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IB Mathematics AI AHL Confidence intervals for the mean of a normal population MAI Study Notes- New Syllabus

IB Mathematics AI AHL Confidence intervals for the mean of a normal population MAI Study Notes

LEARNING OBJECTIVE

  • Confidence intervals for the mean of a normal population. 

Key Concepts: 

  •  Confidence intervals

MAI HL and SL Notes – All topics

CONFIDENCE INTERVALS

A confidence interval (CI) gives a range of plausible values for a population parameter (e.g. mean) based on a sample.

For the population mean μ, a confidence interval is constructed as:

$
\text{Confidence Interval} = \bar{x} \pm (\text{critical value}) \times (\text{standard error})
$

$\bar{x}$: Sample mean
Critical value: Depends on confidence level and distribution (z or t)
Standard error (SE): Estimate of standard deviation of the sample mean

The confidence level (e.g., $95\%$) represents the proportion of all such intervals that would contain the true parameter if the sampling were repeated many times.

Example

A company produces cans of orange soda and wants to check whether they should label them as 330ml. They take a sample of 40 cans and find a sample mean of

$
\bar{x} = 328.4 \text{ ml}, \quad \sigma = 1.6 \text{ ml (known)}
$

Find a $95\%$ confidence interval for the mean volume.

▶️Answer/Explanation

Solution:

This is large enough to use the C.L. Thm, so:
$
Z_{0.975} = 1.96, \quad \text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{1.6}{\sqrt{40}} \approx 0.253
$
$
\text{CI} = \bar{x} \pm z^ \cdot \text{SE} = 328.4 \pm 1.96 \cdot 0.253 \Rightarrow 328.4 \pm 0.496
$

Confidence Interval:
$
\rm{327.90 < \mu < 328.90} \quad \text{(and 330ml is unlikely)}
$

CALCULATING CONFIDENCE INTERVALS

When σ is Known (Using Normal Distribution)

When the population standard deviation $\sigma$ is known, and the population is normally distributed or the sample size is large ($n \geq 30$), we use the standard normal distribution (z-distribution).

Formula:

$
\bar{x} \pm z^ * \left(\frac{\sigma}{\sqrt{n}}\right)
$

$z^*$: Critical value from standard normal distribution for desired confidence level

$90\% → 1.645$
$95\% → 1.960$
$99\% → 2.576$
$\sigma$: Known population standard deviation
$n$: Sample size

When σ is Unknown (Using t-Distribution)

When the population standard deviation is unknown, use the t-distribution with $n – 1$ degrees of freedom.

Assumptions:

Sample is random
Population is normally distributed or sample size is reasonably large

Formula:

$
\bar{x} \pm t^ * \left(\frac{s}{\sqrt{n}}\right)
$

$t^*$: Critical value from t-distribution with $n-1$ degrees of freedom
$s$: Sample standard deviation
$n$: Sample size

Use a t-table or calculator to find the critical value $t^*$.

Example( USING GDC)

A sample of 8 apples is taken, giving the weights (g):

$[152, 147, 149, 153, 150, 148, 154, 151]$

Find a $90\%$ confidence interval for the mean weight.

▶️Answer/Explanation

Solution:

Use GDC, enter into `List1`
Use t-interval with:

`level = 0.90`,
`Sx = unbiased`
`n = 8`
Gives:

$
\rm{148.82 < \mu < 152.43}
$

INTERPRETING CONFIDENCE INTERVALS

A $95\%$ confidence interval means:

“We are $95\%$ confident that the true population mean lies within this interval.”

It does not mean there is a $95\%$ probability that the mean lies in the interval for this one sample.

Key Notes

Wider confidence interval → more uncertainty (can result from smaller samples or higher confidence levels).
Narrower confidence interval → more precision (can result from larger samples or lower confidence levels).
Increasing sample size reduces the margin of error.
Use z-distribution only if σ is known; otherwise, use t-distribution.

Practical Applications of Confidence Intervals

Confidence intervals are used across many fields to draw informed conclusions about populations based on sample data:

  • Quality Control: Manufacturers estimate the average durability or lifespan of a product (e.g., lightbulbs or batteries).
  • Medical Research: Doctors use them to estimate average recovery times or the effectiveness of new treatments.
  • Public Opinion: Pollsters estimate the proportion of a population supporting a political candidate or policy.
  • Environmental Science: Ecologists estimate the average temperature, air quality index, or biodiversity in a region.

Example

An environmental scientist is analyzing carbon dioxide levels in a city. From a random sample of 60 air-quality readings taken over a month, she calculates a $95\%$ confidence interval for the mean $\rm{CO_2}$ concentration to be:

\( (385 \text{ ppm},\ 401 \text{ ppm}) \) . Give Interpretation of the given statement.

▶️Answer/Explanation

Interpretation

We can say with $95\%$ confidence that the true average $\rm{CO_2}$ level in the city during that period lies between 385 ppm and 401 ppm. This means that if the scientist were to repeat the sampling process many times, approximately $95\%$ of the resulting intervals would contain the true mean $\rm{CO_2}$ level.

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