IB Mathematics AI – Topic 4

Types of Validity:

• Face Validity: Does the measure appear to test what it claims to test?
• Content Validity: Does the measure cover all aspects of the concept?
• Construct Validity: Does the measure relate to other variables as expected?
• External Validity: Can results be generalized to other contexts?

Threats to Validity:

• Poor question design: Leading, ambiguous, or biased questions
• Inappropriate sampling method: Sample doesn't represent population
• Measurement errors: Instrument doesn't measure correctly
• Confounding variables: Other factors influencing results

Types of Reliability:

• Test-Retest Reliability: Same results when repeated over time
• Inter-rater Reliability: Different observers produce consistent results
• Internal Consistency: Multiple items measuring same concept produce consistent results

Factors Affecting Reliability:

• Random measurement errors
• Environmental conditions
• Observer/rater variability
• Instrument calibration and precision

Key Relationship:

A measure can be reliable without being valid (consistently wrong), but cannot be valid without being reliable (inconsistent measurements cannot be accurate).

⚠️ Common Pitfalls & Tips:

• High reliability doesn't guarantee validity – consistent but wrong
• Sample size affects reliability – larger samples more reliable
• Random sampling increases validity of generalization
• Pilot studies help identify reliability and validity issues
• Clear operational definitions improve both reliability and validity

Sample Mean (Unbiased Estimator of \(\mu\)):

\[ \bar{x} = \frac{\sum x_i}{n} \]

The sample mean is an unbiased estimator: \(E(\bar{x}) = \mu\)

Sample Variance (Two Formulas):

Biased Sample Variance (NOT used for estimation):

\[ s_n^2 = \frac{\sum (x_i - \bar{x})^2}{n} \]

This underestimates the population variance

Unbiased Sample Variance (used for estimation):

\[ s_{n-1}^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} \]

This is an unbiased estimator: \(E(s_{n-1}^2) = \sigma^2\)

Unbiased Standard Deviation:

\[ s_{n-1} = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \]

Why divide by (n-1)?

Dividing by n produces a biased estimator that systematically underestimates the population variance. Using (n-1) corrects this bias. This is called Bessel's correction.

⚠️ Common Pitfalls & Tips:

• Critical: Use \(n-1\) in denominator for sample variance when estimating population variance
• Your GDC has both: \(\sigma_n\) (population/biased) and \(\sigma_{n-1}\) (sample/unbiased)
• For confidence intervals and t-tests, always use \(s_{n-1}\)
• Larger samples reduce the difference between biased and unbiased estimators
• IB exams typically expect unbiased estimators unless otherwise stated

Solution:

(a) Sample Mean:

\[ \bar{x} = \frac{72 + 85 + 78 + 92 + 83}{5} = \frac{410}{5} = 82 \]

(b) Unbiased Variance:

First, calculate deviations from mean:

\((72-82)^2 = (-10)^2 = 100\)

\((85-82)^2 = (3)^2 = 9\)

\((78-82)^2 = (-4)^2 = 16\)

\((92-82)^2 = (10)^2 = 100\)

\((83-82)^2 = (1)^2 = 1\)

Sum of squared deviations:

\[ \sum (x_i - \bar{x})^2 = 100 + 9 + 16 + 100 + 1 = 226 \]

Unbiased variance (divide by n-1):

\[ s_{n-1}^2 = \frac{226}{5-1} = \frac{226}{4} = 56.5 \]

(c) Unbiased Standard Deviation:

\[ s_{n-1} = \sqrt{56.5} = 7.52 \text{ (3 s.f.)} \]

Using GDC: Enter data in list, use 1-Var Stats, read \(s_x\) (which is \(s_{n-1}\))

Formal Statement:

If \(X_1, X_2, \ldots, X_n\) is a random sample from a population with mean \(\mu\) and standard deviation \(\sigma\), then for large n:

\[ \bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right) \quad \text{approximately} \]

Key Properties:

• Mean of sampling distribution: \(E(\bar{X}) = \mu\)
• Variance of sampling distribution: \(\text{Var}(\bar{X}) = \frac{\sigma^2}{n}\)
• Standard error: \(SE(\bar{X}) = \frac{\sigma}{\sqrt{n}}\)

Standardization:

\[ Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \sim N(0, 1) \]

When Does CLT Apply?

