[PPT] - STAT 113 Normal-Based Inference for Proportions Colin Reimer Dawson PowerPoint Presentation

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

STAT 113 Normal-Based Inference for Proportions

Colin Reimer Dawson

Oberlin College

November 11-12, 2019 1 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Outline

Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test 2 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Limits of Normal Approximation So Far

So far we have still needed to do simulation to calculate the

standard error

We can avoid that with some more theory

3 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Cases to Address

We will need standard errors to do CIs and tests for the following parameters:

1. Single Proportion (now)
2. Single Mean (tomorrow)
3. Difference of Proportions (tomorrow?)
4. Difference of Means (tomorrow?)
5. Mean of Differences (Thursday?)

4 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Analytic Approximations of Sampling Distributions

Param. Stat. Theory SE Distribution df p ˆ p

p(1−p)

n

N(0, 1) – µ ¯ x

s2

n

t n − 1 pA − pB ˆ pA − ˆ pB

SE2

ˆ pA + SE2 ˆ pB

N(0, 1) – µA − µB ¯ xA − ¯ xB

SE2

ˆ pA + SE2 ˆ pB

t min(nA, nB) − 1 µdiff ¯ xdiff

s2

diff

ndiff

t ndiff − 1 ρ r

1−r2

n−2

t n − 2

CI : Observed Statistic ± Standardized Quantile × SE Sandardized Test Statistic : Observed Statistic − Null Param.

SE

5 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Outline

Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test 6 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Sampling Distribution of a Sample Proportion

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 p ^

0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 p ^

0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 p ^

0.0

0.2 0.4 0.6 0.8 1.0 0.00 0.15 p ^

0.0

0.2 0.4 0.6 0.8 1.0 0.00 0.15 p ^

0.0

0.2 0.4 0.6 0.8 1.0 0.00 0.15 p ^

0.0

0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 p ^

0.0

0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 p ^

0.0

0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 p ^

Columns: values of p (left: 0.1, middle: 0.5; right: 0.9)

Rows: values of n (top: 10, middle: 50; bottom: 1000)

Larger samples and more population homogeneity make SE go

down 7 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Sampling Distribution of ˆ p

Condition: Use a Normal approximation with at least 10

expected cases of each outcome: np ≥ 10 n(1 − p) ≥ 10

Mean: p
Standard deviation (standard error):

SEˆ

p =

p(1 − p)

n 8 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Outline

Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test 9 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

CI Summary: Single Proportion

0. Check whether conditions for distribution approximation hold:

nˆ p >= 10 and n(1 − ˆ p) ≥ 10

1. If so, find the mean and SD of the theoretical distribution to

replace the bootstrap distribution

Mean = Sample Statistic = ˆ

p

SD = Standard Error =
ˆ

p(1−ˆ p) n

2. Use the confidence level and a Standard Normal to get

z-scores of the endpoints

95%: z = ±1.96 (≈ 2)
99%: z = ±2.58
90%: z = ±1.64
3. Convert z scores to endpoints on the original scale using the

mean and standard deviation found in step 1. Endpoint = Sample Statistic + z · SE 10 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Example: Kissing Right

Most people are right-handed, and even the right eye is dominant for most people. Developmental biologists have suggested that late-stage human embryos tend to turn their heads to the right. In a study reported in Nature (2003), German bio-psychologist Onur Güntürkün studied kissing couples in public places such as airports, train stations, beaches, and parks. They observed 124 couples, age 13-70 years. For each kissing couple observed, the researchers noted whether the couple leaned their heads to the right or to the

left. Of the 124 couples, 80 leaned right.

Let’s find a 95% confidence interval for p, the proportion of all couples in the target population who would lean right. 11 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Kissing Right: Descriptive Stats and Conditions

Population Parameter (what we want to estimate): p,

proportion of all couples who would lean right

Sample Statistic (what we have): ˆ

p, proportion of couples in this study who leaned right ˆ p = 80/124 = 0.645

Conditions: nˆ

p ≥ 10 and n(1 − ˆ p) ≥ 10 nˆ p = 124 × 0.645 = 80 n(1 − ˆ p) = 124 × 0.355 = 44 so we are okay to use the Normal distribution approximation. 12 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Kissing Right: Standard Error

Standard Error for a proportion: SEˆ

p =

p(1−p)

n

Estimate with

ˆ SE ˆ

p =

ˆ

p(1−ˆ p) n

We have n = 124 and ˆ

p = 0.645: ˆ SE ˆ

p =

ˆ

p(1 − ˆ p) n =

0.645 × 0.355

124 = 0.043 13 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Kissing Right: z score and Confidence Interval

For a 95% interval, the z-scores of the endpoints are the 0.025

and 0.975 quantiles of a standard Normal

zEndpoints <- xqnorm(c(0.025, 0.975), mean = 0, sd = 1)

0.0 0.1 0.2 0.3 0.4 −2 2

density probability

A:0.0250000 B:0.9500000 C:0.0250000

zEndpoints [1] -1.959964 1.959964

The confidence interval is given by

Sample Stat. ± zendpoint · ˆ SE

which is 0.645 ± 1.96 · 0.043, or [0.561, 0.729]
Conclusion: We are 95% confident that between 56.1% and

72.9% of couples would tend to lean right when kissing. 14 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Outline

Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test 15 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

P-values: Single Proportion

Computing P-values using a Normal in place of a randomization distribution:

0. Check whether conditions for distribution approximation hold:

np0 ≥ 10 and n(1 − p0) ≥ 10

1. If so, find the mean and SD of the Normal to replace the

randomization distribution

Mean = Null Parameter Value = p0
SD = Standard Error =
p0(1−p0)

n

2. Convert the observed statistic to its z-score within this Normal

distribution z = Observed Sample Statistic − Null Parameter Standard Error

3. The P-value is the area under the Standard Normal curve past

z (or past z and −z if two-tailed) 16 / 21

SLIDE 17

Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Example: Kissing Right

Most people are right-handed, and even the right eye is dominant for most people. Developmental biologists have suggested that late-stage human embryos tend to turn their heads to the right. In a study reported in Nature (2003), German bio-psychologist Onur Güntürkün studied kissing couples in public places such as airports, train stations, beaches, and parks. They observed 124 couples, age 13-70 years. For each kissing couple observed, the researchers noted whether the couple leaned their heads to the right or to the

left. Of the 124 couples, 80 leaned right.

Let’s assess how strong the evidence is against the null hypothesis that couples are equally likely to lean right and left. 17 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Kissing Right: Descriptive Stats and Conditions

Population Parameter (what our hypotheses are about): p,

proportion of all couples who would lean right

Hypotheses

H0 : p = 0.5 (We write p0 for the null parameter) H1 : p = 0.5

Sample Statistic (what we have): ˆ

p, proportion of couples in this study who leaned right ˆ p = 80/124 = 0.645

Conditions: np0 ≥ 10 and n(1 − p0) ≥ 10

np0 = 124 × 0.5 = 62 n(1 − p0) = 124 × 0.5 = 62 so we are okay to use the Normal distribution approximation. 18 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Kissing Right: Standard Error

Standard Error for a proportion: SEˆ

p =

p(1−p)

n

Estimate with

ˆ SE ˆ

p =

p0(1−p0)

n

We have n = 124 and p0 = 0.5:

ˆ SE ˆ

p =

p0(1 − p0)

n =

0.5 × 0.5

124 = 0.045 19 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Kissing Right: z-score

In place of a randomization distribution, we use a Normal with

mean p0 = 0.5 and standard deviation equal to our estimated standard error: 0.045.

Find the z-score (test statistic) associated with our observed

sample statistic, ˆ p = 0.645 Test Statistic = Observed Statistic − Null Parameter ˆ SE = ˆ p − p0 ˆ SE = 0.645 − 0.5 0.045 = 3.22 20 / 21

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Theoretical SE Distribution of Single Proportion Confidence Interval Hypothesis Test

Kissing Right: P-value and Conclusion

Use the z-score (test statistic) associated with our sample

statistic to find the P-value using a Standard Normal

## two-tailed P.value <- 2 * xpnorm(3.22, mean = 0, sd = 1, lower.tail = FALSE)

z = 3.22 0.0 0.1 0.2 0.3 0.4 −4 −2 2 4 x density

P.value [1] 0.001281906

Conclusion: We have statistically significant evidence