Statistical AnalysisNotes

Vocabulary

Statistic: a number that describes a sample

Parameter: a number that describes a population

We will study how to use statistics to estimate parameters. In other words, we can use information about a small group to infer things about a large group, provided that the small group is fairly representative of the large group.

Let’s look at an example.

In a survey of 50 students at a high school, 32 students said they plan to go to the prom. The school has 720

students.

The statistic is that 32 out of 50 students in the sample say they are going to the prom.

The parameter would be the number of students in the school that plan to go to the prom.

We can use the statistic to estimate the parameter.

Example, continued

We can write a proportion to estimate the parameter:

Now, let’s substitute in this information from the problem:

Cross-multiply to solve for x: x = 460.8

This means that about 461 students plan to attend the dance.

More About Statistics and Parameters

A lottery ticket says the 1 in 3 tickets will be a winner. Michael bought 5 tickets, but hasn’t won a prize yet. He thinks the advertised odds must be incorrect.

However, Michael is basing this on too small of a sample.

Remember that samples must be random and representative of an entire population if we are going to use them to draw VALID conclusions.

Types of Sampling Distributions

Simple Random Sample: members are chosen so that everyone has an equal chance of being selected Example: 100 students’ names are randomly chosen from a

list of all students in the school

Systematic Sample: members are chosen using a pattern Example: every other student is selected from a list of all

students

Stratified Sample: members are randomly chosen from subgroups created from the population Example: 3 students are chosen from each class in school

Types of Sampling Distributions, continued

Cluster Sample: an entire subgroup is surveyed after the subgroup was randomly chosen from the population Example: an entire English class is surveyed

Convenience Sample: members are chosen because they are easily accessible Example: 100 students with first lunch are surveyed

because the surveyor also has first lunch

Self-Selected Sample: members voluntarily participate Example: 100 students show up to take a survey after

school

More About Sampling

Usually cost and accuracy are the most important considerations when choosing how to sample a population

A census is the most accurate survey, but is time-consuming and costly

More accurate sampling methods are usually more difficult to use or are more expensive than others

Simple random samples, stratified samples, and cluster samples are very accurate

Convenience samples and self-selected samples are not very accurate due to the nature of the individuals involved

Let’s practice! Classify the sampling method used in each scenario below. Then, go to the next slide to check your answers.

1. Town officials are deciding whether to build a public park downtown. They decide to ask residents to come to a town hall meeting to vote on the issue.

2. The owner of a movie theater wants to know how customers feel about the new 3D glasses the theater is using. She asks every 25th person who exits the theater each Friday night for a month.

3. The manager of the produce department at a grocery store wants to determine the quality of the pears they receive. She examines several randomly selected oranges from each of the 10 crates.

How well did you do?

1. Self-Selected Sample – the individuals volunteer to come to the meeting to vote on whether or not there should be a park built

2. Systematic Sample – individuals are chosen using a pattern

3. Stratified Sample - pears are chosen randomly from each crate

It’s time to apply what we’ve learned!

We are going to conduct a simulation of random sampling.

To do this, we are going to use a random digits table, that you should have downloaded and printed earlier. If you haven’t done so already, please click on the link to it on the main page and print it now.

A random digit table contains lines of numbers in which all of the digits from 0 to 9 are represented equally.

We will use these digits to randomly choose individuals, and use their information to find a sample mean.

Below are the last names of 20 managers of a set of retail stores. The owner of the store wants to know the opinions of the managers about the stores’ new

pricing system.We will use the random digits table to select 6

managers to be interviewed by the owner.

Baker Helms Myers Smith

Boss Jones Nixon Stamm

Crown King O’Neal Thorn

Cruz Loren Rogers

West

Duram

Moss Sand Wills

First, let’s assign each manager a number.

01 Baker 06 Helms 11 Myers 16 Smith

02 Boss 07 Jones 12 Nixon 17 Stamm

03 Crown 08 King 13 O’Neal 18 Thorn

04 Cruz 09 Loren 14 Rogers 19 West

05 Duram 10 Moss 15 Sand 20 Wills

Now, let’s use line 139 in our table to choose our 6.

Line 139 in the table is 55588 99404 70708 41098 43563 56934 48394 51719

We go through the row to find double-digit numbers that could represent one of our managers. We would see 55, 58, 89, and 94 that are not aligned with our numbers…but then we get 04 which matches Manager Cruz. Cruz will be our first choice to be interviewed.

Continue using the table to identify 5 others. The highlighted numbers represent our other candidates: 70, 70, 84, 10, 98, 43, 56, 35, 69, 34, 48, 39, 45, 17, 19.

We have reached the end of the row and have chosen 4 managers: Cruz (04), Moss (10), Stamm (17), and West (19). We still need 2 more, so we can just continue on to line 140 in the table.

12, 97, 51, 32, 58, 13

The other two managers we will choose are Nixon (12) and O’Neal (13).

More About Using Random Numbers

Unless told a specific row to use, you can choose any row and use it to select individuals.

You can also choose how to identify your individuals. We used two-digit numbers, but you can also choose three-digit numbers, four-digit numbers, etc.

Using random numbers to select our individuals ensures that we get a simple random sample.

Here’s one last example that we can use to practice using the random digits table.

Researchers want to know the average length of a great white shark. Since it would be impossible to measure every great white shark in the world, they recorded the measurements of the lengths of 24 great whites who entered a research facility. The data is summarized on the next slide, where the whales have been numbered, and their weights are given in feet.

Lengths of 24 Sharks

Shark #

Length

Shark #

Length

Shark #

Length

Shark #

Length

01 18.7 07 12.3 13 18.6 19 16.4

02 15.7 08 18.3 14 14.6 20 15.8

03 14.9 09 17.6 15 12.1 21 16.4

04 16.7 10 17.8 16 16.2 22 12.6

05 17.8 11 13.8 17 12.2 23 15.2

06 14.7 12 12.4 18 13.2 24 15.8

Here’s what we need to do.

First, let’s use the random number table to choose 7 sharks. So that we can check our work, let’s start at line 116. Remember to continue on to the next line if you need to.

You should get in order: 14, 03, 10, 22, 06, 11, 16

The mean of our sample would be the average of these lengths. If you add up the lengths of these 7 sharks and divide by 7 you should get 14.93 inches.

In summary,

The random number table can be time-consuming, especially if we need to use a large sample size.

We also need to be able to identify a large number of characteristics of populations, and the random number process limits how much information we can accurately infer.

In our next lesson, we will discuss ways to expand our use of information from a sample in order to more accurately describe an entire population.