Confidence intervals in R Programming - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Calculate 95% confidence interval for mean

What is the output of this R code that calculates a 95% confidence interval for the mean of a numeric vector?

R Programming

data <- c(5, 7, 8, 6, 9)
mean_val <- mean(data)
std_err <- sd(data) / sqrt(length(data))
conf_int <- mean_val + c(-1, 1) * qt(0.975, df=length(data)-1) * std_err
round(conf_int, 2)

A [4.50, 9.50]

B [5.00, 9.00]

C [5.04, 8.96]

D [6.00, 8.00]

💡 Hint

Remember to use the t-distribution quantile for the confidence interval with small samples.

The code calculates the mean and standard error, then uses the t-distribution quantile for 95% confidence with 4 degrees of freedom. The resulting interval is [5.04, 8.96].

🧠 Conceptual

intermediate

Interpretation of confidence interval

Which statement correctly describes the meaning of a 95% confidence interval for a population mean?

A There is a 95% chance the population mean lies within the interval calculated from the sample.

B 95% of the sample data points lie within the interval.

C The population mean is exactly at the midpoint of the interval.

D If we repeat the sampling many times, 95% of the calculated intervals will contain the true population mean.

💡 Hint

Think about what happens if you take many samples and calculate intervals each time.

A 95% confidence interval means that if we repeat the sampling process many times, about 95% of those intervals will contain the true population mean. It does not mean the probability for a single interval.

❓ Predict Output

advanced

Confidence interval for proportion

What is the output of this R code that calculates a 90% confidence interval for a proportion using the normal approximation?

R Programming

x <- 45
n <- 100
p_hat <- x / n
z <- qnorm(0.95)
se <- sqrt(p_hat * (1 - p_hat) / n)
conf_int <- p_hat + c(-1, 1) * z * se
round(conf_int, 3)

A [0.368, 0.532]

B [0.400, 0.500]

C [0.350, 0.550]

D [0.420, 0.480]

💡 Hint

Use the z-score for 90% confidence and calculate standard error for proportion.

The sample proportion is 0.45. The z-score for 90% confidence is about 1.645. The standard error is sqrt(p*(1-p)/n). The interval is [0.368, 0.532].

🔧 Debug

advanced

Identify error in confidence interval code

What error does this R code produce when calculating a 95% confidence interval for the mean?

R Programming

data <- c(10, 12, 14, 16, 18)
mean_val <- mean(data)
std_err <- sd(data) / length(data)
conf_int <- mean_val + c(-1, 1) * qt(0.975, df=length(data)-1) * std_err
conf_int

A The interval is too narrow because standard error is calculated incorrectly.

B SyntaxError due to missing parentheses.

C Runtime error: division by zero.

D The code runs correctly and outputs the confidence interval.

💡 Hint

Check the formula for standard error of the mean.

Standard error should be sd(data) divided by the square root of sample size, not by the sample size itself. This makes the interval too narrow and misleading.

🚀 Application

expert

Calculate and interpret confidence interval for difference of means

Given two independent samples, what is the 95% confidence interval for the difference of their means? Use this R code and select the correct output.

R Programming

group1 <- c(20, 22, 19, 24, 21)
group2 <- c(18, 17, 20, 16, 19)
mean_diff <- mean(group1) - mean(group2)
se_diff <- sqrt(var(group1)/length(group1) + var(group2)/length(group2))
t_val <- qt(0.975, df=min(length(group1), length(group2)) - 1)
conf_int <- mean_diff + c(-1, 1) * t_val * se_diff
round(conf_int, 2)

A [1.00, 5.00]

B [0.11, 6.29]

C [-1.50, 4.50]

D [0.00, 7.00]

💡 Hint

Calculate difference of means, standard error of difference, then use t-value for confidence interval.

The difference of means is 21.2 - 18 = 3.2. The standard error of difference is about 1.11. Using t-value ~2.776 for df=4, the interval is [0.11, 6.29].

Practice

1. What does a 95% confidence interval represent in statistics?

easy

A. The percentage of data points within one standard deviation

B. The probability that the sample mean is exactly 95

C. The range of values that includes 95 data points

D. A range that likely contains the true population parameter 95% of the time

Solution

1. Step 1: Understand confidence interval meaning

   A confidence interval estimates a range where the true population parameter lies based on sample data.

2. Step 2: Interpret 95% confidence level

   A 95% confidence interval means if we repeat sampling many times, 95% of those intervals will contain the true parameter.

3. Final Answer:

   A range that likely contains the true population parameter 95% of the time -> Option D

4. Quick Check:

   Confidence interval = range with true value 95% [OK]

Hint: Confidence interval shows where true value likely lies [OK]

Common Mistakes:

• Confusing confidence interval with probability of sample mean
• Thinking it counts data points inside the interval
• Mixing confidence level with standard deviation coverage

2. Which R function is commonly used to calculate a confidence interval for a sample mean?

easy

A. confint()

B. mean()

C. t.test()

D. summary()

Solution

1. Step 1: Identify function for confidence intervals

   The t.test() function in R calculates confidence intervals for means using t-distribution.

2. Step 2: Compare with other functions

   confint() is for model objects, mean() calculates mean only, summary() gives summary stats but no CI.

3. Final Answer:

   t.test() -> Option C

4. Quick Check:

   Use t.test() for confidence intervals [OK]

Hint: Use t.test() to get confidence intervals in R [OK]

Common Mistakes:

• Using mean() expecting confidence intervals
• Confusing confint() with t.test()
• Assuming summary() returns confidence intervals

3. What is the output of this R code snippet?

set.seed(123)
sample_data <- rnorm(30, mean=50, sd=10)
test_result <- t.test(sample_data, conf.level=0.99)
test_result$conf.int

medium

A. [43.5, 56.5]

B. [42.9, 57.1]

C. [44.2, 55.8]

D. [45.0, 56.0]

Solution

1. Step 1: Generate sample and run t.test()

   The code generates 30 random normal values with mean 50 and sd 10, then calculates 99% confidence interval.

2. Step 2: Calculate 99% confidence interval

   Using t-distribution with 29 degrees of freedom, the interval is approximately [42.9, 57.1].

3. Final Answer:

   [42.9, 57.1] -> Option B

4. Quick Check:

   t.test() 99% CI ≈ [42.9, 57.1] [OK]

Hint: Remember t.test() returns conf.int vector [OK]

Common Mistakes:

• Using 95% CI instead of 99%
• Ignoring set.seed causing different results
• Misreading output bounds

4. Identify the error in this R code that tries to calculate a 90% confidence interval:

data <- c(5, 7, 8, 6, 9)
test <- t.test(data, conf.level=90)
test$conf.int

medium

A. conf.level should be between 0 and 1, not 90

B. t.test() cannot be used on numeric vectors

C. Missing parentheses in t.test call

D. conf.int is not a valid attribute of t.test output

Solution

1. Step 1: Check conf.level argument

   The conf.level argument expects a decimal between 0 and 1, e.g., 0.90 for 90% confidence.

2. Step 2: Identify incorrect usage

   Passing 90 instead of 0.90 causes an error because it's out of valid range.

3. Final Answer:

   conf.level should be between 0 and 1, not 90 -> Option A

4. Quick Check:

   conf.level = 0.90, not 90 [OK]

Hint: conf.level must be decimal like 0.90, not whole number [OK]

Common Mistakes:

• Using conf.level=90 instead of 0.90
• Thinking t.test() needs special input type
• Trying to access non-existent attributes

5. You have two independent samples:

group1 <- c(20, 22, 19, 24, 21)
group2 <- c(30, 28, 35, 32, 31)

How do you calculate a 95% confidence interval for the difference in means between these groups in R?

hard

A. t.test(group1, group2, conf.level=0.95, paired=FALSE)

B. t.test(group1, group2, conf.level=0.95, paired=TRUE)

C. mean(group1) - mean(group2)

D. confint(lm(group1 ~ group2))

Solution

1. Step 1: Identify test type for independent samples

   Since group1 and group2 are independent, use paired=FALSE in t.test().

2. Step 2: Use t.test() with correct parameters

   Call t.test(group1, group2, conf.level=0.95, paired=FALSE) to get the 95% confidence interval for difference in means.

3. Final Answer:

   t.test(group1, group2, conf.level=0.95, paired=FALSE) -> Option A

4. Quick Check:

   Independent samples use paired=FALSE [OK]

Hint: Use paired=FALSE for independent samples in t.test() [OK]

Common Mistakes:

• Using paired=TRUE for independent samples
• Calculating difference of means manually without CI
• Using confint() on lm() incorrectly