-value of 0.0132 is less than 0.05, we reject $H_0$ . There is convincing evidence that the true mean ... with caffeine is less than without ..."
Choosing the procedure: proportion or mean? one sample, paired, or two independent groups? interval or test?

differences in sample means vary around 0 with standard error √((s_1²)/(n_1) + (s_2²)/(n_2)) t = ((x-bar_1 − x-bar_2) − 0)/(√((s_1²)/(n_1) + (s_2²)/(n_2))) with technology df, or the conservative df (smaller n − 1 ), which gives a slightly larger p-value then say what the design allows: cause and effect (random assignment) or generalizing (random sampling)

two-sample t statistic: The test statistic of a two-sample t test: <math class="stat-m"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math><wbr><math class="stat-m"><semantics><mrow><mo form="infix">=</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mover accent="true"><mi>x</mi><mo stretchy="true">¯</mo></mover><mn>1</mn></msub><mo>−</mo><msub><mover accent="true"><mi>x</mi><mo stretchy="true">¯</mo></mover><mn>2</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mn>0</mn></mrow><msqrt><mrow><mfrac><msubsup><mi>s</mi><mn>1</mn><mn>2</mn></msubsup><msub><mi>n</mi><mn>1</mn></msub></mfrac><mo>+</mo><mfrac><msubsup><mi>s</mi><mn>2</mn><mn>2</mn></msubsup><msub><mi>n</mi><mn>2</mn></msub></mfrac></mrow></msqrt></mfrac></mrow><annotation encoding="application/x-tex">= \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}</annotation></semantics></math>, the observed difference in sample means measured in standard errors. Its p-value comes from a t distribution with technology df or the conservative df.