faculty.montgomerycollege.edufaculty.montgomerycollege.edu/maronne/Math117A-LOCK-STATIS… · Web viewNote: mu = 4.29. That is, the sum of the number of letters of the 268 words

Section 3.1: Sampling Distributions1) Inference2) Population3) Sample 4) Parameter (Is it variable or fixed?)5) Statistic (Is it variable or fixed?)6) Best estimate or point estimate7) Sampling distribution of the proportion

a. What is it?b. Describe

i. Shapeii. Center (mean)

iii. Variability (standard error)8) Effect of sample size

- Sample size and standard error9) Importance of random sampling10) Simulating with STATKEY

HOW DO WE DO IT? Go to sampling distributions for proportions, edit proportion, change sample size, generate samples

11) Simulating with the CALCULATORHOW DO WE DO IT? Use randBin(n, p) which is in the MATH PRB menu; you obtain a count, divide by sample size to get p-hat

NING OBJECTIVESading this module, you

Section 6.1 – Describe the Distribution of a Sample Proportion

1) Use formulas to describe the distribution of sample proportionsa. The centerb. The standard errorc. The shape (justify)2) Simulations via STATKEY and CALCULATOR

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Section 3.1: Using Sample Statistics to estimate Population ParametersWe are continuously bombarded with survey results. Were they obtained from the complete population or from a sample?Should we trust these reports? Here are some examples.

1) The data shows the average price of gasoline in some states – January 2019

2) From a survey conducted in December 2018 – (survey of 1114 American adults)

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Section 3.1: Inference

The process of estimating population ________________________ using sample ____________________________ is called __________________

Recall the notation:For quantitative variableCalculate means

For categorical variableCalculate proportions

Sample Statistic x-bar p-hatPopulation Parameter mu p

Thinking about inference

1) Will the sample statistic (x-bar or p-hat) be equal to the population parameter (mu or p)?

2) Will a different sample produce the same sample statistic (x-bar or p-hat)?

3) How do we know if the observed sample statistic is a good estimate for the parameter?

a. Around what value do they vary?b. By how much they vary? What is the expected variability?

The objective of this section is to study the variability of sample statistics.We need to convince ourselves that when we use a sample statistic to estimate a population parameter, we make a sampling error and this error can be made smaller under certain conditions.

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Section 3.1: Understanding Sampling Distributions via Simulations

3) Recall the simulation from section 1.2 – Lincoln’s Gettysburg Address

Objective: Estimate the average number of letters per word in Lincoln’s Gettysburg Address (Why did we do an estimate instead of calculating the exact average?)

Population – cases: 268 words in the Lincoln’s Gettysburg AddressVariable: number of letters per word

How did we do it? We selected a random sample of size 10, we counted the number of letters in each of the 10 words selected and we found the average x-bar. We used the x-bar as an estimate for mu

Was the obtained sample mean exactly the same as the population parameter? Were all sample means the same?

The dot-plots of x-bars for random samples versus biased samples are shown below

Which sampling technique produces better estimates?

Note the importance of random sampling!

Note: mu = 4.29. That is, the sum of the number of letters of the 268 words divided by 268 = 4.29. In our simulation, the x-bars from random samples varied around the parameter mu = 4.29. The sample statistics x-bars obtained were not necessarily equal to mu, but they were “close” to mu. When we estimate a parameter we expect a sampling error (difference between x-bar and mu); however, as we observed in our simulation, the error is not big; and, it can be made smaller under certain conditions!

Section 3.1: Sampling Distribution of the Proportion via Simulations4

4) Below you find the gender of 90 students from my classes. We would like to estimate the proportion of female students in this group of 90 students

A) Select 10 students at random and calculate the proportion of females in your sample. B) Select 20 students at random and calculate the proportion of females in your sample.

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Section 3.1: Sampling Distribution of the Proportion via Simulations

Problem 4 continues – Recording the results of your simulation

C) Results of your simulation: Dot plot of the p-hats from samples of size 10

____|____|____|____|____|____|____|____|____|____|____|____|___

D) What do you observe?E) What is the parameter?F) Are all p-hats obtained the same? Are they the same as the parameter? Is the sampling error large?

