Algebra 2 Sampling, Surveys and Experiments Worksheet Solutions (PDF)

The population is all 1,200 students. The sample is the 120 students asked.

Population: 1,200 students; sample: 120 students

People at a rock concert are more likely to prefer rock, so they do not represent all teens. The sample was chosen for convenience.

Biased: the sample overrepresents rock fans

Only method B gives every student an equal chance of selection.

Method B

(a) Residents are asked: survey. (b) The groups choose themselves and nothing is assigned: observational study. (c) Treatments are assigned at random: experiment.

(a) survey; (b) observational study; (c) experiment

Random assignment tends to make the groups similar in every other way, such as age and health. Then a difference in results is likely caused by the treatment and not by something else.

To balance other variables and allow cause-and-effect conclusions

\(\frac{45}{150} = 0.30\). \(0.30 \cdot 2400 = 720\).

30%, about 720 students

\(\frac{1}{\sqrt{100}} = 0.10\) and \(\frac{1}{\sqrt{400}} = 0.05\).

10 percentage points and 5 percentage points

\(\frac{1}{\sqrt{625}} = \frac{1}{25} = 0.04\).

\(62\% \pm 4\%\) gives 58% to 66%.

About 58% to 66%

Let digits 0, 1, 2, 3 stand for rain (4 of 10 digits is 40%) and 4 to 9 stand for no rain.

Generate 7 random digits for one week, count the digits from 0 to 3, and repeat many times.

0-3 = rain, 4-9 = dry; 7 digits per week

The bars are for 0.35, 0.40, 0.45, 0.50, 0.55, and 0.60 as left edges, with counts 1, 3, 6, 5, 3, 2.

Samples of 0.60 or more: the last bar, 2 samples. \(\frac{2}{20} = 0.10\).

10%

A result that occurs in only 1% of random samples is unlikely to be due to chance if the population proportion were 0.5.

The population proportion is probably greater than 0.5

\(\frac{2}{200} = 0.01\): such a result occurs about 1% of the time with a fair coin.

Probably not fair

Differences of 0.6 or more occurred in the last bar only: 7 of 100. A difference as large as 0.8 is even rarer, so it is below 7%.

A difference this large rarely happens by chance alone.

Unusual: the app probably improved sleep

The interval is centered at 0.52 with half-width \(\frac{0.58 – 0.46}{2} = 0.06\).

About 0.06, or 6 percentage points

\(\frac{1}{\sqrt{10}} \approx 0.32\), so the true value could be anywhere from about 58% to 100%.

The sample is too small, and the method of choosing the dentists is unknown.

Margin of error about 32 percentage points; weak evidence

This is a stratified random sample: the population is split into groups (grades) and a random sample is taken from each.

It guarantees every grade is represented.

Stratified random sample

People chose whether to drink coffee, so the study is observational. Other variables, such as exercise or income, may differ between the groups and explain the result.

Observational: confounding variables are possible

Randomly assign 20 volunteers to eat the energy bar (treatment) and 20 to eat a plain bar (control). Time everyone’s 100-meter sprint.

The response variable is the sprint time; compare the group averages.

Random assignment, 20 vs 20, response: sprint time

\(\frac{1}{\sqrt{n}} = 0.05\) gives \(\sqrt{n} = 20\), so \(n = 400\).

400 people, four times as many

The margin of error depends on the sample size, not on the population size. Samples of the same size give about the same margin of error.

No; the same sample size gives the same margin of error

\(2 \cdot \frac{6}{6} = 2\) cm.

\(24 \pm 2\) gives 22 to 26.

About 22 cm to 26 cm