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S&DS 101 Final
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Gravity
Terms in this set (36)
rbinom(): generate random numbers from a binomial distribution
2. dbinom(): create the binomial density function Pr(X = k; n, π)
3. pbinom(): create the cumulative distribution function Pr(X ≤ k; n, π)pbinom() is the cumulative sum of dbinom()
generate random numbers from a binomial distribution
parametric vs nonparametric tests
parametric tests include underlying assumptions about the population
95% confidence interval
95% of the confidence intervals will contain the parameter
parameter
numerical summary of a population
standard error
sd of sampling distribution (the average distance from the statistic to the parameter)
statistical test
uses data from a sample to assess a claim about a population
null and alternative hypothesis
null: the hypothesis you wish to test, usually the "current norm"
alternative: a statement about the population mean (mu), says that the population mean is different from the value specified under the null hypothesis
null distribution
the sampling distribution of outcomes for a test statistic under the assumption that the null hypothesis is true
p-value
The probability of observing a test statistic as extreme as, or more extreme than, the statistic obtained from a sample, under the assumption that the null hypothesis is true.
density curve
A mathematical model used to describe the overall pattern of the distribution of a random variable.
normal density curve
symmetric about the mean μ; has standard deviation σ; highest point at X = μ; total area under the curve = 1; values of the random variable X on x-axis; probabilities are represented by areas under the curve
t-distribution
A distribution specified by degrees of freedom used to model test statistics for the sample mean, differences between sample means, etc. where IT (' s) is (are) unknown
sampling distribution
takes many samples from the population and creates a
descriptive statistics
describe the sample of data we have
inferential statistics
use the sample to make claims about the properties of the population/process
population
all individuals/objects of interest
pi, mu, sigma, rho, beta
sample
a subset of the population
p-hat, x-bar, s, r, b
simple random sampling/ random selection
each member of the population has an equal chance of being selected. Good because it generalizes from the sample to the population
What do you do with outliers?
Investigate, describe, but can't pretend they don't exist
Steps of a hypothesis test
0) Make a plot
1) State null and alternative
2) Calculate observed stat
3) Create null distribution
4) Calculate the p-value
5) make a judgement (if using Neyman-Pearson)
Code for randomizing null distributions in step 3 of the hypothesis test for PROPORTIONS?
rbinom(number of simulations, size of sample, probability)
Code for randomizing null distributions in step 3 of the hypothesis test for COMPARING MEANS?
Use a for loop!
Code for parametric null distributions for PROPORTIONS?
dbinom(x_range, size, prob)/size
to get p-value use pbinom(obs_num, size, prob)
Code for parametric null distributions for COMPARING 2 MEANS?
dnorm(x_range, mu, sigma)
p-value <- pt(t_stat, df)
df = min(n1, n2)-1
parametric test for more than 2 means?
One-Way ANOVA (analysis of variance).
Use F-statistic
Bootstrap
Creates a bootstrap distribution by sampling with replacement from the sample.
Bootstrap SE
the standard deviation of the bootstrap distribution
SEM
sample standard deviation/ sqrt(n)
standard error of a proportion
sqrt(p-hat(1-p-hat)/n)
3 problems with Neyman-Pearson?
1) we are interested in the results of a specific experiment, not whether we are right most of the time
2) arbitrary thresholds for alpha levels
3) running many tests can give rise to a high number of type 1 errors
rbinom()
generate random numbers from a binomial distribution
dbinom()
create the binomial density function Pr(X = k; n, π)
pbinom()
create the cumulative distribution function Pr(X ≤ k; n, π)
is the cumulative sum of dbinom()
Steps for creating a null distribution
1) Combine data from both groups
2) Shuffle data
3) Randomly select 10 points to be the 'null' treatment group
4) Take the remaining points to the 'null' control group.
5) Compute the statistic of interest on these 'null' groups
6) Repeat 10,000 Emes to get a null distribution
Writing a for loop for difference in 2 means
null_dist <- NULL for (i in 1:10000) {
shuff_data <- sample(combo_data)
shuff_light <- shuff_data[1:9]
shuff_dark <- shuff_data[10:17]
null_dist[i] <- mean(shuff_light) - mean(shuff_dark)
}
To get the p-value
1 sided: p_val <- sum(null_dist >= obs_stat)/10000
2 sided: p_val<-sum(abs(null_distr)>=abs(obs_stat))/10000
p-val 1 and 2 sided for parametric comparison of two means
1: p_val <- pt(obs_stat, df, lower.tail = FALSE)
2: p_val <- 2 * pt(abs(obs_stat), df, lower.tail = FALSE)
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