G) Results of your simulation: Dot plot of the p-hats from samples of size 20

____|____|____|____|____|____|____|____|____|____|____|____|___

H) Comment on the effect of sample size.

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Section 6.1 – D: Describing Distribution of Sample Proportions p-hatWe will be working with categorical variables only

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Describing the Distribution of Sample Proportions

When choosing random samples of size n from a population with a proportion p, the distribution of the sample proportions has the following characteristics:

Center (mean): The center is equal to the population proportion, p.

Spread/Variability: The standard error is .

Central Limit TheoremShape: If np ≥10 (number of YES) and n(1− p) ≥10 (number of NOs), the distribution is reasonably normal.

Summarizing with the notation for normal distributions:

If np ≥10 and n(1− p) ≥10,

Effect of the sample size – Based in what you learned in chapter 3, The larger the sample size n, the …………………………

the variability (standard error)

Based on the formula,If we divide by a larger number, the quotient is ……………………

And the standard error is ……………………….

Section 3.1: Describing the Distribution of Sample Proportions p-hatSimulate with STATKEY

5) Back to the example: 61.1% of the students in a population are female. PART 1 - If we select random samples of 10 students from this population,

A) Use the formulas from section 6.1 to describe the distribution of sample proportions p-hat.i. Shape, (ii) Center (mean of p-hats), (iii) Spread (standard error)

B) Do a simulation with STATKEY – Go to STATKEY – Sampling Distribution – Proportions Edit proportion (0.611), change Sample size to 10 – Generate samples

C) Sketch distribution labeling one, two and three standard errors around the mean

PART 2 - If we select random samples of 20 students from this population, D) Use the formulas from section 6.1 to describe the distribution of sample proportions p-hat.

i. Shape, (ii) Center (mean of p-hats), (iii) Spread (standard error)E) Do a simulation with STATKEY – change sample size to 20F) Sketch distribution labeling one, two and three standard errors around the meanG) Comment on the effect of sample size

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Section 3.1: Describing the Distribution of Sample Proportions p-hatObserving the Variability of Sample Proportions – Using WEB APPLETS

http://www.rossmanchance.com/applets/OneProp/OneProp.htm?candy=16) Example: Reese’s Pieces – 40% of Reese’s pieces are orange

a) The cases are _________________. The category of interest is ______________b) The population proportion (parameter) is _____________c) Select samples of size 25. Count the number of orange pieces in the sample, determine the sample

proportion and construct dotplot. Repeat this 10 times. Note – this applet use pi, instead of p for the population proportion.

d) Use the formulas from section 6.1 to describe the distribution of sample proportions p-hat.i. Shape, (ii) Center (mean of p-hats), (iii) Spread (standard error)

e) Sketch distribution labeling one, two and three standard errors around the meanf) Do a simulation with STATKEY – change proportion and sample size – generate a few p-hats and place them

on the sketch

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http://www.rossmanchance.com/applets/OneProp/OneProp.htm?candy=1

Section 3.1 and Section 6.1 – Describing Distribution of Sample Proportions Simulate with STAKEY and with the CALCULATOR

7) Example: The last census reported that 27.5% of US adults at least 25 years old have a college bachelor’s degree or higher.

The cases/population are ________________ The category of interest is _____________The population proportion (parameter) is _____________

If we select at random 200 US adults 25+ years old, count how many have a college bachelor’s degree or higher and calculate the corresponding proportion,


B) Sketch distribution labeling one, two and three standard errors around the meanC) Simulate with STATKEY – change proportion and sample size – generate a few p-hats and place them on

the sketchD) Simulate with the calculator . Use randBin(n,p) which is in the MATH PRB menu.

Press MATH, Arrow to PRB, Select 7:randBin( ; Enter sample size and parameter p(in older calculators that have a horizontal format you need to separate with a comma)The obtained result is a COUNT; (how many have the specific characteristic). You need to calculate the proportion by dividing by the sample size.

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Section 3.1 and Section 6.1 – Describing Distribution of Sample Proportions

8) Example: According to the MC Fall 2017 student profile, 53.4% of MC students are female.

