Unit 4 · Topic 4.4 Beta

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.

Practice 2: Collect DataPractice 4: Interpret Results

Part 1 · Hook

Why this matters

A delivery company promises 30 minutes on average. A manager thinks it is longer. Before anyone computes a number, the test has to be set up: which parameter, which hypotheses, which direction, and whether the conditions let a t procedure be trusted. Most lost points on inference questions happen here, not in the arithmetic.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Which pair of hypotheses is written correctly?

$H_0: p = 0.5$ , $H_a: p > 0.5$
$H_0: \hat{p} = 0.5$ , $H_a: \hat{p} > 0.5$
$H_0: p > 0.5$ , $H_a: p = 0.5$
$H_0: p = 0.56$ , $H_a: p > 0.56$ , using the sample result

Show the answer

Hypotheses are about the parameter, the null holds the equals sign, and the null value comes from the question.

Correct: $H_0: p = 0.5$ , $H_a: p > 0.5$ :
$H_0: \hat{p} = 0.5$ , $H_a: \hat{p} > 0.5$ :
$H_0: p > 0.5$ , $H_a: p = 0.5$ :
$H_0: p = 0.56$ , $H_a: p > 0.56$ , using the sample result:

2. For paired data, a confidence interval for the mean difference is built from

the differences within each pair, as one sample
the two groups as separate samples
the larger of the two means
the before values only

Show the answer

Subtract within each pair, then use a one-sample t procedure on the differences.

Correct: the differences within each pair, as one sample:
the two groups as separate samples:
the larger of the two means:
the before values only:

Part 4 · See it

See it first

Left: a slope chart of resting heart rate for 12 gym members, before and after a breathing class, on a scale from 60 to 90 beats per minute; 11 lines slope down and one slopes up. Right: a dotplot of the 12 differences, before minus after: -1, 1, 1, 1, 2, 3, 3, 3, 5, 5, 5, 6, with mean 2.83 and standard deviation 2.12. All but one difference is above 0. — Resting heart rates of 12 gym members before and after a breathing class, joined by a line for each member, and the dotplot of the 12 differences (before minus after). The test uses the differences. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

Start from the question, before the datait gives the parameter, the hypothesized mean $\mu_0$ and the direction of $H_a$
Write $H_0: \mu = \mu_0$ and $H_a$ with $>$ , $<$ or $\ne$ define $\mu$ in context: "the true mean ... of all ..."
If each individual gives two linked measurementsthe data are paired: test $H_0: \mu_d = 0$ on the differences, with the order of subtraction stated
Check random, 10% and normal/large sample in wordsa one-sample (or paired) t test can be trusted, with $\text{df} = n - 1$

Part 6 · Key ideas

Key ideas

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.

Part 7 · Misconception

A common mistake

The wrong idea: The sample mean was below 30, so the alternative should be $H_a: \mu < 30$ .

What actually happens: The alternative comes from the research question asked before the data. If the question is "different from 30", the alternative is $\mu \ne 30$ whatever the sample shows.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

Delivery times

A delivery company advertises a mean delivery time of 30 minutes in one city. A manager suspects the true mean is longer. She takes a random sample of 45 of last month's thousands of deliveries in the city and records each delivery time (invented data).

The study at a glance
Claim	Suspicion	Sample	Variable
mean delivery time is 30 minutes	the mean is greater than 30	45 deliveries, chosen at random	delivery time in minutes

1. Which hypotheses should the manager test?

$H_0: \mu = 30$ , $H_a: \mu \ne 30$
$H_0: \mu = 30$ , $H_a: \mu > 30$
$H_0: \mu > 30$ , $H_a: \mu = 30$
$H_0: \bar{x} = 30$ , $H_a: \bar{x} > 30$

Show the answer

$H_0: \mu = 30$ and $H_a: \mu > 30$ , where $\mu$ is the true mean delivery time of all the company's deliveries in the city. The hypothesized mean is $\mu_0 = 30$ .

$H_0: \mu = 30$ , $H_a: \mu \ne 30$ : The manager suspects longer times before seeing the data, so the alternative is one-sided: greater than 30.
Correct: $H_0: \mu = 30$ , $H_a: \mu > 30$ : The null is the advertised value; the alternative is the manager's suspicion, longer deliveries.
$H_0: \mu > 30$ , $H_a: \mu = 30$ : The null hypothesis holds the equals sign and the claim being tested; the suspicion belongs in $H_a$ .
$H_0: \bar{x} = 30$ , $H_a: \bar{x} > 30$ : Hypotheses are about the population mean $\mu$ , never the sample mean $\bar{x}$ .

