Confidence IntervalsDr. D’Agostino McGowan1 / 21

confidence intervals

If we use the same sampling method to select different samples and computed an interval estimate for each sample, we would expect the true population parameter ( $β_{1}$ ) to fall within the interval estimates 95% of the time.

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What it is NOT

95% CI: (5-9)

We would expect the true population parameter to fall within 5 to 9 95% of the time

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What it is NOT

95% CI: (5-9)

We would expect the true population parameter to fall within 5 to 9 95% of the time

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What it is NOT

95% CI: (5-9)

~~We would expect the true population parameter to fall within 5 - 9 95% of the time~~

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What it is NOT

95% CI: (5-9)

~~We would expect the true population parameter to fall within 5 - 9 95% of the time~~

Why?

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What it is NOT

95% CI: (5-9)

~~We would expect the true population parameter to fall within 5 - 9 95% of the time~~

Why?

The "true population parameter" is FIXED!

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The true parameter is FIXED!

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The true parameter is FIXED!

*when we are talking about confidence intervals, which rely on Frequentist theory. If you take a Bayesian inference class, you will learn about credible intervals which have different assumptions

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Example

We are interested in the relationship between Age and Wage. To demonstrate what a confidence interval is, I am going to construct a "truth" for the relationship in Lucy-land.

$W a g e = 2 \times A g e + ϵ$

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Example

We are interested in the relationship between Age and Wage. To demonstrate what a confidence interval is, I am going to construct a "truth" for the relationship in Lucy-land.

$W a g e = 2 \times A g e + ϵ$

What is the "true parameter" for the relationship between Age and Wage?

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Example

We are interested in the relationship between Age and Wage. To demonstrate what a confidence interval is, I am going to construct a "truth" for the relationship in Lucy-land.

$W a g e = 2 \times A g e + ϵ$

What is the "true parameter" for the relationship between Age and Wage?

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Example

We are interested in the relationship between Age and Wage. To demonstrate what a confidence interval is, I am going to construct a "truth" for the relationship in Lucy-land.

$W a g e = 2 \times A g e + ϵ$

Age ~ Normal(30, 10)
$ϵ$ ~ Normal(0, 10)
Sample $n = 100$

set.seed(7)
n <- 100
sample <- data.frame(
  Age = rnorm(n, 30, 10)
)
sample$Wage <- 2 * sample$Age + rnorm(n, 0, 10)

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Example

We are interested in the relationship between Age and Wage. To demonstrate what a confidence interval is, I am going to construct a "truth" for the relationship in Lucy-land.

$W a g e = 2 \times A g e + ϵ$

head(sample)

##        Age      Wage
## 1 52.87247 110.93553
## 2 18.03228  41.93996
## 3 23.05707  45.32082
## 4 25.87707  40.01053
## 5 20.29327  43.67375
## 6 20.52720  25.01562

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Example

We are interested in the relationship between Age and Wage. To demonstrate what a confidence interval is, I am going to construct a "truth" for the relationship in Lucy-land.

$W a g e = 2 \times A g e + ϵ$

n <- 100
sample2 <- data.frame(
  Age = rnorm(n, 30, 10)
)
sample2$Wage <- 2 * sample2$Age + rnorm(n, 0, 10)

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Example

We are interested in the relationship between Age and Wage. To demonstrate what a confidence interval is, I am going to construct a "truth" for the relationship in Lucy-land.

$W a g e = 2 \times A g e + ϵ$

head(sample)

##        Age      Wage
## 1 52.87247 110.93553
## 2 18.03228  41.93996
## 3 23.05707  45.32082
## 4 25.87707  40.01053
## 5 20.29327  43.67375
## 6 20.52720  25.01562

head(sample2)

##        Age      Wage
## 1 50.23344 105.71950
## 2 38.62492  74.49258
## 3 29.75091  60.04891
## 4 36.00635  68.13020
## 5 42.16481  80.15938
## 6 18.23468  24.82762

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Example

We are interested in the relationship between Age and Wage. To demonstrate what a confidence interval is, I am going to construct a "truth" for the relationship in Lucy-land.

$W a g e = 2 \times A g e + ϵ$ Fit a linear model on the sample

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Example

We are interested in the relationship between Age and Wage. To demonstrate what a confidence interval is, I am going to construct a "truth" for the relationship in Lucy-land.

