confidence interval for sample mean

Question 1

Why do sample estimates vary?

Answer 1

Sample estimates from the same population vary because different random samples produce different values.

Question 2

How can we assess if a sample represents the population?

Answer 2

If estimates from different samples are similar → a particular estimate is likely close to the true parameter.

If estimates differ greatly → difficult to assume any estimate is close to the true parameter.

Question 3

What does a sample estimate provide about the population parameter?

Answer 3

A sample estimate provides a point estimate of the true parameter.

Question 4

How is uncertainty in a sample estimate represented?

Answer 4

Standard error (SE): SD of the sampling distribution

Confidence interval (CI): Range where the true parameter likely lies

Question 5

what is a confidence interval

Answer 5

an interval (defined by a lower/upper limit) within which the true value of the population parameter is stated to lie with a specified confidence

Question 6

How can we transform X∼N(μ,σ) into a standard normal variable?

Answer 6

Z = (X - μ) / σ

This standardises 𝑋 to follow N(0,1)

Question 7

What percentage of
𝑍 values lie between -1.96 and 1.96 in a standard normal distribution?

Answer 7

95% of randomly generated values lie in this range:

Pr(−1.96≤Z≤1.96)=0.95

Question 8

What happens to the sampling distribution as the sample size increases?

Answer 8

The distribution becomes more symmetric (normal) around 𝜇.

Variability depends on population variability (𝜎) and sample size (𝑛).

Question 9

How is the standard error (SE) calculated?

Answer 9

SE(μ^)= σ / sqrt(n)
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Question 10

What is the formula for the distribution of the sample mean?

Answer 10

μ^∼N(μ, (σ^2/n)

Question 11

How do we transform 𝜇^ into a standard normal variable?

Answer 11

Z= (μ^−μ) / (σ/sqrtn)
​
​

Question 12

What is the formula for a 95% confidence interval when 𝜎 is known?

Answer 12

μ^±1.96×σ/sqrtn

Question 13

What does the Central Limit Theorem (CLT) state?

Answer 13

If sufficiently many samples are drawn, the sample mean follows a normal distribution, regardless of the population distribution.

Question 14

What if the population standard deviation (𝜎) is unknown?

Answer 14

Use the sample standard deviation (s) and the t-distribution instead of the normal distribution.

Question 15

How is the confidence interval calculated when
𝜎 is unknown?

Answer 15

μ^±t(1−α/2,df=n−1) × s/sqrtn
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Question 16

What are the key properties of the t-distribution?

Answer 16

Similar in shape to the normal distribution

Centered at zero.

Shape depends on degrees of freedom (df=n−1)

shape changes with sample size e.g. has heavier tails for small samples compared to the normal distribution

Question 17

How does the similarity between the 𝑡-distribution and the standard normal distribution depend on sample size

Answer 17

The 𝑡-distribution is more spread out for small 𝑛.

As 𝑛 increases, the 𝑡-distribution approaches the normal distribution.

For 𝑛=1000, 95% of the 𝑡-distribution falls within ±1.96, similar to the normal distribution.

Question 18

What are the key differences between the normal and 𝑡-distributions?

Answer 18

The normal distribution has a fixed shape.

The 𝑡-distribution has heavier tails, especially for small sample sizes.

The 𝑡-distribution approaches the normal as 𝑛 increases.

Question 19

How does the 95% coverage change between normal and 𝑡-distributions?

Answer 19

Normal: 95% falls within ±1.96 standard deviations.

𝑡-distribution with df=10: Only 92% falls within ±1.96, requiring ±2.23 for 95% coverage.

Question 20

What is the formula for a confidence interval using the 𝑡-distribution?

Answer 20

μ^±(t(1−α/2,df=n−1)×SE(μ^))

Question 21

What does each term in the confidence interval formula represent?

Answer 21

𝜇^ : Sample mean (estimate of the true mean)

t(1−α/2,n−1): 𝑡-value for the desired confidence level

SE (μ^): Standard error of the estimate

Question 22

What two factors are needed to calculate a confidence interval?

Answer 22

Confidence level (choice of 𝛼)

Standard error (SE(μ^), which measures uncertainty)

Question 23

How do we find the
𝑡-multiplier for a 95% confidence interval?

Answer 23

Choose 𝛼=0.05

Compute 1−α/2=0.975.

Find 𝑡 such that 𝑃(𝑇≤𝑡)=0.975 for T∼tdf=n−1

Question 24

How does the choice of confidence level affect the confidence interval?

Answer 24

Higher confidence levels (e.g., 99%) → wider intervals → more likely to contain the true parameter.

Lower confidence levels (e.g., 90%) → narrower intervals → less likely to contain the true parameter.

Common choices: 90%, 95%, and 99%.

Question 25

What does “confidence” mean in a confidence interval?

Answer 25

Confidence refers to a long-run process, not a single interval.

If we repeatedly take samples, 95% of confidence intervals will contain the true value (for a 95% CI).

If many researchers independently study the same population, 95% of them will capture the true parameter.

Question 26

How do you calculate the 95% confidence interval for a sample?
Sample Mean μ^=9.56
Sample SD s=1.13
Sample size 𝑛 = 20
t-multiplier=2.09

Answer 26

SE = 0.253
CI = (9.03,10.09)

Question 27

What is an empirical (bootstrap-based) confidence interval?