• Sample size guideline: Generally n ≥ 30 is sufficient
• If population is already normal, CLT applies for any n
• If population is moderately skewed, n ≥ 30 usually sufficient
• If population is heavily skewed, may need n ≥ 100

Implications:

• Allows use of normal distribution methods even when population is not normal
• Justifies confidence intervals and hypothesis tests for large samples
• As n increases, sampling distribution becomes more normal
• As n increases, standard error decreases (more precise estimates)

⚠️ Common Pitfalls & Tips:

• CLT applies to the distribution of sample means, not individual observations
• Don't confuse \(\sigma\) (population SD) with \(\sigma/\sqrt{n}\) (standard error)
• Larger samples → smaller standard error → narrower confidence intervals
• CLT doesn't require population to be normal, just sample size to be large
• Rule of thumb: n ≥ 30 for most applications

Solution:

Given: \(\mu = 45\) minutes, \(\sigma = 12\) minutes, \(n = 36\)

(a) Distribution of Sample Mean:

Since n = 36 ≥ 30, by the Central Limit Theorem:

Mean of sampling distribution: \(E(\bar{X}) = \mu = 45\)

Standard error: \(SE = \frac{\sigma}{\sqrt{n}} = \frac{12}{\sqrt{36}} = \frac{12}{6} = 2\)

Variance: \(\frac{\sigma^2}{n} = \frac{144}{36} = 4\)

\[ \bar{X} \sim N(45, 4) \quad \text{or} \quad \bar{X} \sim N(45, 2^2) \]

(b) P(\(\bar{X}\) > 48):

Using GDC: normalcdf(48, 1×10⁹⁹, 45, 2)

Or using standardization:

\[ Z = \frac{48 - 45}{2} = \frac{3}{2} = 1.5 \]

\[ P(\bar{X} > 48) = P(Z > 1.5) = 1 - P(Z < 1.5) = 1 - 0.9332 = 0.0668 \]

Answer: 0.0668 or 6.68%

(c) P(43 < \(\bar{X}\) < 47):

Using GDC: normalcdf(43, 47, 45, 2)

\[ P(43 < \bar{X} < 47) = 0.683 \text{ or } 68.3\% \]

Interpretation: Due to CLT, even though we don't know the original distribution, we can use the normal distribution for the sample mean because n ≥ 30. There's about a 68% chance the average time for 36 people falls within 2 minutes of the population mean.

Interpretation:

A 95% confidence interval means: If we repeated the sampling process many times and constructed a confidence interval each time, approximately 95% of these intervals would contain the true population parameter.

Common Misconceptions:

• WRONG: "There is a 95% probability that \(\mu\) is in this interval"
• WRONG: "95% of the data falls in this interval"
• CORRECT: "We are 95% confident that this interval contains \(\mu\)"
• CORRECT: "95% of such intervals will contain the true parameter"

General Form:

\[ \text{Point Estimate} \pm \text{Margin of Error} \]

\[ \text{or} \quad \bar{x} \pm (\text{critical value}) \times SE \]

Formula:

\[ \bar{x} - z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} < \mu < \bar{x} + z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} \]

Where:

• \(\bar{x}\) = sample mean
• \(z_{\alpha/2}\) = critical z-value
• \(\sigma\) = population standard deviation
• \(n\) = sample size

Common Critical Values:

Formula:

\[ \bar{x} - t_{\alpha/2} \cdot \frac{s_{n-1}}{\sqrt{n}} < \mu < \bar{x} + t_{\alpha/2} \cdot \frac{s_{n-1}}{\sqrt{n}} \]

Where:

• \(\bar{x}\) = sample mean
• \(t_{\alpha/2}\) = critical t-value with df = n-1
• \(s_{n-1}\) = unbiased sample standard deviation
• \(n\) = sample size

Degrees of Freedom:

\[ df = n - 1 \]

Properties of t-Distribution:

• Symmetric, bell-shaped (like normal)
• Heavier tails than normal distribution
• As df increases, approaches standard normal
• More spread out than normal for small samples

⚠️ Common Pitfalls & Tips:

• Always use GDC for confidence intervals in exams
• z-interval requires known \(\sigma\); t-interval uses sample \(s_{n-1}\)
• Wider confidence level → wider interval (more confident, less precise)
• Larger sample → narrower interval (more precise)
• GDC: ZInterval for known σ, TInterval for unknown σ
• t-interval is always wider than z-interval (accounts for extra uncertainty)

Solution:

Given: \(\sigma = 8\) cm (known), \(\bar{x} = 175\) cm, \(n = 50\), confidence level = 95%

(a) 95% Confidence Interval:

Since \(\sigma\) is known, use z-interval.

