Unit 4: Inference for Quantitative Data: Means

z Procedures Versus t Procedures

Use a z procedure for a population mean only when the population standard deviation $\sigma$ is known:

$$z = \frac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}.$$

In most real problems, $\sigma$ is unknown, so use the sample standard deviation $s$ and a t-distribution:

$$t = \frac{\bar{x}-\mu_0}{s/\sqrt{n}}.$$

The t-distribution is symmetric and bell-shaped like the normal distribution, but it has heavier tails. As degrees of freedom increase, the t-distribution approaches the standard normal distribution.

Key infoWhy t is wider

Replacing the unknown population standard deviation $\sigma$ with the sample standard deviation $s$ adds uncertainty. The t-distribution accounts for that extra uncertainty with heavier tails, especially when $n$ is small.

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Sampling Distributions For Means

For a quantitative variable with population mean $\mu$ and standard deviation $\sigma$, the sample mean $\bar{x}$ has

$$\mu_{\bar{x}}=\mu$$

and

$$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}.$$

If the population is normal, the sampling distribution of $\bar{x}$ is normal for any sample size. If the population is not normal, the Central Limit Theorem says the sampling distribution of $\bar{x}$ becomes approximately normal as $n$ gets large, assuming independence.

For two independent sample means,

$$\mu_{\bar{x}_1-\bar{x}_2}=\mu_1-\mu_2$$

and

$$\sigma_{\bar{x}_1-\bar{x}_2} =\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}.$$

When $\sigma_1$ and $\sigma_2$ are unknown, use $s_1$ and $s_2$ in the standard error for t procedures.

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One-Sample t-Interval For A Mean

Use a one-sample t-interval to estimate a population mean $\mu$:

$$\bar{x} \pm t^*\frac{s}{\sqrt{n}}.$$

Degrees of freedom:

$$df = n-1.$$

Conditions

1. Random sample or random assignment.
2. Independence: if sampling without replacement, $n \le 0.10N$.
3. Normal/large sample condition: population is approximately normal, or sample size is large enough for the Central Limit Theorem. Check graphs for strong skew and outliers when $n$ is small.

Interpretation: “We are ___% confident that the true population mean ___ is between ___ and ___.”

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One-Sample t-Test For A Mean

Use a one-sample t-test for a claim about one population mean:

$$H_0:\mu=\mu_0.$$

The test statistic is

$$t = \frac{\bar{x}-\mu_0}{s/\sqrt{n}},$$

with

$$df=n-1.$$

The alternative may be $\mu>\mu_0$, $\mu<\mu_0$, or $\mu\ne\mu_0$. The p-value is found from the t-distribution with the correct degrees of freedom.

On the examMean tests

State the parameter as a mean of the population, not a mean of the sample. For example: “Let $\mu$ be the true mean wait time for all customers at this store.”

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Two-Sample t-Interval For The Difference In Means

Use a two-sample t-interval to estimate $\mu_1-\mu_2$ for two independent groups:

$$(\bar{x}_1-\bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}.$$

Conditions

1. Random samples or random assignment.
2. The two samples/groups are independent.
3. Independence within each group: $n_1 \le 0.10N_1$ and $n_2 \le 0.10N_2$ if sampling without replacement.
4. Normal/large sample condition for each group.

Degrees of freedom can be found with technology. If doing by hand, use the conservative choice:

$$df = \min(n_1-1,\ n_2-1).$$

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Two-Sample t-Test For The Difference In Means

For independent samples, test

$$H_0:\mu_1-\mu_2=0$$

with

$$t = \frac{(\bar{x}_1-\bar{x}_2)-0} {\sqrt{s_1^2/n_1+s_2^2/n_2}}.$$

Use technology for degrees of freedom unless told otherwise. AP problems often care more about setup, conditions, and conclusion than hand-calculating df.

Warning

Do not pool variances unless a problem specifically says to use a pooled two-sample t procedure. Modern AP Statistics generally uses unpooled two-sample t procedures.

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Matched Pairs t Procedures

A matched pairs design compares paired observations: before/after measurements on the same subject, twins, matched individuals, or two treatments applied to each unit in random order.

For matched pairs, convert the data to differences:

$$d_i = \text{value}_{1,i} - \text{value}_{2,i}.$$

Then run a one-sample t procedure on the differences.

Warning

Matched pairs data are not two independent samples. Once the observations are naturally paired, subtract within each pair and analyze the list of differences with a one-sample t procedure.

Interval:

$$\bar{d} \pm t^*\frac{s_d}{\sqrt{n}}.$$

Test statistic:

$$t = \frac{\bar{d}-\mu_{d,0}}{s_d/\sqrt{n}}.$$

Degrees of freedom:

$$df=n-1.$$

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Choosing The Correct Mean Procedure

Situation	Procedure
One quantitative sample, $\sigma$ unknown	One-sample t
One quantitative sample, $\sigma$ known	One-sample z
Two independent quantitative samples	Two-sample t
Paired quantitative measurements	Matched pairs t

If the data are categorical counts or proportions, use Unit 6 or Unit 8 instead.

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Confidence Intervals And Tests Together

A two-sided hypothesis test at significance level $\alpha$ corresponds to a $(1-\alpha)100\%$ confidence interval. If the null value is outside the interval, reject $H_0$. If the null value is inside the interval, fail to reject $H_0$.

For one-sided tests, this direct interval comparison requires more care, but the logic is still connected: values far from the interval’s plausible range are less compatible with the data.

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Calculator Notes

Common calculator tools:

• TInterval: one-sample t confidence interval.
• T-Test: one-sample t test.
• 2-SampTInt: two-sample t confidence interval.
• 2-SampTTest: two-sample t test.
• For matched pairs, enter the list of differences and use TInterval or T-Test.

Calculator output should be translated into statistical language: parameter, conditions, statistic, interval or p-value, and conclusion in context.

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Working Checklist

Working checklist

1. Identify the parameter: $\mu$, $\mu_1-\mu_2$, or $\mu_d$.
2. Decide whether the samples are independent or paired.
3. Check random, independence, and normal/large sample conditions.
4. Use t unless population $\sigma$ is known.
5. Compute the interval or test statistic/p-value.
6. Conclude in context, including units.

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Key Equations

Key equations

Idea	Equation
One-sample t interval	$\bar{x}\pm t^*s/\sqrt{n}$
One-sample t test	$t=(\bar{x}-\mu_0)/(s/\sqrt{n})$
Two-sample t interval	$(\bar{x}_1-\bar{x}_2)\pm t^*\sqrt{s_1^2/n_1+s_2^2/n_2}$
Two-sample t test	$t=\frac{(\bar{x}_1-\bar{x}_2)-0}{\sqrt{s_1^2/n_1+s_2^2/n_2}}$
Matched pairs interval	$\bar{d}\pm t^*s_d/\sqrt{n}$
Matched pairs test	$t=(\bar{d}-\mu_{d,0})/(s_d/\sqrt{n})$