vault backup: 2024-02-06 17:54:50

education/math/Matrices.md

@@ -0,0 +1,14 @@
+$$ 
+\begin{bmatrix}
+1 & 2 & 3\\
+4 & 5 & 6
+\end{bmatrix}
+$$
+To write a system of equations as a matrix, you use the coefficients of each term. The solutions for each equation might be written with a bar separating:
+
+$$ 
+\begin{bmatrix}
+1 & 2 & 3|4\\
+4 & 5 & 6|5
+\end{bmatrix}
+$$

education/statistics/Hypothesis Tests.md

@@ -7,10 +7,12 @@ This test can be used if:
 - Can use a box with tickets of 1s and zeros to represent the population
 If an observed value is too many SEs away from the expected value, it is hard to explain by chance.
 
-Start by finding a null and alternative hypothesis, eg:
+1. Start by finding a null and alternative hypothesis:
-- Null: *x* is *y*. This is often given in the problem
+	- Null: *x* is *y*. This is often given in the problem
-- Alternative: If you're being asked to determine if something has changed, you're determining whether or not *x* is equal to. If you're being asked to find the more than, or less than, it's a one sided test.
+	- Alternative: If you're being asked to determine if something has changed, you're determining whether or not *x* is equal to. If you're being asked to find the more than, or less than, it's a one sided test.
-Then find the SE, take the EV and the observed value, and find the $z$ score. You can use this $z$ score combined with something like $normalcdf$ to find the amount that is outside of the expected range. If that total amount is less than 5%, than the null hypothesis should be rejected. If that total amount is more than 5%, the difference is too small, and it should not be rejected. 
+2. Then find the SE. This is usually found with: $\frac{SD}{\sqrt{num_{draws}}}$. 
+3. The EV (Expected Value) is usually given as the population %. Then with the above info, you can find the $z$ score with the formula $z = \frac{expected_\% - observed_\%}{SE_\%}$.
+4. You can use this $z$ score combined with something like $normalcdf$ to find the amount that is outside of the expected range. If that total amount is less than 5%, than the null hypothesis should be rejected. If that total amount is more than 5%, the difference is too small, and it should not be rejected. 
 Then you can provide a conclusion based off of either the null hypothesis, or the alternative hypothesis.
 
 | Term | Description |
@@ -20,8 +22,19 @@ Then you can provide a conclusion based off of either the null hypothesis, or th
 | One-tailed test | Use when the alternative hypothesis says that the % of 1s is *less than* or *greater than* expected. It's one sided, because the area of importance on a distribution only has one side, and extends all the way outwards, away from the normal curve. |
 | Two tailed test | Use when something is *not equal* to the expected. It's called a two tailed test because the area of significance has two sides. You can find the likelihood of ending up on one side, and the likelihood of ending up on another side, and adding them together (or multiplying by 2 if it's the same on each). |
 ## z tests for averages
-This test will look very similar to a z test for percentages, it still requires that a large, random, sample was given.
+This test will look very similar to a z test for percentages, it still requires that a large, random, sample was given ($>30$).
 
+## Two sample z tests for averages
+These tests are still very similar to a normal z test. In order to conduct a two sample z-test, the two samples being used must be independent from each other. Each sample must be large ($>30$), and a simple random sample. 
+
+The two sample z-statistic is computed from:
+- the sizes of the two samples
+- the averages of the two samples
+- the SDs of the two samples
+$$ \frac{observed_{diff} - expected_{diff}}{SE_{diff}}$$
+The diff is the difference between the two samples, and can be found by subtracting one from the other.
+$$ SE_{diff} = \sqrt{a^2 + b^2} $$
+The above formula is used where $a$ and $b$ are the $SE_{ave}$ of each sample. 
 ## t tests for averages
 This test is used when you have a small sample size ($<30$). 
 The only major differences used with a *t* test is that you use SD+.
@@ -30,11 +43,11 @@ With a small sample size, the standard deviation will be relatively higher, so t
 $$ SD_+ = \sqrt{\frac{size\space sample}{sample\space size}}*SD$$
 This found value is then used in all further calculations where you would normally use the $SD$ in a z score test.
 $$ t = \frac{obs_{ave} - EV_{ave}}{SE_{ave}} $$
-The student curve is then used instead of the normal curve. It is similar, but has more area under the tails. 
+The student/t curve is then used instead of the normal curve. It is similar, but has more area under the tails. 
 
 Degrees of freedom ($df$) can be found by subtracting 1 from the sample size. The lower the degree of freedom, the greater the difference between the student curve and the normal curve.
 
-The equivalent of $normalcdf$ for a t test is $tcdf$. 
+The equivalent of $normalcdf$ for a t test is $tcdf$. This function returns a percentage.
 ## P Value
 The chance of observing at least a sample statistic, or something more extreme, if the null hypothesis is true.
 If the p-value is less than *5*%, reject the null hypothesis.

education/statistics/Sampling.md

@@ -8,6 +8,7 @@
 | Sample | A portion or subset of the population |
 | Parameter | A number that describes a characteristic of an entire *population* (*10%* of US senators voted for something) |
 | Statistic | A number that describes a *sample* characteristic (*71%* of Americans feel that ...) |
+
 > A global consumer survey reported that 6% of US taxpayers used or owned cryptocurrency in 2020. The US government is interested in knowing if this percentage has increased. The University of Chicago surveys 1,004 taxpayers and finds that 13% have used or owned crypto in the past year (2021)
 
 In the above example: