Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Inference & Hypotheses
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
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Confident
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The null hypothesis H0 states the baseline assumption, usually that no effect or relationship exists. If we reject the null hypothesis, we accept an alternative hypothesis H1.
p-value: The probability of getting results as surprising (or more) as the observation if the null hypothesis were true.
Significance level (α): The cutoff we choose in advance. If the p-value is below α, we reject the null hypothesis.
A χ² goodness of fit test compares measured data to expected frequencies, and returns a p-value that captures the likelihood of equal or greater deviation from the expected frequencies. On a calculator:
Enter in L1 the observed frequencies
Enter in L2 the expected frequencies
Find the χ2 GOF-Test on your calculator, with
Observed: L1
Expected: L2
df: (n−1), where n is the number of items in either list
The calculator returns the p-value, which we interpret as usual for a hypothesis test. It also returns the value of χ2, which we can compare to a critical value if it is given.
When the total number of observations is fixed, and we have n different categories, we only have n−1 degrees of freedom since we can find one entry by subtracting the sum of the other entries from the total.
The critical value for a χ² test is a threshold we are given, against which we compare the value of χ² for our data. If our χ² is larger than the critical value, we reject H0.
A χ2 test can also be used to test whether categorical variables are related, for example, does favorite movie depend on gender? It works by comparing how far off the observed data is from what we would expect if the variables were not related (H0).
On a calculator:
Enter the observed frequencies in a matrix (table)
Enter the expected frequencies in a separate matrix
Navigate to χ2-Test on your calculator, and enter the observed and expected matrices you just filled.
The calculator returns the χ2 value and the p value.
Given a null hypothesis H0:μ=μ0, we can have any of the following alternative hypotheses
The first two alternative hypotheses are called one-tailed since we only care how far the sample mean, xˉ, is from μ0 in one direction. The last hypothesis is two-tailed because we care how far xˉ is from μ0 regardless of direction.
We can perform a t-test for a single sample against a known mean by on a calculator:
Enter the sample data into a list.
Navigate to T-Test on a calculator.
Select "DATA" and enter the name of the list where sample is stored.
Select the tail type depending on what our alternative hypothesis is (μ0 is the population mean):
=μ0 for a change in mean
<μ0 for a decrease in mean
>μ0 for an increase in mean
Hit calculate, and interpret the p-value as usual.
To compare the means of two samples using a T-test, we use a calculator:
Enter each sample in its own list.
Navigate to 2-SampTTest.
Select "Data", then enter the names of the lists containing the samples.
Select the tail type depending on what our alternative hypothesis is:
μ1=μ2 for different means
<μ2 for first list mean smaller than second
>μ2 for first list mean greater than second
Set "Pooled" to true.
The calculator reports the t-value and p-value, which we interpret as usual.