The deck is fair. The deck is not fair.

Say, “the deck isn’t fair.”

Oops.(false positive) Right!

Say, “the deck is fair”

Right! Oops.(false negative)

One of these will be true about your population…but you don’t

know which one.

 

The deck is fair. The deck is not fair.

Say, “the deck isn’t fair.”

Oops.(false positive) Right!

Say, “the deck is fair” Right! Oops.

(false negative)

You’ll make one of these claims, based on your sample

 

The deck is fair. The deck is not fair.

Say, “the deck isn’t fair.”

Oops.(false positive) Right!

Say, “the deck is fair” Right! Oops.

(false negative)

 

The deck is fair. The deck is not fair.

Say, “the deck isn’t fair.”

Oops.(false positive) Right!

Say, “the deck is fair” Right! Oops.

(false negative)

 

The deck is fair. The deck is not fair.

Say, “the deck isn’t fair.”

Oops.(false positive) Right!

Say, “the deck is fair” Right! Oops.

(false negative)

 

The deck is fair. The deck is not fair.

Say, “the deck isn’t fair.”

Oops.(false positive) Right!

Say, “the deck is fair” Right! Oops.

(false negative)

Why are we even HERE?

Unbiased Estimators are statistics which accurately target the parameters they aim

to target (catch them “on average”).

In the interest of time…

(10 weeks time I would like to have)

Each individual statistic will most likely be “off” a bit in value from the parameter

(“sampling error”)…for example…

…so, in MTH 244, we will learn to build “wings” around our statistics to mitigate as

much error as possible…

To properly estimate a parameter from a sample, you need to form a confidence interval (CI)…

3. Give the center its MOE to help alleviate any sampling error.

2. Find the center of interest of your sample…in MTH 244, that’ll be either a percentage or an average.

1. Start with a random sample…

(remember 2?)

Let’s play!

The Wonderful Binomial Distribution!(fixed number of trials n, constant probability of success p)

Population Average Population Standard Deviation

= np 𝛔=√𝐧𝐩𝐪

Sampling Based on the W.B.D.!(population believed to have constant, but unknown, probability of success p)

Sample Average Standard Error Margin of Error

𝐒𝐄=√ �̂� �̂�𝐧�̂�=𝐱𝐧 𝐌𝐄=𝐳𝐜𝐫𝐢𝐭𝐢𝐜𝐚𝐥 √ �̂� �̂�𝐧

Confidence (area) “critical” z values68% 190% 1.64595% 2 (actually, 1.96)99% 2.575

99.7% 3

Maybe this’ll Help…

Let’s do a quiz!

The BIG idea of Confidence Intervals (for any statistic)!

i.e. – “Sean, why are you always griping about the media?”

But, um…why “z”?

• You might recall (from MTH 243) that binomial distributions’ histograms begin to look like bell curves if 1) p 0.5 and 2) n is very large.

• You might recall (from MTH 243) that binomial distributions’ histograms begin to look like bell curves if 1) p 0.5 and 2) n is very large.

• You might recall (from MTH 243) that binomial distributions’ histograms begin to look like bell curves if 1) p 0.5 and 2) n is very large.

• You might recall (from MTH 243) that binomial distributions’ histograms begin to look like bell curves if 1) p 0.5 and 2) n is very large.

• You might recall (from MTH 243) that binomial distributions’ histograms begin to look like bell curves if 1) p 0.5 and 2) n is very large.

• You might recall (from MTH 243) that binomial distributions’ histograms begin to look like bell curves if 1) p 0.5 and 2) n is very large.

• So, for a given value of p (close to 0.5), as n , the distribution resembles a normal distribution more and more. But, what if p varies? In the next few slides, n = 20.

p = 0.5

p = 0.4

p = 0.3

p = 0.2

p = 0.1

• So, as p begins to vary significantly from 0.5, the distributions skew more and more. But…watch what happens when n , even when p = 0.1 (highly skewed)...

n = 20

np = 20(0.1) = 2

n = 40

np = 40(0.1) = 4

n = 60

np = 60(0.1) = 6

n = 80

np = 80(0.1) = 8

n = 100

np = 100(0.1) = 10

• So long as we can ensure that our sample is big enough (that is, np 5 and nq 5), our methods of proportional CI’s will be valid. It’s an even better fit when np 10 and nq 10.

(For a formal proof, check the enrichment page of the website)

January 15, 2014Gallup Daily: Government Itself Still Cited as Top U.S. ProblemAmericans start the new year with a variety of national concerns on their minds. Although none is dominant, the government, at 21%, leads the list of what Americans consider the most important problem facing the country. The economy closely follows at 18%, and then unemployment/jobs and healthcare, each at 16%. Results for this Gallup poll are based on telephone interviews conducted Jan. 5-8, 2014, with a random sample of 1,018 adults, aged 18 and older, living in all 50 U.S. states and the District of Columbia. For results based on the total sample of national adults, the margin of sampling error is ±4 percentage points at the 95% confidence level.

January 15, 2014Gallup Daily: Government Itself Still Cited as Top U.S. ProblemAmericans start the new year with a variety of national concerns on their minds. Although none is dominant, the government, at 21%, leads the list of what Americans consider the most important problem facing the country. The economy closely follows at 18%, and then unemployment/jobs and healthcare, each at 16%. Results for this Gallup poll are based on telephone interviews conducted Jan. 5-8, 2014, with a random sample of 1,018 adults, aged 18 and older, living in all 50 U.S. states and the District of Columbia. For results based on the total sample of national adults, the margin of sampling error is ±4 percentage points at the 95% confidence level.

