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.