Back to Exercise: Understand statistics as a process for inference

Exercises: Understand Statistics as a Process for Making Inferences

Work through each section in order. For each scenario, read carefully and use the correct vocabulary (population, parameter, sample, statistic) and notation. Explain your reasoning where a written response is asked for.

Grade 10·21 problems·~40 min·Common Core Math - HS Statistics and Probability·group·hss-ic-a-1

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Recall / Warm-Up

A school district wants to know the true average number of hours of sleep per night for all 8,000 of its students. Which of these is the population?

All 8,000 students in the district

The 150 students who were actually surveyed

The average sleep computed from the survey

The true average sleep of all 8,000 students

In a survey, 39 of 60 randomly chosen Lincoln High students said they would re-enroll in an elective, giving $\hat{p} = 39/60 = 0.65$ . The value $0.65$ is best described as a:

parameter

statistic

population

sample

A random sample of 80 workers has a mean commute time of $\bar{x} = 27$ minutes. A student concludes, "So the mean commute time for all 50,000 workers is exactly 27 minutes." What is wrong with this conclusion?

Nothing — the sample mean always equals the population mean.

The sample statistic $\bar{x}$ is an estimate of the parameter $\mu$ , not the exact value of $\mu$ ; a different sample would give a different $\bar{x}$ .

The student should have used the median instead of the mean.

The sample of 80 is too small to compute any mean.

Which procedure produces a genuinely random sample of a school's students?

Surveying the first 30 students who walk into the cafeteria.

Assigning every student a number and using a random-number generator to pick 30 numbers.

Asking for volunteers to fill out an online form.

Surveying everyone in one homeroom class.

Fluency Practice

A nutritionist wants to know the true mean daily sugar intake of all 2,000 students at a high school. She randomly selects 50 students and finds their mean intake is 68 grams.

Identify each of the following and give the correct symbol where one applies ( $\mu$ , $\bar{x}$ , $p$ , or $\hat{p}$ ):
(a) the population
(b) the parameter
(c) the sample
(d) the statistic

A researcher poses the question "What proportion of all town voters support the new park?", randomly polls 400 voters, finds 220 in favor, and concludes the town proportion is "around 0.55." In the four-step inference process, computing $220/400 = 0.55$ corresponds to which step?

Step 1: pose a question about a parameter

Step 2: draw a random sample

Step 3: compute a statistic

Step 4: infer about the parameter

A problem states: "A bag contains 60% red marbles. If you draw 20 marbles, how many do you expect to be red?" Compared with statistical inference, which direction does this reasoning run?

Population to sample — this is probability (we know the population and predict the sample).

Sample to population — this is inference (we observe a sample and estimate the population).

It runs in neither direction; it is just arithmetic.

Sample to sample — it compares two samples.

Poll X surveys 1,000,000 people who chose to call in to a radio show. Poll Y surveys 600 people selected by a random-number draw. Which poll's result should be trusted more as an estimate of the true population opinion, and why?

Poll X, because a sample of one million is far larger and therefore more representative.

Poll Y, because random selection removes bias; Poll X is a voluntary-response sample whose large size cannot fix its bias.

Poll X, because larger samples always have smaller bias.

Neither — sample size and selection method do not affect trustworthiness.

Three different random samples of 60 students are drawn from the same Lincoln High population. They give $\hat{p} = 0.62$ , $\hat{p} = 0.68$ , and $\hat{p} = 0.65$ . The true proportion $p$ for the whole school did not change between samples.

(a) Why do the three samples give three different values?
(b) What is this sample-to-sample variation called?

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