Answer 27

A non-parametric method for estimating confidence intervals when:

The sample size is small

The population is skewed

The normality assumption for sample means is questionable

Question 28

What are the 5 steps for generating a bootstrap confidence interval?

Answer 28

Randomly select 𝑛 values from the original sample with replacement.

Compute the sample mean for this bootstrap sample.

Repeat steps 1 & 2 many times (e.g., 1000 times).

Examine the distribution of bootstrap means.

Find the central 95% range (2.5th and 97.5th percentiles) to define the confidence interval.

Question 29

What are the 3 advantages of bootstrap confidence intervals?

Answer 29

No assumption of normality in the population.

Can estimate CIs for any statistic (mean, median, etc.).

Useful for small or skewed samples.

Question 30

What are the 2 disadvantages of bootstrap confidence intervals?

Answer 30

Computationally intensive (requires many resamples).

Coverage issues (may not always capture the true parameter correctly).

Question 31

How do we estimate the difference between two group means?

Answer 31

Difference(A−B)= μ^A − μ^B

Same parent population → Difference is due to sampling variability.

Different parent populations → True difference exists beyond sampling variability.

Question 32

How do we determine if the difference between two groups is meaningful?

Answer 32

If difference ≈ 0, likely due to sampling variability.

If difference is large, likely a real difference between populations.

Confidence intervals help quantify uncertainty in the difference.

Question 33

What is the formula for a confidence interval for the difference between two group means?

Answer 33

(μ^1− μ^2)±(t(1−α/2),df×se(μ^1− μ^2))

Question 34

What does the standard error of the difference measure?

Answer 34

It quantifies the uncertainty in the difference between two group means by combining the variability of both samples

Question 35

What is the formula for the pooled standard deviation
𝑠𝑝 when assuming equal variances?

Answer 35

sp = sqrt[((n1 - 1)s1^2 + (n2−1)s2^2)) / (n1 + n2 -2)]

Question 36

How do you calculate the standard error of the difference in means when assuming equal variances?

Answer 36

se(μ^1− μ^2) = sp sqrt[(1/n1) +(1/n2)]

Question 37

How do you determine the degrees of freedom when assuming equal variances?

Answer 37

df=(n1+n2−2)

Question 38

How is the 95% confidence interval for the difference in means constructed? (6)

Answer 38

Compute the sample means: 𝜇^1−𝜇^2

Calculate the pooled standard deviation: sp
​
Find the standard error
𝑠𝑒(𝜇^1−𝜇^2)

Get the 𝑡-multiplier from the
𝑡-distribution.

Compute the margin of error: 𝑡×𝑠𝑒

Find the CI:
(𝜇^1−𝜇^2)±marginoferror

Question 39

Example: What is the 95% CI for the difference in group means given the following data?

Group 1: n1=10, μ^1=157.2,s1=6.349

Group 2: 𝑛2=15, 𝜇^2=171.4, 𝑠2=6.335

𝑡0.975,𝑑𝑓=23=2.069

Answer 39

(8.9,19.6)

Question 40

How do we interpret a confidence interval for the difference in means? (for a 95% CI of (8.9, 19.6)

Answer 40

A 95% CI of (8.9,19.6) means we are 95% confident that the true difference in population means is between 8.9 and 19.6. If the CI does not contain 0, it suggests a significant difference between groups.

Question 41

What formula is used for the standard error of the difference in means when variances are unequal?

Answer 41

se(μ^1− μ^2) = sp sqrt[(1/n1) +(1/n2)]

Question 42

What is the hand-calculated formula for degrees of freedom when assuming unequal variances?

Answer 42

df=min(n1−1, n2−1)

Question 43

What is the Welch-Satterthwaite formula for degrees of freedom?

Answer 43

more precise but complex estimate.

𝑑𝑓𝑤 is smaller than the df under equal standard deviation, so for the same 𝛼
level, the 𝑡-multiplier will be larger.

done using R

Question 44

What 3 assumptions are made when calculating a confidence interval using the 𝑡-distribution?

Answer 44

Independence of data (within and between groups).

Normality of data (assess using histograms or Shapiro-Wilk test).

Equal or unequal variances (test using an 𝐹-test or variance test).

Question 45

When should we assume equal vs. unequal variances in a two-sample 𝑡-test?

Answer 45

Assume equal variances if the sample standard deviations are similar.

Assume unequal variances if standard deviations differ significantly (Welch’s 𝑡-test is more conservative).

Use an 𝐹-test or Levene’s test to formally compare variances.

Question 46

What is an empirical (bootstrap) confidence interval for a difference in means? (5)

Answer 46

A non-parametric method that does not assume normality. Steps:

Resample each group with replacement.

Compute the sample means for each resample.

Calculate the difference between means.

Repeat many times (e.g., 1000+ iterations).

Find the 2.5 and 97.5 percentiles of the distribution of differences.

Question 47

When is an empirical bootstrap CI useful?

Answer 47

When sample sizes are small.

When data are highly skewed.

When the normality assumption is questionable.

When computing a confidence interval for non-standard statistics (e.g., median).