Using GDC:

STAT → TESTS → ZInterval

Input: Stats, \(\sigma = 8\), \(\bar{x} = 175\), \(n = 50\), C-Level = 0.95

Calculate

Manual calculation:

For 95% confidence, \(z_{\alpha/2} = 1.96\)

Standard error: \(SE = \frac{\sigma}{\sqrt{n}} = \frac{8}{\sqrt{50}} = \frac{8}{7.071} = 1.131\) cm

Margin of error: \(ME = z_{\alpha/2} \times SE = 1.96 \times 1.131 = 2.217\) cm

Confidence interval:

\[ 175 - 2.217 < \mu < 175 + 2.217 \]

\[ 172.8 < \mu < 177.2 \text{ cm} \]

Or: (172.8, 177.2) cm

(b) Interpretation:

We are 95% confident that the true mean height of the population of adult males lies between 172.8 cm and 177.2 cm. If we repeated this sampling process many times, approximately 95% of the confidence intervals constructed would contain the true population mean.

(c) To narrow the interval:

• Increase sample size: Larger n reduces standard error
• Decrease confidence level: Lower confidence (e.g., 90% instead of 95%) gives narrower interval
• Reduce population variability: Not usually controllable

Solution:

(a) Sample Statistics:

Using GDC: Enter data in list, use 1-Var Stats

Sample mean: \(\bar{x} = 130.5\) units

Unbiased standard deviation: \(s_{n-1} = 8.75\) units (approximately)

Sample size: \(n = 10\)

(b) 90% Confidence Interval:

Since \(\sigma\) is unknown, use t-interval.

Degrees of freedom: \(df = n - 1 = 10 - 1 = 9\)

Using GDC:

STAT → TESTS → TInterval

Input: Data (list) or Stats, C-Level = 0.90

Calculate

Manual calculation:

For 90% confidence with df = 9, \(t_{\alpha/2} = 1.833\) (from t-table or GDC)

Standard error: \(SE = \frac{s_{n-1}}{\sqrt{n}} = \frac{8.75}{\sqrt{10}} = \frac{8.75}{3.162} = 2.767\)

Margin of error: \(ME = 1.833 \times 2.767 = 5.072\)

Confidence interval:

\[ 130.5 - 5.072 < \mu < 130.5 + 5.072 \]

\[ 125.4 < \mu < 135.6 \text{ units} \]

GDC gives: (125.4, 135.6) units

(c) Assumption Required:

For a t-interval with small sample size (n = 10), we must assume that the population distribution is approximately normal. With larger samples (n ≥ 30), the Central Limit Theorem allows us to relax this assumption.

Width of Interval:

\[ \text{Width} = 2 \times \text{Margin of Error} = 2 \times z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} \]

Three Key Factors:

1. Confidence Level (C):

• Higher confidence level → larger critical value → wider interval
• 90% CI is narrower than 95% CI, which is narrower than 99% CI
• Trade-off: More confidence requires wider interval

2. Sample Size (n):

• Larger sample → smaller standard error → narrower interval
• Relationship: \(SE \propto \frac{1}{\sqrt{n}}\)
• To halve the width, need 4 times the sample size
• Most effective way to improve precision

3. Population Variability (\(\sigma\)):

• More variable population → wider interval
• Usually cannot control this factor
• Homogeneous populations give narrower intervals

Summary Table:

Unbiased Estimators

• Mean: \(\bar{x} = \frac{\sum x_i}{n}\)
• Variance: \(s_{n-1}^2 = \frac{\sum(x_i-\bar{x})^2}{n-1}\)
• Use \(n-1\) for estimation

Central Limit Theorem

• \(\bar{X} \sim N(\mu, \sigma^2/n)\)
• Applies for n ≥ 30
• \(SE = \sigma/\sqrt{n}\)

z-Interval (σ known)

• \(\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}\)
• 95% CI: use z = 1.96
• Use when σ known

t-Interval (σ unknown)

• \(\bar{x} \pm t_{\alpha/2} \cdot \frac{s_{n-1}}{\sqrt{n}}\)
• df = n - 1
• Wider than z-interval

↓ Estimating population mean μ

↓ Estimating population proportion p

→ Use 1-PropZInterval on GDC

❌ INCORRECT Interpretations:

• "There is a 95% probability that μ is between 172 and 178"
• "95% of the population has height between 172 and 178"
• "95% of sample means fall in this interval"
• "The probability that this interval contains μ is 0.95"

✓ CORRECT Interpretations:

• "We are 95% confident that the true mean height is between 172 and 178 cm"
• "If we repeated this process many times, 95% of intervals would contain the true mean"
• "This interval was constructed using a method that captures the true mean 95% of the time"