Lecture 4 and 5 Flashcards
State the null and alternative hypothesis when testing variances from two independent
populations.
- Null Hypothesis H0): The null hypothesis states that the variances of the two populations are equal.
H0:σ1=σ2
Alternative Hypothesis (H1 or Ha): The alternative hypothesis suggests that the variances of the two populations are not equal.
H1:σ1≠σ2
Is called a f-test for variances
State the null and alternative hypothesis when testing for difference of means from two
independent populations. Either One-tail or two-tail.
- I aim performing a two tailed test
- I aim performing a two tailed test
- Null Hypothesis H0): The null hypothesis states that there is no significant difference between the means of the two populations.
H0:μ1=μ2 - Alternative Hypothesis (H1 or Ha): The alternative hypothesis suggests that there is a significant difference between the means of the two populations.
H1:μ1=μ2
Both the equal-variances and unequal variances techniques require that populations be
normally distributed. How can you check if this requirement is satisfied?
With a histogram and it have a peak with evenly distributed sides.
Design a study (or explain a study) where you will apply difference of means test to
compare two independent samples. Also state the hypothesis (You can use an example
from lecture slides).
Study Title: “A Comparative Analysis of Math Test Scores”
Research Question: Do students who receive additional math tutoring perform significantly better on a math test compared to students who do not receive tutoring?
Hypotheses:
1. Null Hypothesis (H0): There is no significant difference in the mean math test scores between students who receive math tutoring and students who do not receive math tutoring.
0:Tutor=No TutorH0:μTutor=μNo Tutor
2. Alternative Hypothesis (H1 or Ha): Students who receive math tutoring have a significantly higher mean math test score compared to students who do not receive math tutoring.
1:Tutor>No TutorH1:μTutor>μNo Tutor
- What are the steps in hypothesis testing?
State the null-hypothesis
- State the alternative-hypothesis
- Select a significance level
- Collect data and calculate test statistics, use t-test or chi-squared test.
- Determine the critical region
- compare the test statistic to critical region, can you reject the null-hypothesis
- interpret the result
- What is the meaning of null-hypothesis? What is the objective of hypothesis testing?
The meaning of null-hypothesis is to prove that nothing have change
- The objective og hypothesis testing is to evaluate the evidence provided by sample data to determine whether there is enough statistical support to reject the null hypothesis in favor of an alternative hypothesis
Can you describe when to use what type of test (e.g., two-tailed, left-tailed, right-tailed)
for testing a hypothesis?
- For a two-tailed test: H1: μ ≠ μ0 (indicating a significant difference, with no specific direction).
- For a right-tailed test: H1: μ > μ0 (indicating a significant increase or positive effect).
- For a left-tailed test: H1: μ < μ0 (indicating a significant decrease or negative effect).
Can you give an example of when to run a paired t-test and when to run an independent t-
test?
paired t-test: You want to determine whether a new exercise program has a significant effect on the weight of a group of individuals. You measure the weight of each individual before they start the exercise program (pre-test) and again after several weeks of the program (post-test).
- independent t-test: You want to determine if there is a significant difference in the mean test scores between two different groups of students.
What confidence intervals are used for? How can you increase or decrease the width of a
confidence interval?
- Estimation the range within which a population parameter such as a mean or proportion, is likely to lie.
- Samplesize, increasing the sample size typically narrows the width of the confidence
- Confidence level, a higher confidence level results in wider intervals because you are aiming to be more certain.
Why cannot we use linear regression models when dependent variables is binary (0/1) or
choice variable?
Because linear regression depends on the data is normal distributed and in the real world data is often not normal distributed.
Explain Chi-square goodness of fit tests application, provide an example
Let’s consider a simple example in the context of a quality control scenario:
Scenario: A factory produces light bulbs, and they claim that 30% of their bulbs are of type A, 50% are of type B, and 20% are of type C. To ensure quality control, an inspector randomly selects 100 light bulbs and records the type of each bulb. The goal is to test if the observed distribution of light bulb types matches the factory’s claims.
Hypotheses:
* Null Hypothesis (�0H0): The observed distribution of light bulb types is consistent with the factory’s claims.
* Alternative Hypothesis (�1H1): The observed distribution of light bulb types is not consistent with the factory’s claims.
Data:
* Observed counts: Type A (32 bulbs), Type B (48 bulbs), Type C (20 bulbs)
Analysis:
1. Calculate the expected counts based on the factory’s claims: 100 bulbs * 30% = 30 bulbs for Type A, 100 bulbs * 50% = 50 bulbs for Type B, and 100 bulbs * 20% = 20 bulbs for Type C.
2. Set up a chi-square goodness of fit test with three categories (Type A, Type B, Type C).
3. Calculate the chi-square test statistic based on the observed and expected counts.
4. Determine the degrees of freedom (df) for the test, which is (number of categories - 1).
5. Find the critical chi-square value from a chi-square distribution table for a chosen significance level (e.g., 0.05) and df.
6. Compare the calculated chi-square statistic with the critical value. If the calculated statistic is greater than the critical value, you reject the null hypothesis, indicating that the observed distribution significantly differs from the expected distribution.
