Why statistical tests validate hypotheses in R Programming - Challenge Your Understanding

Challenge - 5 Problems

❓ Predict Output

intermediate

Understanding p-value interpretation in hypothesis testing

What is the output of this R code that performs a t-test on two small samples?

R Programming

group1 <- c(5, 7, 8, 6, 9)
group2 <- c(10, 12, 11, 13, 14)
test <- t.test(group1, group2)
print(test$p.value < 0.05)

A FALSE

B TRUE

C Error: object 'test' not found

D NA

💡 Hint

Check if the p-value is less than 0.05 to decide significance.

The t-test compares the means of two groups. Since group2 values are higher, the p-value is small, so the test returns TRUE for significance.

🧠 Conceptual

intermediate

Why do we reject the null hypothesis when p-value is low?

Which statement best explains why a low p-value leads to rejecting the null hypothesis?

A A low p-value means the observed data is very unlikely if the null hypothesis is true.

B A low p-value proves the alternative hypothesis is true with 100% certainty.

C A low p-value means the sample size is too small to trust results.

D A low p-value means the null hypothesis is definitely false.

💡 Hint

Think about what the p-value measures about the data under the null hypothesis.

The p-value measures how likely the observed data is if the null hypothesis is true. A low p-value means this is unlikely, so we reject the null.

🔧 Debug

advanced

Identify the error in this hypothesis test code

What error does this R code produce when running a t-test?

R Programming

x <- c(1, 2, 3, 4, 5)
y <- c(2, 3, 4, 5)
t.test(x, y)

A Error: 't.test' requires both samples to have the same length

B Error: non-numeric argument to binary operator

C No error; code runs and returns a p-value

D Error: sample sizes must be greater than 1

💡 Hint

Check if t.test requires equal length samples.

The t.test function in R can handle samples of different lengths. It runs without error and returns a p-value.

❓ Predict Output

advanced

Output of a one-sided hypothesis test in R

What is the output of this R code testing if mean of x is greater than 5?

R Programming

x <- c(6, 7, 8, 9, 10)
test <- t.test(x, mu=5, alternative='greater')
print(test$p.value < 0.05)

A TRUE

B FALSE

C Error: invalid alternative hypothesis

D NA

💡 Hint

Check if the sample mean is significantly greater than 5.

The sample mean is above 5, so the one-sided test returns a low p-value, making the output TRUE.

🧠 Conceptual

expert

Why do statistical tests rely on probability distributions?

Which option best explains why statistical tests use probability distributions to validate hypotheses?

A Because probability distributions eliminate all uncertainty in data.

B Because probability distributions guarantee the alternative hypothesis is true.

C Because probability distributions are only used for large sample sizes.

D Because probability distributions model the expected behavior of data under the null hypothesis, allowing calculation of p-values.

💡 Hint

Think about how we decide if data is unusual under the null hypothesis.

Statistical tests use probability distributions to understand what data looks like if the null hypothesis is true. This helps calculate p-values to decide if data is unusual enough to reject the null.

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