$W a g e = 2 \times A g e + ϵ$ Fit a linear model on the sample

term	estimate	std.error	statistic	p.value	conf.low	conf.high
(Intercept)	0.9950715	3.2797052	0.3034027	0.7622261	-5.513397	7.503540
Age	2.0098559	0.0999765	20.1032929	0.0000000	1.811456	2.208256

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Example

We are interested in the relationship between Age and Wage. To demonstrate what a confidence interval is, I am going to construct a "truth" for the relationship in Lucy-land.

$W a g e = 2 \times A g e + ϵ$ Fit a linear model on the sample

term	estimate	std.error	statistic	p.value	conf.low	conf.high
(Intercept)	0.9950715	3.2797052	0.3034027	0.7622261	-5.513397	7.503540
Age	2.0098559	0.0999765	20.1032929	0.0000000	1.811456	2.208256

95% CI: 1.81, 2.21

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Example

What percent of the time does the "true parameter" fall within this interval?

term	estimate	std.error	statistic	p.value	conf.low	conf.high
(Intercept)	0.9950715	3.2797052	0.3034027	0.7622261	-5.513397	7.503540
Age	2.0098559	0.0999765	20.1032929	0.0000000	1.811456	2.208256

95% CI: 1.81, 2.21

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Example

What percent of the time does 2 fall within this interval?

term	estimate	std.error	statistic	p.value	conf.low	conf.high
(Intercept)	0.9950715	3.2797052	0.3034027	0.7622261	-5.513397	7.503540
Age	2.0098559	0.0999765	20.1032929	0.0000000	1.811456	2.208256

95% CI: 1.81, 2.21

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Example

We are interested in the relationship between Age and Wage. To demonstrate what a confidence interval is, I am going to construct a "truth" for the relationship in Lucy-land.

$W a g e = 2 \times A g e + ϵ$ Fit a linear model on the sample2

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Example

We are interested in the relationship between Age and Wage. To demonstrate what a confidence interval is, I am going to construct a "truth" for the relationship in Lucy-land.

$W a g e = 2 \times A g e + ϵ$ Fit a linear model on the sample2

term	estimate	std.error	statistic	p.value	conf.low	conf.high
(Intercept)	-2.819931	3.0542175	-0.9232909	0.3581233	-8.880926	3.241064
Age	2.078026	0.0968743	21.4507408	0.0000000	1.885782	2.270270

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Example

We are interested in the relationship between Age and Wage. To demonstrate what a confidence interval is, I am going to construct a "truth" for the relationship in Lucy-land.

$W a g e = 2 \times A g e + ϵ$ Fit a linear model on the sample2

term	estimate	std.error	statistic	p.value	conf.low	conf.high
(Intercept)	-2.819931	3.0542175	-0.9232909	0.3581233	-8.880926	3.241064
Age	2.078026	0.0968743	21.4507408	0.0000000	1.885782	2.270270

95% CI: 1.89, 2.27

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Example

What percent of the time does 2 fall within this interval?

term	estimate	std.error	statistic	p.value	conf.low	conf.high
(Intercept)	-2.819931	3.0542175	-0.9232909	0.3581233	-8.880926	3.241064
Age	2.078026	0.0968743	21.4507408	0.0000000	1.885782	2.270270

95% CI: 1.89, 2.27

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Example

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Example

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Example

What percent of the intervals contain the "true parameter"?

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Example

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Example

What percent of the intervals contain the "true parameter"

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Example

What percent of the intervals contain the "true parameter"

48 / 50 = 96%

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Example

95 / 100 = 95%

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  Dr. Lucy D'Agostino McGowan

 Applicaton ExerciseWatch me code this up in the next lecture video
Code along with me, or repeat the same steps to build an understanding about confidence intervals
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Confidence Intervals

Dr. D’Agostino McGowan

confidence intervals

What it is NOT

What it is NOT

What it is NOT

What it is NOT

What it is NOT

The true parameter is FIXED!

The true parameter is FIXED!

Example

Example

Example

Example

Example

Example

Example

Example

Example

Example

95% CI: 1.81, 2.21

Example

95% CI: 1.81, 2.21

Example

95% CI: 1.81, 2.21

Example

Example

Example

95% CI: 1.89, 2.27

Example

95% CI: 1.89, 2.27

Example

Example

Example

Example

Example

Example

48 / 50 = 96%

Example

95 / 100 = 95%

Applicaton Exercise

confidence intervals

Help

`Applicaton Exercise`