5-Minute Check on Activity 7-45-Minute Check on Activity 7-45-Minute Check on Activity 7-45-Minute Check on Activity 7-4

Click the mouse button or press the Space Bar to display the answers.Click the mouse button or press the Space Bar to display the answers.

Given the following data: 8, 1, 9, 2, 3, 4, 4, 5, 5, 6, 7, 8, 8

1. Find the mean

2. Find the median

3. Find the mid-range

4. Find the mode

5. What are the four descriptors of a distribution?

6. What are the three types of shape?

7. Which measures of center are resistant?

A CB DMean Median Mid-range Mode

Sum=70 70 / 13 = 5.38

Middle value of 13 is the 7th: 5

(min+max)/2 = (1 + 9) / 2 = 5

8 occurs the most often

Shape, Outliers, Center, Spread

Skewed left, symmetric, skewed right

B D

Activity 7 - 5

Sampling a Population

Objectives

• Know the difference between a census and a sample

• Identify the characteristics of a simple random sample

• Know what bias in a sample means

• Be able to select a simple random sample, when possible

• Identify how the size of a sample affects the result

Vocabulary• Census – when every individual in a population is

measured

• Sample – is a subset of the population (not the whole)

• Sampling – the process of collecting data from some fraction of the population

• Bias – simply means that some individuals are somehow favored over others in the population

• Simple Random Sample (SRS) – the result of a sampling method that assures every possible sample of the same size has an equal probability of being selected

ActivitySuppose you have a need to know the average weight of adults in the United States, or the mean age of blue crabs in the Chesapeake Bay, or the median household income in the Commonwealth of Virginia. To determine such measures of central tendency, it is not usually practical to obtain the data for every individual in the population.

Why is it not possible to conduct a census on the populations described above?US adults - costly as the census is done only every 10 years and still misses people

Blue crabs - we would have to empty the Bay to get them all

Household income - similar to the census; plus people don’t tell you the truth always about how much money they make

Activity contSince a sample can be of any size, how would you compare the practicality and usefulness of samples of size 1, 25, 2000, and 2 million when attempting to estimate the mean household income in Virginia.

Size n=1 easy to do, but what does it really show?

Size n=25 still easy to do, and it shows more data so it would be more significant

Size n=1 harder to do, but the results would be even more significant

Size n=1 very hard to do – nearing census like, but the results be have the strongest significance

Census versus a Sample

• Census – every member of a population is measured– expensive to do (and time consuming)– US census is done once every 10 years

• Sample– subset of the population is measured– inferences about the entire population are made

from the measurements gathered in the sample– telephone polls usually sample about 1000 people

A Few Thoughts on Sampling

Larger samples are better and more accurate, but there are always trade-offs in time and money.

Exact samples needed to get a certain accuracy can be determined by a mathematical formula (beyond our course). These numbers are most often seen in surveys reported in the media (with accuracy listed as percentage points).

Four Sources of Data

• Observational Studies– Census– Existing sources

• Previous study’s data• Data collections from other activities

– Survey sampling

• Designed experiments

Simple Random Sampling (SRS)

• Simple random sampling (SRS)– Most important sampling technique we study

• Many of the inference techniques have it as a requirement

• Often times it is assumed or given in the problem

– All possible samples of a given size must be equally likely

SRS and Random # Table

• In order to have students get the same results in a SRS, questions ask students to use a random number table

Bias

• Bias – nonsampling error introduced by giving preference to selecting some individuals over others, by giving preference to some answers by wording the questions a particular way, etc

• Key is systematically favoring something (not being a random selection)

Sources of Bias

• Undercoverage results from an incomplete frame on the surveyor’s part

• Nonresponse can be from either the surveyor (can’t find the person) or the person’s unwillingness to answer

• Response bias (lies) can result from either the respondent or the influence of the interviewer

Sampling Error Non sampling Error

Iceberg Sampling Process

Sampling Process

AnalysisProcess

AnalysisProcess

Interviewer errors

Collection Execution

Data-entry (input) errors

DesignerIncomplete FrameQuestionnaire Design

Poorly worded questions Inflammatory words Question order Response order

SubjectNonresponseMisrepresented answers

sample gives incomplete information about the population

Errors in Sampling

Sources of Nonsampling Error• Incomplete Frame• Nonresponse• Questionnaire Design

– Poorly worded questions– Inflammatory words– Question order– Response order

• Data Collection– Interviewer error– Misrepresented answers– Data-entry (input) errors

Sampling ExampleThe table below shows the current grades for all 20 students in Mr. Horton’s economics class.

ID Name Grade ID Name Grade

1 Adams 78 11 Lee 67

2 Baker 84 12 Maloney 91

3 Cooper 82 13 Nelson 69

4 Davenport 95 14 O’Brien 74

5 Elacqua 71 15 Park 81

6 Flanagan 83 16 Reeves 79

7 Grant 97 17 Snider 62

8 Haught 65 18 Thoreson 88

9 Jacobs 80 19 Vilece 93

10 Kim 78 20 Whiting 63

Setting Things Up

1. Enter the desired seed, 15

2. Press STO. An arrow appears in the display

3. Press MATH, highlight PRB, select rand (item #1)

4. Press ENTER. The seed is now set

Use randint(lower bound, upper bound, # needed)

Sampling Example conta) Compute the mean and median for the class

b) What do these values represent?

c) Use your calculator to generate 10 random numbers

d) Use the first 3 distinct (non repeating) as an SRS of Mr Horton’s class and calculate the mean and median

e) Repeat steps b and c with a sample size of 5

Mean = 79 and Median = 79.5 (??)

Population values; for mean,

4, 17, 8, 1, 7, 6, 5, 7, 19, 20

95, 62, 65 mean = 74 median = 65

18, 5, 16, 4, 1, 10, 20, 18, 2, 9

88, 71, 79, 95, 78 mean = 82.2 median = 79

Sampling Example contf) Repeat steps b and c with a sample size of 10 (note:

you probably will have to generate 15 random numbers)

g) What do you do if you get too many repeated numbers?

h) What do we notice as sample size increases?

i) Why did we all get the same “random” numbers?

11, 11, 15, 3, 10, 2, 1, 17, 1, 10, 17, 15, 18, 10, 20 3, 19, 13, ….

67, 81, 82, 78, 84, 78, 62, 88, 63, 93 mean = 77.6 median = 79.5

just get some more and keep eliminating numbers already chosen

The sample values should approach the population values

We started off with the same seed value and the calculator produces pseudorandom numbers (see www.random.org)

Summary and Homework

• Summary– Census measures every individual in a population

– Sample is a fraction of the population

– Sampling is the process of collecting a sample from the population

– Bias occurs when some individuals have a better chance of being selected than other

– Simple Random Sample (SRS) is the best type of sample

– Populations have parameters (, )

– Samples have statistics (x-bar and s) that estimate parameters

• Homework– pg 834 – 837; problems 1, 2, 5, 6