Confidence intervals in R Programming - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is a confidence interval?

A confidence interval is a range of values that likely contains the true value of a population parameter. It shows how sure we are about an estimate from sample data.

beginner

In R, which function is commonly used to calculate a confidence interval for the mean of a numeric vector?

You can use the t.test() function on the numeric vector. It returns a confidence interval for the mean by default.

beginner

What does a 95% confidence interval mean in simple terms?

It means if we repeat the sampling many times, about 95% of those intervals will contain the true population value.

intermediate

How do you interpret a confidence interval that is very wide?

A wide confidence interval means there is more uncertainty about the estimate. It can happen with small samples or high data variability.

intermediate

Write a simple R code snippet to calculate a 99% confidence interval for the mean of a numeric vector data.

result <- t.test(data, conf.level = 0.99) result$conf.int This code runs a t-test and extracts the 99% confidence interval for the mean.

What does the confidence level (e.g., 95%) represent in a confidence interval?

A The percentage of intervals that will contain the true parameter if we repeat the experiment many times

B The probability that the true parameter is inside the calculated interval for this sample

C The percentage of data points inside the interval

D The chance that the sample mean equals the population mean

The confidence level means that if we repeat the sampling many times, that percentage of intervals will contain the true parameter.

Which R function can you use to get a confidence interval for a sample mean?

A mean()

B summary()

C t.test()

D confint()

t.test() calculates confidence intervals for the mean by default.

If a confidence interval for a mean is very narrow, what does it suggest?

A High uncertainty about the estimate

B The sample size is too small

C The data is biased

D Low variability or large sample size

A narrow interval usually means the estimate is precise due to low variability or a large sample.

What happens to the width of a confidence interval if you increase the confidence level from 95% to 99%?

A It becomes wider

B It stays the same

C It becomes narrower

D It becomes zero

Higher confidence levels require wider intervals to be more sure the true value is inside.

Which of these is NOT a reason for a wide confidence interval?

A Small sample size

B Large sample size

C High confidence level

D High data variability

A large sample size usually makes the interval narrower, not wider.

Explain what a confidence interval is and how you would calculate it in R for a sample mean.

Think about how sample data estimates a population value and how R helps with this.

Describe how the width of a confidence interval changes with sample size and confidence level.

Consider what makes us more or less sure about our estimate.

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