Why statistical tests validate hypotheses in R Programming - Quick Recap

Recall & Review

beginner

What is the main purpose of a statistical test in hypothesis validation?

A statistical test helps decide if the data supports or rejects a hypothesis by measuring how likely the observed results are if the hypothesis is true.

beginner

What does a p-value represent in hypothesis testing?

The p-value shows the probability of getting results at least as extreme as the observed ones, assuming the null hypothesis is true.

intermediate

Why do we set a significance level (alpha) before testing a hypothesis?

The significance level is a threshold to decide when to reject the null hypothesis, usually set at 0.05, meaning we accept a 5% chance of wrongly rejecting it.

intermediate

How does a statistical test use sample data to validate a hypothesis?

It compares the sample data to what we expect under the null hypothesis and calculates a test statistic to see if the data is unusual enough to reject the null.

beginner

What role does the null hypothesis play in statistical testing?

The null hypothesis represents the default assumption (no effect or no difference) that the test tries to challenge using data.

What does a low p-value indicate in hypothesis testing?

A The data is unlikely under the null hypothesis

B The null hypothesis is definitely true

C The sample size is too small

D The test was performed incorrectly

A low p-value means the observed data is unlikely if the null hypothesis is true, suggesting we may reject it.

What is the null hypothesis usually assumed to be?

A The data is biased

B There is a strong effect

C The alternative hypothesis

D There is no effect or difference

The null hypothesis usually states there is no effect or difference, serving as the starting assumption.

Why do we choose a significance level like 0.05?

A To limit the chance of wrongly rejecting the null hypothesis

B To guarantee the null hypothesis is true

C To increase the sample size

D To avoid calculating p-values

The significance level sets how much risk we accept for wrongly rejecting the null hypothesis.

What does a statistical test compare to validate a hypothesis?

A Two unrelated datasets

B Observed data vs expected data under null hypothesis

C The hypothesis vs the alternative hypothesis

D Sample size vs population size

The test compares observed data to what is expected if the null hypothesis is true.

What happens if the p-value is greater than the significance level?

A The test is invalid

B We reject the null hypothesis

C We fail to reject the null hypothesis

D The sample size must be increased

If p-value is greater than alpha, we do not have enough evidence to reject the null hypothesis.

Explain in your own words why statistical tests help validate hypotheses.

Think about how we decide if data supports or rejects an idea.

Describe the steps involved in performing a statistical test to validate a hypothesis.

Imagine you are explaining the process to a friend who has never done it before.

Practice

1. Why do we use statistical tests in R when working with data?
t.test() is an example of such a test. What is its main purpose?

easy

A. To create graphs and charts from data

B. To clean and organize data before analysis

C. To check if the data supports a specific idea or hypothesis

D. To store data in a database

Solution

1. Step 1: Understand the role of statistical tests

   Statistical tests like t.test() are used to check if the data supports a hypothesis or if observed results could be due to chance.

2. Step 2: Identify the main purpose of t.test()

   The t.test() compares means between groups to validate if differences are statistically significant, supporting or rejecting a hypothesis.

3. Final Answer:

   To check if the data supports a specific idea or hypothesis -> Option C

4. Quick Check:

   Statistical tests validate hypotheses = To check if the data supports a specific idea or hypothesis [OK]

Hint: Statistical tests check if data supports your idea [OK]

Common Mistakes:

• Thinking tests create graphs
• Confusing data cleaning with hypothesis testing
• Assuming tests store data

2. Which of the following is the correct way to perform a t-test in R to compare two groups stored in vectors group1 and group2?

easy

A. t.test(group1, group2)

B. t_test(group1, group2)

C. ttest(group1, group2)

D. test.t(group1, group2)

Solution

1. Step 1: Recall the correct function name in R

   The correct function for a t-test in R is t.test(), with a dot between 't' and 'test'.

2. Step 2: Check the syntax for comparing two groups

   Calling t.test(group1, group2) compares the two vectors correctly.

3. Final Answer:

   t.test(group1, group2) -> Option A

4. Quick Check:

   Correct t-test syntax = t.test(group1, group2) [OK]

Hint: Remember R uses dot in function names like t.test() [OK]

Common Mistakes:

• Using underscores instead of dots in function names
• Misspelling the function name
• Using incorrect function order

3. What will be the output of this R code?

set.seed(123)
group1 <- rnorm(10, mean=5)
group2 <- rnorm(10, mean=5.5)
test_result <- t.test(group1, group2)
round(test_result$p.value, 3)

medium

A. 0.001

B. 0.500

C. 0.900

D. 0.406

Solution

1. Step 1: Understand the code generating data

   Two groups of 10 random numbers are created with means 5 and 5.5, so they differ slightly.

2. Step 2: Perform t-test and check p-value

   The t-test compares these groups; the p-value shows the chance difference is random. With these means, p-value is about 0.406.

3. Final Answer:

   0.406 -> Option D

4. Quick Check:

   p-value ≈ 0.406 means difference likely due to chance = 0.406 [OK]

Hint: p-value > 0.05 means groups do not differ significantly; check rounded p-value [OK]

Common Mistakes:

• Ignoring set.seed causing different results
• Confusing p-value with test statistic
• Rounding incorrectly

4. Identify the error in this R code that tries to validate a hypothesis using a t-test:

data1 <- c(2, 3, 5, 7)
data2 <- c(3, 4, 6, 8)
test <- t.test(data1 + data2)

medium

A. Missing parentheses around vectors

B. Using '+' inside t.test instead of two separate vectors

C. t.test cannot be used with numeric vectors

D. Data vectors are too short for t.test

Solution

1. Step 1: Analyze the t.test function call

   The code uses t.test(data1 + data2), which adds vectors element-wise, producing a single vector.

2. Step 2: Understand correct usage of t.test

   t.test expects two separate vectors as arguments to compare groups, not their sum.

3. Final Answer:

   Using '+' inside t.test instead of two separate vectors -> Option B

4. Quick Check:

   t.test needs two groups, not their sum [OK]

Hint: Pass two vectors separately to t.test, not their sum [OK]

Common Mistakes:

• Adding vectors inside t.test
• Thinking t.test works on one combined vector
• Ignoring function argument rules

5. You want to check if a new teaching method improves test scores. You have scores before and after the method for 15 students stored in vectors before and after. Which R code correctly validates the hypothesis that scores improved?

hard

A. t.test(before, after, paired=TRUE, alternative='greater')

B. t.test(before, after, paired=FALSE, alternative='two.sided')

C. t.test(before, after, paired=FALSE, alternative='greater')

D. t.test(before, after, paired=TRUE, alternative='less')

Solution

1. Step 1: Identify paired data and hypothesis direction

   Scores before and after for same students means paired data. Improvement means after > before, so test if after is greater than before.

2. Step 2: Choose correct t.test parameters

   Use paired=TRUE for paired data. Since we want to check if scores improved (after > before), use alternative='greater'.

3. Final Answer:

   t.test(before, after, paired=TRUE, alternative='greater') -> Option A

4. Quick Check:

   Paired test with 'greater' alternative = t.test(before, after, paired=TRUE, alternative='greater') [OK]

Hint: Use paired=TRUE and alternative='greater' to test improvement [OK]

Common Mistakes:

• Not using paired=TRUE for same subjects
• Choosing wrong alternative hypothesis
• Using unpaired test for paired data