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FSA Algebra 2 EOC Benchmark And Intervention

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Algebra 2 includes a lot of the same concepts that Algebra 1 does: numbers, variables, operations...the list goes on. In fact, the two FSA Algebra EOC exams share the same reporting categories, so dust off those fond (or not-so-fond) memories of algebra and modeling, functions and modeling, and statistics and the number system. Shmoop's Benchmark and Intervention product can help students drill down into the differences with a full-length benchmark assessment that can be administered throughout the year, standard-specific review, and targeted practice questions to promote mastery. Algebra 2 mastery.

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Sample Content

Understand statistics as a process for making inferences about population parameters based on a random sample from that population (MAFS.912.S.IC.1.1)

We've been studying the different ways that a set or event can be analyzed in terms of certain probabilities, but now it's time to step back and analyze the sets and events themselves a little more. Two of the main tools we use for this kind of work are population parameters and sample statistics.

A population parameter tells us something about a population. A parameter could be something like puppies are cute, elephants are large, or little brothers are annoying.

A sample statistic is a bit different. It's an estimate of a population parameter. We get that estimate by sampling part of the population.


A cute population or a cute sample? (Source)

To know a population parameter, we have to check the entire population. We would have to evaluate every puppy for cuteness, measure every elephant for largeness, and evaluate every little brother in the world on a finely tuned annoyingness scale.

It's often impractical to analyze each and every member of a population, which is why we use sample statistics to estimate parameters. As much as we would like to look at all the cute puppies in the world, we just don't have the time.

When we make a declaration like, "Puppies are cute," we're making an inference based on a population sample. We may observe that our friend's little puppies are cute and infer that all puppies are cute based on that small sample.

However, to get an accurate sample statistic, it's important that we obtain a proper sample. We want it to be non-biased, random, and sufficiently large.

If we only ask puppy owners about their puppies, they're likely to be biased.

If we only ask pet shop customers looking to buy a new puppy, they're likely to be biased as well. They're probably there for a reason related to puppy cuteness.

If we only ask three random people about their puppies, we may actually find three people with ugly puppies, and then we'd have to conclude that puppies are not cute. It sounds absurd, but that's what happens with inadequate sample sizes.

The point is that sampling methods are important. Here are a few types of methods used by survey takers.

In non-probability methods, we can't be sure of the likelihood that every element, or type of response, will be chosen in a sample, or even that it will be chosen at all.

For example, let's say we post a question online asking people their opinion of puppies. We don't know the sample size because we don't know how many people will respond. Because the response is voluntary, we're also likely to get more responses from people with a strong opinion (really love or really hate puppies).

In probability-based methods, we set the sample size and characteristics to make sure that they're representative of the population; we know the population (N) and the sample size (n). We choose a size that gives us a certain confidence level and keep the sample as random as possible.

To have a simple random sample, we need to make sure that each item has an equal chance of being sampled and that all possible samples of n items are equally likely. If the first part is true but the second part is not, the sample is stratified—which may be what we want. Maybe we know that 75% of our population has a bias that we need to account for ahead of time.

We may want to group our population in certain ways. For example, if we randomly went door-to-door in our neighborhood asking about puppies, our conclusion is only valid to our neighborhood. We may want to compare puppies in our neighborhood to puppies across town, in Cincinnati, or in Paris. Getting data may be tough in that case, but we paint a broader picture.

If we ask exactly 100 people in our neighborhood, 100 people in Paris, and 100 people in Cincinnati about their puppies, we have a stratified sample. If we ask any 300 people that could be located in our neighborhood, or Paris, or Cincinnati, we have a simple random sample.

We could also stratify our sample with respect to a subset of the population. In our example, this makes more sense. There are far more people with puppies in Paris or Cincinnati than in our neighborhood because there are more people in Paris and Cincinnati period.

If the ratio of Paris to Cincinnati to our neighborhood were 20 to 10 to 1, we would sample twice as many people in Paris as we would in Cincinnati, and 20 times as many people in Paris as we would in our neighborhood. At this point, we're just looking for an excuse to go to Paris.