January 15, 2014Gallup Daily: Government Itself Still Cited as Top U.S. ProblemAmericans start the new year with a variety of national concerns on their minds. Although none is dominant, the government, at 21%, leads the list of what Americans consider the most important problem facing the country. The economy closely follows at 18%, and then unemployment/jobs and healthcare, each at 16%. Results for this Gallup poll are based on telephone interviews conducted Jan. 5-8, 2014, with a random sample of 1,018 adults, aged 18 and older, living in all 50 U.S. states and the District of Columbia. For results based on the total sample of national adults, the margin of sampling error is ±4 percentage points at the 95% confidence level.

January 15, 2014Gallup Daily: Government Itself Still Cited as Top U.S. ProblemAmericans start the new year with a variety of national concerns on their minds. Although none is dominant, the government, at 21%, leads the list of what Americans consider the most important problem facing the country. The economy closely follows at 18%, and then unemployment/jobs and healthcare, each at 16%. Results for this Gallup poll are based on telephone interviews conducted Jan. 5-8, 2014, with a random sample of 1,018 adults, aged 18 and older, living in all 50 U.S. states and the District of Columbia. For results based on the total sample of national adults, the margin of sampling error is ±4 percentage points at the 95% confidence level.

January 15, 2014Gallup Daily: Government Itself Still Cited as Top U.S. ProblemAmericans start the new year with a variety of national concerns on their minds. Although none is dominant, the government, at 21%, leads the list of what Americans consider the most important problem facing the country. The economy closely follows at 18%, and then unemployment/jobs and healthcare, each at 16%. Results for this Gallup poll are based on telephone interviews conducted Jan. 5-8, 2014, with a random sample of 1,018 adults, aged 18 and older, living in all 50 U.S. states and the District of Columbia. For results based on the total sample of national adults, the margin of sampling error is ±4 percentage points at the 95% confidence level.

January 15, 2014Gallup Daily: Government Itself Still Cited as Top U.S. ProblemAmericans start the new year with a variety of national concerns on their minds. Although none is dominant, the government, at 21%, leads the list of what Americans consider the most important problem facing the country. The economy closely follows at 18%, and then unemployment/jobs and healthcare, each at 16%. Results for this Gallup poll are based on telephone interviews conducted Jan. 5-8, 2014, with a random sample of 1,018 adults, aged 18 and older, living in all 50 U.S. states and the District of Columbia. For results based on the total sample of national adults, the margin of sampling error is ±4 percentage points at the 95% confidence level.

January 15, 2014Gallup Daily: Government Itself Still Cited as Top U.S. ProblemAmericans start the new year with a variety of national concerns on their minds. Although none is dominant, the government, at 21%, leads the list of what Americans consider the most important problem facing the country. The economy closely follows at 18%, and then unemployment/jobs and healthcare, each at 16%. Results for this Gallup poll are based on telephone interviews conducted Jan. 5-8, 2014, with a random sample of 1,018 adults, aged 18 and older, living in all 50 U.S. states and the District of Columbia. For results based on the total sample of national adults, the margin of sampling error is ±4 percentage points at the 95% confidence level.

January 15, 2014Gallup Daily: Government Itself Still Cited as Top U.S. ProblemAmericans start the new year with a variety of national concerns on their minds. Although none is dominant, the government, at 21%, leads the list of what Americans consider the most important problem facing the country. The economy closely follows at 18%, and then unemployment/jobs and healthcare, each at 16%. Results for this Gallup poll are based on telephone interviews conducted Jan. 5-8, 2014, with a random sample of 1,018 adults, aged 18 and older, living in all 50 U.S. states and the District of Columbia. For results based on the total sample of national adults, the margin of sampling error is ±4 percentage points at the 95% confidence level.

January 15, 2014Rasmussen Reports: Right Direction or Wrong TrackFor the second week in a row, 29% of Likely U.S. Voters think the country is heading in the right direction, according to a new Rasmussen Reports national telephone survey for the week ending January 12. In early October during the federal government shutdown, confidence in the country’s course fell to 13%, the lowest finding in five years. A year ago, 36% said the country was heading in the right direction. The national telephone survey of 3,500 likely voters was conducted by Rasmussen Reports on January 6-12, 2014. The margin of sampling error for the survey is +/- 2 percentage points with a 95% level of confidence.

January 15, 2014Rasmussen Reports: Right Direction or Wrong TrackFor the second week in a row, 29% of Likely U.S. Voters think the country is heading in the right direction, according to a new Rasmussen Reports national telephone survey for the week ending January 12. In early October during the federal government shutdown, confidence in the country’s course fell to 13%, the lowest finding in five years. A year ago, 36% said the country was heading in the right direction. The national telephone survey of 3,500 likely voters was conducted by Rasmussen Reports on January 6-12, 2014. The margin of sampling error for the survey is +/- 2 percentage points with a 95% level of confidence.

December 1, 2008 (date honestly doesn’t matter – this happens all the time) LA Times: Muslims - India’s New “Untouchables”According to the report, produced by a committee led by a former Indian chief justice, Rajender Sachar, Muslims are now worse off than the Dalit caste, or those called “untouchables”. Some 52% of Muslim men were unemployed, compared with 47% of Dalit men.

Hmmmmmmmmmmm...

Muslim Unemployment

Caste Unemployment

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

Indian Unemployment (n = 100)

Percentage @ 95% confidence

Muslim Unemployment

Caste Unemployment

0.3 0.35 0.4 0.45 0.5 0.55 0.6

Indian Unemployment (n = 500)

Percentage @ 95% confidence

Muslim Unemployment

Caste Unemployment

0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56

Indian Unemployment (n = 1000)

Percentage @ 95% confidence

Muslim Unemployment

Caste Unemployment

0.44 0.46 0.48 0.5 0.52 0.54

Indian Unemployment (n = 2000)

Percentage @ 95% confidence

“So – how big a sample do you need?”