2. Why is a one-sided alternative appropriate here?

The sample mean will turn out to be greater than 30
One-sided tests are easier to reject with
The manager suspected longer delivery times before collecting data
The sample size is larger than 30

Show the answer

A one-sided $H_a$ is set by a directional question stated in advance ("is the mean longer than advertised?").

The sample mean will turn out to be greater than 30: Choosing the direction from the sample result is a common error; the question decides it.
One-sided tests are easier to reject with: Ease of rejecting is not a reason; the research question is.
Correct: The manager suspected longer delivery times before collecting data: The direction of $H_a$ comes from the question asked before the data, not from the sample result.
The sample size is larger than 30: Sample size affects the conditions, not the direction of the alternative.

3. The manager has not graphed the delivery times. Is the normal/large sample condition met?

Yes, because $n = 45 \ge 30$
No, because delivery times are probably skewed
Yes, because 45 is less than 10% of last month's deliveries
No, because a graph of the sample is required for each test

Show the answer

Normal/large sample: $n = 45 \ge 30$ . (Still, a quick graph is good practice, to catch extreme outliers.)

Correct: Yes, because $n = 45 \ge 30$ : With a sample of at least 30, the central limit theorem makes the sampling distribution of $\bar{x}$ approximately normal.
No, because delivery times are probably skewed: Some skew is fine with $n = 45$ ; only very strong skew or extreme outliers would worry us.
Yes, because 45 is less than 10% of last month's deliveries: That is the 10% condition, which is about independence.
No, because a graph of the sample is required for each test: A graph of the sample is required only when $n < 30$ and the population is not known to be normal.

4. A student writes " $H_0: \bar{x} = 30$ " for this study. What is wrong?

It should use 45 instead of 30
Nothing, $\bar{x}$ and $\mu$ mean the same thing
It should be a two-sided statement
It is about the sample mean, which is known once the data are in

Show the answer

Use $\mu$ (the parameter) in hypotheses, never $\bar{x}$ .

It should use 45 instead of 30: 45 is the sample size, not the hypothesized mean.
Nothing, $\bar{x}$ and $\mu$ mean the same thing: $\bar{x}$ is a statistic from one sample; $\mu$ is the population parameter.
It should be a two-sided statement: The null hypothesis is an equality; the problem is the symbol, not the sides.
Correct: It is about the sample mean, which is known once the data are in: Hypotheses describe the population. Write $H_0: \mu = 30$ , with $\mu$ defined in context.

5. A sleep researcher wants to know whether college students at a large university sleep a mean amount different from 8 hours a night. Which hypotheses fit?

$H_0: \mu = 8$ , $H_a: \mu \ne 8$
$H_0: \mu = 8$ , $H_a: \mu < 8$
$H_0: \mu \ne 8$ , $H_a: \mu = 8$
$H_0: \mu = 7.4$ , $H_a: \mu \ne 7.4$

Show the answer

$H_0: \mu = 8$ , $H_a: \mu \ne 8$ , where $\mu$ is the true mean nightly sleep of students at the university.

Correct: $H_0: \mu = 8$ , $H_a: \mu \ne 8$ : "Different from" points to a two-sided alternative, with 8 hours as the hypothesized mean.
$H_0: \mu = 8$ , $H_a: \mu < 8$ : The question asks about any difference, not only less sleep.
$H_0: \mu \ne 8$ , $H_a: \mu = 8$ : The equality belongs in the null hypothesis.
$H_0: \mu = 7.4$ , $H_a: \mu \ne 7.4$ : The hypothesized value comes from the question (8 hours), not from a sample result.

6. A study measures reaction time for 20 sets of identical twins; one twin in each set is randomly assigned to drink coffee and the other water. Which analysis fits?

A test comparing two independent groups of 20
A paired t test on the 20 within-twin differences
A one-sample t test on the 20 coffee times alone
A one-proportion z test on the share of faster coffee twins

Show the answer

Matched pairs design: compute coffee minus water within each twin pair and test $H_0: \mu_d = 0$ with a paired t test (df 19).

A test comparing two independent groups of 20: The twins are matched, so the two groups are not independent.
Correct: A paired t test on the 20 within-twin differences: Twins are matched pairs, so each pair gives one difference (coffee minus water), and the test is on the mean difference.
A one-sample t test on the 20 coffee times alone: The water twins carry the comparison; leaving them out loses the point of the design.
A one-proportion z test on the share of faster coffee twins: That throws away the size of each difference; the reaction times are quantitative.

Part 9 · Summary

Summary

To set up a t test for a mean, name the parameter in context, write

H_0: \mu = \mu_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

\mu_d

with a paired t test on the differences, saying which way you subtracted.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic