Parameters and confidence inter v als FOU N DATION S OF IN FE R E N - - PowerPoint PPT Presentation

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Parameters and confidence inter v als FOU N DATION S OF IN FE R E N - - PowerPoint PPT Presentation

Apr 28, 2023 513 likes •991 views

Parameters and confidence inter v als FOU N DATION S OF IN FE R E N C E Jo Hardin Instr u ctor Research q u estions H y pothesis test Con dence inter v al Under w hich diet plan w ill participants lose Ho w m u ch sho u ld participants e x

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Parameters and confidence intervals

FOU N DATION S OF IN FE R E N C E

Jo Hardin

Instructor

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FOUNDATIONS OF INFERENCE

Research questions

Hypothesis test Condence interval Under which diet plan will participants lose more weight on average? How much should participants expect to lose on average? Which of two car manufacturers are users more likely to recommend to their friends? What percent of users are likely to recommend Subaru to their friends? Are education level and average income linearly related? For each additional year of education, what is the predicted average income?

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FOUNDATIONS OF INFERENCE

Parameter

A parameter is a numerical value from the population Examples (continued): The true average amount all dieters will lose on a particular program The proportion of individuals in a population who recommend Subaru cars The average income of all individuals in the population with a particular education level

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FOUNDATIONS OF INFERENCE

Confidence interval

Range of numbers that (hopefully) captures the true parameter "95% condent that between 12% and 34% of the entire population recommends Subarus"

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Let's practice!

FOU N DATION S OF IN FE R E N C E

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Bootstrapping

FOU N DATION S OF IN FE R E N C E

Jo Hardin

Instructor

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FOUNDATIONS OF INFERENCE

Hypothesis testing

How do samples from the null population vary? Statistic, proportion of successes in sample → Parameter, proportion of successes in population → p

p ^

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FOUNDATIONS OF INFERENCE

Confidence intervals

No null population, unlike in hypothesis testing How do p and vary?

p ^

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FOUNDATIONS OF INFERENCE

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FOUNDATIONS OF INFERENCE

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FOUNDATIONS OF INFERENCE

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FOUNDATIONS OF INFERENCE

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FOUNDATIONS OF INFERENCE

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FOUNDATIONS OF INFERENCE

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FOUNDATIONS OF INFERENCE

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FOUNDATIONS OF INFERENCE

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FOUNDATIONS OF INFERENCE

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FOUNDATIONS OF INFERENCE

Polling

# Original data Source: local data frame [30 x 3] flip_num flip <int> <chr> 1 1 H 2 2 H 3 3 H 4 4 T 5 5 H 6 6 H # ... with 24 more rows

Original data Candidate X Total voters Proportion X 17 30 0.5667

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FOUNDATIONS OF INFERENCE

Polling

# First resample Source: local data frame [30 x 3] replicate flip_num flip <dbl> <int> <chr> 1 1 7 H 2 1 17 T 3 1 13 H 4 1 14 H 5 1 24 H 6 1 28 T # ... with 24 more rows

First resample Candidate X Total voters Proportion X 17 30 0.5667 14 30 0.4667

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FOUNDATIONS OF INFERENCE

Polling

# Second resample Source: local data frame [30 x 3] replicate flip_num flip <dbl> <int> <chr> 1 2 21 H 2 2 19 T 3 2 25 H 4 2 24 T 5 2 21 H 6 2 28 T 7 2 13 H 8 2 23 H 9 2 24 T 10 2 24 T # ... with 20 more rows

Second resample Candidate X Total voters Proportion X 17 30 0.5667 14 30 0.4667 18 30 0.6

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FOUNDATIONS OF INFERENCE

Polling

# Third resample Source: local data frame [30 x 3] replicate flip_num flip <dbl> <int> <chr> 1 3 6 H 2 3 19 H 3 3 1 H 4 3 24 T 5 3 11 H 6 3 28 T 7 3 16 H 8 3 13 H 9 3 21 T 10 3 29 H # ... with 20 more rows

Third resample Candidate X Total voters Proportion X 17 30 0.5667 14 30 0.4667 18 30 0.6 12 30 0.4

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FOUNDATIONS OF INFERENCE

Standard error

Obtained standard error of 0.09 by resampling many times Describes how the statistic varies around parameter Bootstrap provides an approximation of the standard error

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FOUNDATIONS OF INFERENCE

Variability of p-hat from the population

# Compute p-hat for each poll ex1_props <- recommend %>% group_by(poll) %>% summarize(prop_yes = mean(vote == "yes")) # Variability of p-hat ex1_props %>% summarize(sd(prop_yes)) # A tibble: 1 × 1 `sd(prop_yes)` <dbl> 1 0.08523512

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FOUNDATIONS OF INFERENCE

Variability of p-hat from the sample (bootstrapping)

# Select one poll from which to resample

  • ne_poll <- all_polls %>%

filter(poll ==1) %>% select(vote) # Compute p-hat for each resampled poll ex2_props <- one_poll %>% specify(response = vote, success = "yes") %>% generate(reps = 1000, type = "bootstrap") # Variability of p-hat ex2_props %>% summarize(sd(stat)) # A tibble: 1 × 1 `sd(stat)` <dbl> 1 0.08691885

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Let's practice!

FOU N DATION S OF IN FE R E N C E

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Variability in p-hat

FOU N DATION S OF IN FE R E N C E

Jo Hardin

Instructor

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FOUNDATIONS OF INFERENCE

How far are the data from the parameter?

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FOUNDATIONS OF INFERENCE

How far are the data from the parameter?

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FOUNDATIONS OF INFERENCE

How far are the data from the parameter?

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FOUNDATIONS OF INFERENCE

Standard error of p-hat

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FOUNDATIONS OF INFERENCE

Empirical rule

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FOUNDATIONS OF INFERENCE

Empirical rule

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FOUNDATIONS OF INFERENCE

Empirical rule

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Let's practice!

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Interpreting CIs and technical conditions

FOU N DATION S OF IN FE R E N C E

Jo Hardin

Instructor

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FOUNDATIONS OF INFERENCE

Creating CIs

# Compare confidence intervals

  • ne_poll_boot %>% summarize(

lower = p_hat - 2 * sd(prop_yes_boot), upper = p_hat + 2 * sd(prop_yes_boot)) # A tibble: 1 × 2 lower upper <dbl> <dbl> 1 0.536148 0.863852 # Find 2.5% and 97.5% of p-hat vals

  • ne_poll_boot %>% summarize(

q025_prop = quantile(prop_yes_boot, p = .025), q975_prop = quantile(prop_yes_boot, p = .975)) # A tibble: 1 × 2 q025_prop q975_prop <dbl> <dbl> 1 0.5333333 0.8333333

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FOUNDATIONS OF INFERENCE

Motivating CIs

Goal is to nd the parameter when all we know is the statistic Never know whether the sample you collected actually contains the true parameter

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FOUNDATIONS OF INFERENCE

Interpreting the CIs

Bootstrap t-CI: (0.536, 0.864) Percentile interval: (0.533, 0.833) We are 95% condent that the true proportion of people planning to vote for candidate X is between 0.536 and 0.864 (or 0.533 and 0.833)

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FOUNDATIONS OF INFERENCE

Technical conditions

Sampling distribution of the statistic is reasonably symmetric and bell-shaped Sample size is reasonably large Variability of resampled proportions

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Let's practice!

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Summary of statistical inference

FOU N DATION S OF IN FE R E N C E

Jo Hardin

Instructor

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FOUNDATIONS OF INFERENCE

Inference

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FOUNDATIONS OF INFERENCE

Testing

H : There is no gender discrimination in hiring H : Men are more likely to be promoted than women

A

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FOUNDATIONS OF INFERENCE

Estimation

What proportion of the voters will select candidate X?

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FOUNDATIONS OF INFERENCE

Bootstrapping

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Congratulations!

FOU N DATION S OF IN FE R E N C E