The cases/population are ____________________ The variable (category of interest) is ______________The population proportion (parameter) p = ______________

If we repeatedly select random samples of 100 MC students and determine the proportion of female students in each of the samples,




Press MATH, Arrow to PRB, Select 7:randBin( ; Enter sample size and parameter p(in older calculators that have a horizontal format you need to separate with a comma)

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Section 3.1 and Section 6.1 – Describing Distribution of Sample Proportions Observing the Variability of sample proportions using STAKEY

9) Example: Proportion Speaking a Language other than English in New MexicoFrom the 2010 US Census, we learn that 36.2% of the residents of New Mexico speak a language other than English at home.


If we repeatedly select random samples of 100 residents of New Mexico and calculate the proportion of the sample that speaks a language other than English at home,





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Section 3.1 and Section 6.1 – Describing Distribution of Sample Proportions

Importance of Sample SizeDo we always obtain a normal distribution?

10) Use some of the examples from the prior pages and use STATKEY to explore the distribution of sample proportions for small sample sizes. Do we always obtain a normal distribution?

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Section 3.1: Sampling Distribution of a Proportion

11) Example: In the Fall 2014, it was reported that 13.1% of MC students were Hispanic.


This is what I got via STATKEY. Do you understand what this means?a) Explain the meaning of the following simulation.

b) Use the formulas from section 6.1 to describe the distribution of sample proportions p-hat.i. Shape, (ii) Center (mean of p-hats), (iii) Spread (standard error)

c) Now you try it. Do you understand what you are doing?

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From Section 3.1: Sampling Distribution – SummaryRead section 3.1 of the book to complete the following:

d) What is a sampling distribution? Complete the following:If we select many samples of the same size from the same population and compute the sample statistic for each of the selected samples, we generate the _______________________________________ of the statistic.A sampling distribution shows how the sample statistics (x-bar or p-hat) vary from sample to sample.

e) Shape and Center of a sampling distribution: If samples are ____________________________

and the sample size is ______________________________ the sampling distribution will be (describe the shape

here) _______________________________________ and centered at ________________________________.

From the 95% rule we can conclude that about ________________ of sample statistics will be within

___________________________________ from the parameter.

f) What is the standard error? Circle the correct choice below:a) The standard deviation of the populationb) The standard deviation of the samplec) The standard deviation of the distribution of sample means (or sample proportions)

g) Effect of sample size on the standard error: What happens with the standard error of the sampling distribution as the sample size increases?

h) Sample Size Matters! As the sample size _______________________ the variability of sample statistics tend to _______________________and sample statistics tend to be closer to the _________________________ of the ______________________________. Show sketches that reinforce this concept.

i) Why is it important to select Random Samples? Refer here to the section 1.2 – Gettysburg

example; the simulations done in class and sketch dot-plot here

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Sections 3.1 and 6.1 - Describe the Distribution of Sample Proportions

What to do by Thursday 2/28?

1) Read the notes from classIN WILEY PLUS DO THE FOLLOWING:2) Watch again, Section 3.1 – Videos – Proportions3) Watch: Section 3.1 – Interactive Video4) Solve the problems in the assignment – Section 3.1 – Sampling Distribution – PHAT

5) Complete the notes from class6) From the book: section 6.1, book, pages 404 – 407

1. Read the examples and2. Solve all problems 6.1-6.10 that are listed in pages 406/407

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Social Media Use in 2018Majority of Americans now use Facebook, YouTube

% of U.S. adults who say they use the following social media sites online or on their cellphone. Based on a sample of 2002 Adults (18+) living in the U.S.

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Due next class – Select one category, and:1) Write an inferential statement.2) Use the formulas from section 6.1 to describe the distribution of sample proportions p-hat for samples of size 100.

i. Shape, (ii) Center (mean of p-hats), (iii) Spread (standard error)3) Sketch distribution labeling one, two and three standard errors around the mean4) Simulate with STATKEY – change proportion and sample size – generate a few p-hats and place them on the sketch5) Simulate with the calculator. Use randBin(n,p) which is in the MATH PRB menu.


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faculty.montgomerycollege.edufaculty.montgomerycollege.edu/maronne/Math117A-LOCK-STATIS… · Web viewNote: mu = 4.29. That is, the sum of the number of letters of the 268 words