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ISTEP+ Grade 7 - Math: Test Prep & Practice22 chapters | 171 lessons | 8 flashcard sets

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson explains random samples and how to use data from a random sample of a population to make predictions about the whole. We use a real world example to illustrate this process.

Suppose a math tutoring company, MathToots, has 1000 tutors. Mathtoots wants to know their level of job satisfaction, so they put Bob from HR on the task of finding this information.

It would be an extremely long and tedious task for Bob to interview all 1000 tutors. Instead, he decides to interview 20 of the tutors and then make generalizations about the group from his findings. How should Bob choose the 20 tutors?

If he chooses tutors that he knows enjoy or dislike their job, then the data would be skewed because it would be biased. Bob should choose a sample in such a way that each tutor is equally likely to be chosen.

Bob quickly has his computer assign random numbers to each of the tutors, so he won't see their names. He then prints off each of these numbers (1000 of them, one for each tutor) and puts them in a large basket. Finally, he puts on a pair of vision blocking sunglasses and selects 20 of the numbers out of the bowl. The 20 numbers represent the tutors he will interview.

Phew! That was a process, but we know for sure that the sample of tutors is unbiased and random.

In mathematics, we call the 20 tutors a random sample, where a **random sample** of *a* elements of a population is a sample of the population that is random, and every possible sample of *a* elements of that population is equally likely.

After Bob gets his 20 tutors, he interviews them. The results are as follows:

- 15 of the tutors have high job satisfaction.
- 4 of the tutors have medium job satisfaction.
- 1 of the tutors has low job satisfaction.

From this, he makes the following generalizations about the entire population of 1000 tutors.

- 15/20 = 0.75, so 15 is 75% of 20. About 75% of the 1000 tutors, or 750 tutors, have high job satisfaction.
- 4/20 = 0.20, so 4 is 20% of 20. About 20% of the 1000 tutors, or 200 tutors, have medium job satisfaction.
- 1/20 = 0.05, so 1 is 5% of 20. About 5% of the 1000 tutors, or 50 tutors, have low job satisfaction.

Bob reports the data to his boss, Karen, so she can draw inferences about the tutors. For example, she may consider how many tutors may quit in the near future or how to implement programs to increase job satisfaction.

In mathematics, we call this **making predictions** using random sampling. We basically take data from a random sample of a population and make predictions about the whole population based on that data.

Because the sample of the population is random in random sampling, there is a lot of room for variation (especially when the sample is very small). For instance, suppose Bob had some extra time, so he used 150 random tutors instead of just 20. The results would vary from the sample of 20 tutors, but they would be more accurate because the sample size is larger.

Similarly, if the random sample of 20 tutors was a different group of tutors, the results may have been a bit different. Though random sampling allows for unbiased samples, the data from different random samples can vary. Let's take a look at another example of making predictions from random samples.

Even though only a small percentage of MathToot's tutors have low job satisfaction, Karen decides to implement a rewards program with a gift certificate to a store when a tutor's students show improvement. She has four stores to choose from, and she wants to know which one is the most popular with the tutors.

Ah-ha! Another opportunity to use random sampling to make predictions! She has Bob, her go-to guru for random sampling, draw a random sample of 50 tutors and poll them on which store they prefer. The results are shown in the table.

Store | Number of Tutors |

A | 28 |

B | 6 |

C | 14 |

D | 2 |

Let's use this data from the random sample of 50 tutors to predict how many of the total population of tutors would prefer each store. To do this, we first figure out what percent of the 50 tutors liked each store. This is shown in the image.

From this, we can make the following predictions about the entire population:

- 56% of the tutors prefer store A, so 0.56 × 1000 = 560 tutors prefer store A.
- 12% of the tutors prefer store B, so 0.12 × 1000 = 120 tutors prefer store B.
- 28% of the tutors prefer store C, so 0.28 × 1000 = 280 tutors prefer store C.
- 4% of the tutors prefer store D, so 0.04 × 1000 = 40 tutors prefer store D.

Based on this data, Karen decides to get the gift certificates at store A. Pretty neat that we can make so many predictions using a random sample like this, huh?

A **random sample** of *a* elements of a population is a sample of the population that is unbiased, random, and every possible sample of *a* elements of that population is equally likely.

We can use data from random samples of a population to **make predictions** about the population as a whole by drawing inferences from our findings. Of course, there will be variations based on the size of the sample and differing aspects of each random sample, but we can do this because random samples are random and unbiased. We know they most accurately represent the population as a whole.

As we've seen, being able to make predictions about a population as a whole from random samples is quite useful in various real-world applications, so let's keep this process in mind the next time we want to analyze or make predictions about a large population.

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ISTEP+ Grade 7 - Math: Test Prep & Practice22 chapters | 171 lessons | 8 flashcard sets

- Descriptive & Inferential Statistics: Definition, Differences & Examples 5:11
- Difference between Populations & Samples in Statistics 3:24
- What is Random Sampling? - Definition, Conditions & Measures 5:55
- Using Data from a Random Sample to Make Predictions
- Go to ISTEP+ Grade 7 Math: Populations & Statistics

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