Confidence Intervals (Part I)

Normal Distribution

A continuous, symmetric, bell-shaped probability distribution in which values near the mean (μ) occur most frequently and values farther away become increasingly rare.

Fully described by its mean (μ) and standard deviation (σ)

f(x) = (1/(σ((2π)^1/2)))e^(-((x - μ)^2)/(2σ^2))

Equation for the probability density of a normal distribution.

Central Limit Theorem (CLT)

A fundamental statistical principle stating that, for a sufficiently large random sample, the distribution of the sample mean (X̄) becomes approximately normal, regardless of the shape of the population distribution.

The mean of this sampling distribution is μ, and its standard deviation (standard error) is σ_X̄ = (σ/((n)^1/2))

Confidence Interval

An interval around the sample mean that is likely to contains the true population mean.

It expresses how confident we are that μ lies within a given range.

Sample Mean (X̄)

The arithmetic mean of a random sample.

Under the Central Limit Theorem, it is approximately normally distributed with mean μ and standard deviation (σ_X̄) = (σ/((n)^1/2))

Fluctuate

(Verb)

Rise and fall irregularly in number or amount.

Standard deviation of the Sample Mean (σ_(X̄))

Measures how much the sample mean X̄ fluctuates from sample to sample.

(σ_(X̄)) = (σ/((n)^1/2))

Formula to calculate the Standard Deviation of the sample mean

95% Confidence Interval for μ

Covers μ for about 95% of all random samples.

The middle 95% of the normal distribution.

X̄ ± 1.96(σ_(X̄))

Formula for the 95% Confidence Interval for μ

68% Confidence Interval for μ

Covers μ for about 68% of all random samples; narrower but less reliable than the 95% Confidence Interval.

X̄ ± 1.00(σ_(X̄))

Formula for the 68% Confidence Interval for μ

μ

Symbol that represents the population mean.

A 95% Confidence Interval means that if we repeated sampling many times, about 95% of those intervals would cover μ.

Correct interpretation of the confidence level

That there is a 95% probability that μ lies in this one interval.

Incorrect interpretation of the confidence level

Wider Interval (Less Precision)

Higher Confidence Level = ?

Narrower Interval (More Precision)

Lower Confidence Level = ?

95% Confidence Interval, 99.7% Confidence Interval.

68% Confidence Interval is narrower than ___________________ which is narrower than _____________________.

Tail of a Distribution

Refers to the extreme ends of the curve.

The far left and the far right parts

Where values are much smaller or much larger than the mean.

rare, unlikely outcomes

In a normal distribution, the tails represents ____ or ________________.

Left Tail

The tail of the normal distribution that contains very low values.

Right Tail

The tail of the normal distribution that contains very high values.

α

The significance level used in confidence interval calculations.

Represents the total area in both tails of the normal distribution that lies outside the confidence interval.

The probability of the interval failing to contain the true population mean (μ)

α/2

Half of the significance level.

Represents the area in one tail of the normal distribution beyond the confidence limits

z-score

A standardized value that indicates how many standard deviations a data point or sample mean is from the mean (μ) of a distribution.

z = ((x - μ)/(σ))

Formula for the z-score for individual values

z = (X̄ - μ)/(σ_(X̄))

Formula for the z-score of the sample means.

z_(α/2)

Critical z-score that cuts off an area of α/2 in the right hand tail of the standard normal curve.

It marks the point beyond which α/2 of the probability lies.

X̄ ± (z_(α/2))(σ_x) = X̄ ± (z_(α/2))(σ/(n)^1/2)

General formula for 100(1 - α)% confidence interval

Determining the confidence interval of a given interval

If CI = (x, y)

Solve

(y - x) = u

y - u - u = v

z_(α/2) = (y - v)/(σ/((n)^1/2) = z-score

Look at the z-score in the table and you shall get your final answer after taking that value and subtracting it by one.

reliability

Confidence level measures the ___________ of the method, not the probability that μ is in the interval.

“We are 90% confident that μ is in (14.73, 14.91)” ✅

“The probability that μ is between 14.73 and 14.91 is 90%” ❌

Higher Confidence (Wider Interval)

Less Precision = ?

Lower Confidence (Narrower Interval)

More Precision = ?

95% Confidence Level

Most common confidence level to balance both Higher confidence and lower confidence?

Concept of Repeated Confidence Intervals

• While many confidence intervals are created (one for each rock)

• Each interval estimates its rock’s true mean weight.

• Even though the interval differ from sample to sample,

• They collectively show how the confidence method behaves over many repetitions.

Yes, it is true

Is it true that,

A 95% confidence level means that if we repeatedly constructed confidence intervals using the same method, about 95% of them will include the true mean (μ). It will measure the reliability of the method and not the probability for a single interval.

Bernoulli trial, Binomial distribution

For modeling the confidence interval success, each interval can be viewed as a _______________, success if it cover μ, failure if it doesn’t.

When many intervals are created (like 250), the total number follows a _____________________.

normal distribution

When the number of trials is large, the binomial distribution behaves similarly to a _________________, making it easy to approximate probabilities for many confidence intervals at once.

confidence

By modeling and approximating, we can calculate the probability that a certain number of intervals will cover μ. This shows how often the __________ method is expected to succeed in practice.

One-sided Confidence Interval

Used when only one bound (lower or upper) is of interest.

X̄ - 1.645(σ_X̄)

Formula to calculate the 95% lower bound.

X̄ + 2.33(σ_X̄)

Formula to calculate the 99% upper bound.

A one sided 95% bound fails only on one side (5% tail), while a two-sided 95% interval can fail on either side (2.5% + 2.5%). Hence, the one-sided bound is higher.

Why is one-sided bound > two-sided lower limit

random sample

Formula of X̄ ± (z_(α/2))(σ_(X̄)) is only valid if data are a ____________

Cyclic Pattern Violation

The patten shown in the repeated process indicating that the sample is not random.

This pattern happens possibly because of the dependence between runs or periodic external effects.

Confidence Interval formula should not be used.

Trend Violation

• Yields increase with time

• Indicating that the sample is not random

• This happens possibly because of the operator learning effect or due to time drift

• In such cases Confidence Interval formula is invalid

• Unless only the stable portion of the data is used.

Random Sample

Sample that is independent and identically distributed. Confidence Interval formula is valid to be used for such samples.

Non-random Sample

Sample where the time or process trend is present.

Using Confidence Interval formulas for such samples may misrepresent uncertainty.

s

Symbol that represents the sample standard deviation in a t-statistic.

t-score

Measures how many estimated standard errors a sample mean is from the population mean when the population standard deviation (σ) is unknown and the sample size is small (n ≤ 30)