How to perform a hypothesis test for population proportion (generic shit)?
- State null and alternative hypotheses, defining any parameters, and stating the significance level.
- State the distribution for assuming the null hypothesis is true.
- Calculate the probability of finding the test statistic or more extreme using (2). This is the p-value
- Compare the p-value to the significance level.
- If the p-value is** smaller** than the significance level, reject null hypothesis.
- Conclusion: There is (in)significant evidence to suggest that ____, therefore we (don’t) reject the null hypothesis.
How to perform a hypothesis test for a mean?
Where x̄ is the sample mean, and μ is the population mean
- Define the variable you are using and their distribution
x ~ N (μ, a) - State the null and alternative hypotheses
H_0: μ = b
H_1: μ >/< b - State the test statistic x̄ and its distribution
x̄ ~ N (b, a/n) - Find the p value of getting the result you did in the sample or ** more extreme **
- Compare to the significance level and write a conclusion
a is the variance of the population, n is the size of the sample
How do you perform a hypothesis test for correlation coefficients (either spearman’s or pearson’s)?
- State the null and alternative hypotheses
null: no correlation
alternative: some, positive/negative correlation - Write down the significance level, value of n, and therefore the critical value (from the correct table in the data booklet)
- Compare the correlation coefficient to the critical value (might need to compare the modulus of the CC)
- Conclude (i.e. there is evidence for a (positive/negative) correlation or there is no correlation)
(2) Be careful about whether it is one tail or two tail
conclusion and hypothesis tests should be in context to the question
How do you perform a single-sample sign test?
- State the null and alternative hypotheses
H_0 : median is A
H_1 : median is not A - State your variable X (whether it is the number of results above or below A)
- State the distribution of X
X ~ B (n, 0.5) - Find the p-value (the probability of getting your result of X or more extreme)
- Compare the p-value to the significance level and give a conclusion
If one tail, only look at the values in the direction of H_1
How do you perform a single-sample Wilcoxon signed-rank test?
Step 1: Calculating the test statistic
1. Find the differences from the median
2. Rank the modulus of the differences
3. W+ is the sum of all the positive ranks and W- the sum of all the negative ranks
4. T is the smaller value of W+ and W-
Step 2: Hypothesis test
1. State the null and alternative hypotheses
H_0 : median is A
H_1 : median is not A
2. State the test statistic (found in part 1)
3. State the value of n, the significance level, and therefore the critical value (from the correct table in the data booklet)
4. Compare T to the critical value; we are looking for smaller than the CV
5. Conclude (in context!)
Be careful finding the CV: is it two tail or one tail?
How would you perform a paired-sample test?
Step 1: Calculating the test statistic
1. Format a table with each data point in each pair, the difference, |d|, and rank
2. Fill in the table (duh)
3. Find W+ (sum of positive ranks) and W- (sum of negative ranks)
4. T is the smaller of W+ and W-
Step 2: The hypothesis test
1. State the null and alternative hypothesis
H_0 : the distribution is the same
H_1 : the distribution is not the same
2. State the value of n, the significance level, and therefore the critical value (from the correct table in the data booklet).
3. Compare T to the CV
4. In this test we are looking for T to be smaller than CV in order to reject the null hypothesis
5. Write a conclusion (don’t be lazy)
Which difference you take much be stated (i.e. stating d = a - b
How would you perform a Wilcoxon Rank Sum test?
Step 1: Calculate the test statistic, T
1. Rank all the values in both samples in order of increasing size (the lowest being 1)
2. R_m is the sum of the ranks in m and R_n is the sum of the ranks in n
3. T is the smaller of R_m and [ m(m + n + 1) - R_m ]
Step 2: The hypothesis test
1. State the null and alternative hypotheses
H_0 : The median of the two groups is the same i.e. median_1 = median_2
H_1 : The median of the two groups is not the same
2. State the value of T you found in part 1
3. State the value of m, n, and the significance level. Find and state the relevant critical value from the correct table in the data booklet)
4. Compare T to the CV
5. Conclude; T < CV leads to a rejection of the null hypothesis
When would you use a Wilcoxon rank sum / unpaired sample test?
- Used for unpaired data
- Two groups with different n (called m and n where m < n)
- Two different populations
When approximating using the normal what do you need to do?
Normal approximations can be used for Wilcoxon signed rank and rank sum
note: the approximations are different in each case
Add the continuity correction:
If T is less than the mean of the normal distribution, add 0.5
If T is more than the mean of the normal distribution, subtract 0.5
How do you perform a hypothesis test for the mean of a large sample?
This is where we use the CLT!
CLT = Central Limit Theorem
- Define the variables (including units), and give the distribution
“X = lifetime of a bulb (hours), X ~ N(A, B^2)” - State the null and alternative hypotheses
H_0 : μ = A
H_1 : μ >/< A - State the distribution of the mean, we can use the CLT because n > 25 and is therefore approximately normally distributed.
x̄ ~ N( A, B^2/n) - Find the p-value; the probability of getting your result or more extreme
- Compare the p-value to the significance level and make a conclusion
Reject H_0 if [ p-value < significance level ]
How do you perform a Chi^2 Test?
Step 1: Calculation of the Chi^2 Statistic
1. Build a contingency table
2. In the observed table, add an extra column and row, and total each row and column
3. Build the expected frequency contingency table:
* calculating the probability of each row and column, multiplying them for each box’s probability.
* Multiply each box’s probability by the total of all the boxes in the observed (bottom right of the extended observed table).
4. Merge any columns or rows if there is an expected frequency** less than 5**
5. Calculate Chi^2 which is the sum of [(O-E)^2/E] for each box in the table
Step 2: Testing using Chi^2 Statistic
1. State the null and alternative hypotheses, as well as the significance level
H_0 : The two variables are independent
H_1 : The two variables are dependent
2. Calculate the degrees of freedom: v = (n-1)(m-1)
3. Find the critical value according to the stated significance level
4. Compare the Chi^2 statistic with the CV
5. T > CV ==> reject H_0
6. Conclusion (go on add some context)
large values of chi^2 indicate a large differece between O and E
expected frequencies do not have to be integers
What change do you need to make when calculated Chi^2 for a 2x2 table?
The Yates Correction:
Chi^2 = Sum of [(O-E-0.5)^2/E]
How do you perform a goodness-of-fit test?
Step 1: Calculating the Chi^2 statistic
1. Determine the distribution for the expected frequencies
2. Calculate the expected frequencies using the probability calculated from the distribution and multiply by the total
3. Perform the Chi^2 calculation as normal with these expected frequencies
Step 2: Perform the hypothesis test
1. State null and alternative hypotheses
H_0 : the data does follow this distribution
H_1 : the data does not follow this distribution
2. Calculate the Chi^2 statistic (see part 1)
3. Calculate the degrees of freedom [ v = number of bins - 1 - number of constraints ]
4. State the critical value found from the table in the data book
5. A larger Chi^2 than the CV would lead to a rejection of the null hypothesis ==> the distribution is not a good fit
be careful with the number of constraints
How do you find the degrees of freedom in a goodness-of-fit tests?
v = number of bins (data points?) - 1 - number of constraints/ assumptions
If you have to calculate a mean, standard deviation, or proportion from the data, and use that value in predicting the expected values, these all add to the number of constraints.
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Differential Equations14
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Proof1
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Circular Motion7
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Centre of Mass1
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Hypothesis testing14
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Partial fractions (potentially going to later be part of integration deck)1
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Other stats (combination of linear variables)12
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Combinatorics + Probability29
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Statistical Sampling and Data Collection3
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Matrices9
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volume of revolution formulae2
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Probability Distributions8
