Estimation: Point & Interval Estimates, Confidence Intervals, Sample Size

TAKEAWAYS:

• Understand the difference between point estimation (single-value guesses) and interval estimation (ranges with confidence).
• Learn how to construct confidence intervals for population means (known/unknown variance) and proportions using the t-distribution and z-distribution.
• Master the formula for sample size determination to ensure reliable estimates in surveys or experiments.
• Compare maximum likelihood estimation (MLE) and method of moments (MoM) for parameter estimation.
• Apply estimation techniques to real-world problems like quality control in manufacturing or polling in elections.
• Recognize the trade-off between confidence level and margin of error in interval estimation.

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What is Estimation?

Estimation is the process of inferring population parameters (e.g., mean, variance, proportion) from sample statistics. It is divided into two main types:

1. Point Estimation: Provides a single value as an estimate (e.g., sample mean $\bar{x}$ for population mean $\mu$).
2. Interval Estimation: Provides a range of values (confidence interval) within which the true parameter likely lies.

Why Estimation?

In engineering, we rarely have access to the entire population (e.g., testing every battery in a factory). Instead, we rely on samples to estimate population characteristics. For example:

• A Daraz seller might estimate the average delivery time for a product based on a sample of orders.
• NTC could estimate the average electricity consumption of households in Kathmandu using a sample survey.

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Point Estimation

Point estimation uses sample statistics to estimate population parameters. Common estimators include:

• Mean: $\hat{\mu }=\bar{x}=\frac{1}{n}{\sum }_{i=1}^n{x}_i$
• Variance: ${\hat{\sigma }}^2=s^2=\frac{1}{n-1}{\sum }_{i=1}^n(x_i-\bar{x}{)}^2$
• Proportion: $\hat{p}=\frac{X}{n}$, where $X$ is the number of successes in $n$ trials.

Properties of Good Estimators

A good estimator should be:

1. Unbiased: $E(\hat{\theta })=\theta$ (e.g., $\bar{x}$ is unbiased for $\mu$).
2. Consistent: Converges to the true parameter as sample size $n\to \infty$.
3. Efficient: Has the smallest variance among unbiased estimators.

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Interval Estimation: Confidence Intervals

A confidence interval (CI) is a range $[L,U]$ such that: $$P(L\le \theta \le U)=1-\alpha$$ where:

• $1-\alpha$ is the confidence level (e.g., 95%).
• $\alpha$ is the significance level (e.g., 0.05).

Confidence Interval for Population Mean ($\mu$)

Case 1: Known Population Variance (${\sigma }^2$)

Use the z-distribution: $$\bar{x}\pm z_{\alpha /2}\cdot \frac{\sigma }{\sqrt{n}}$$ Example: A factory produces light bulbs with a known lifetime variance ${\sigma }^2=25$ (hours). A sample of 36 bulbs has a mean lifetime of 1000 hours. Construct a 95% CI for $\mu$.

Case 2: Unknown Population Variance (Use $t$-distribution)

Use the t-distribution with $n-1$ degrees of freedom: $$\bar{x}\pm t_{\alpha /2,n-1}\cdot \frac{s}{\sqrt{n}}$$ Example: A sample of 10 solar panel efficiencies has $\bar{x}=18\%$ and $s=2\%$. Find a 90% CI for $\mu$.

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Confidence Interval for Population Proportion ($p$)

For a proportion, use the z-distribution: $$\hat{p}\pm z_{\alpha /2}\cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$ Example: In a survey of 500 voters, 300 support a candidate. Find a 99% CI for the true proportion $p$.

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Sample Size Determination

The sample size $n$ depends on:

• Desired confidence level ($1-\alpha$).
• Margin of error (ME).
• Population variance (${\sigma }^2$) or proportion ($p$).

For Population Mean ($\mu$):

$$n={\left(\frac{z_{\alpha /2}\cdot \sigma }{\text{ME}}\right)}^2$$ Example: How many light bulbs must be tested to estimate $\mu$ within 2 hours with 95% confidence, given $\sigma =5$?

For Population Proportion ($p$):

$$n={\left(\frac{z_{\alpha /2}}{2\cdot \text{ME}}\right)}^2\cdot p(1-p)$$ Example: How many voters must be surveyed to estimate $p$ within 3% with 95% confidence? Assume $p=0.5$ (worst case).

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Methods of Estimation

1. Maximum Likelihood Estimation (MLE)

MLE finds the parameter value that maximizes the likelihood function $L(\theta |x)$. Example: For a normal distribution $X\sim N(\mu ,{\sigma }^2)$, the MLE of $\mu$ is the sample mean $\bar{x}$.

2. Method of Moments (MoM)

MoM equates sample moments to population moments and solves for $\theta$. Example: For an exponential distribution with PDF $f(x)=\lambda e^{-\lambda x}$, the first moment is: $$E(X)=\frac{1}{\lambda }=\bar{x}⟹\hat{\lambda }=\frac{1}{\bar{x}}$$

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In the Real World

1. eSewa and Khalti:

   • These apps estimate the average transaction time for digital payments using confidence intervals. For example, Khalti might sample 1000 transactions to estimate the mean processing time within 0.5 seconds with 95% confidence. This helps them set realistic user expectations and optimize server load.

2. Daraz Order Fulfillment:

   • Daraz uses interval estimation to predict delivery times. If a sample of 500 orders in Kathmandu has a mean delivery time of 3 days with a standard deviation of 1 day, they can construct a 90% CI to communicate to customers: "Your order will arrive in 2.5 to 3.5 days, 90% of the time."

3. NTC Electricity Demand Forecasting:

   • NTC estimates the average monthly electricity consumption of households in Pokhara using sample surveys. For instance, if a sample of 200 households shows a mean consumption of 250 kWh with a standard deviation of 50 kWh, they can build a confidence interval to plan grid capacity. A 95% CI might show the true mean lies between 240 and 260 kWh, guiding infrastructure investments.

4. Bank Loan Interest Rates (NMB, Global IME):

   • Banks estimate the average default rate for loans using historical data. For example, if 5% of 1000 loans defaulted in a sample, a 99% CI for the true default rate $p$ might be [3.5%, 6.5%]. This interval helps banks set interest rates to cover risks.

5. Pathao Driver Earnings:

   • Pathao estimates the average daily earnings of drivers in Lalitpur by sampling 300 drivers. If the sample mean is $25 with a standard deviation of $5, a 90% CI would be [$23.80, $26.20]. This helps drivers plan finances and Pathao adjust incentives.

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Worked Example: Traffic Congestion Estimation (Kathmandu)

Scenario: The Kathmandu Metropolitan City (KMC) wants to estimate the average daily traffic delay at a busy intersection. A sample of 40 drivers reports delays with $\bar{x}=20$ minutes and $s=5$ minutes. Construct a 95% CI for the true mean delay $\mu$.

Solution:

1. Use the t-distribution (since $\sigma$ is unknown).
2. Degrees of freedom: $n-1=39$.
3. From the t-table, $t_{0.025,39}\approx 2.023$.
4. Margin of error (ME): $$\text{ME}=t_{\alpha /2}\cdot \frac{s}{\sqrt{n}}=2.023\cdot \frac{5}{\sqrt{40}}=1.60$$
5. Confidence interval: $$\mu \in [20-1.60,20+1.60]=[18.40,21.60]\text{ minutes}$$ Interpretation: KMC can be 95% confident that the true average daily delay at this intersection is between 18.40 and 21.60 minutes. This estimate can be used to justify traffic signal adjustments or alternative route planning.

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Advantages and Disadvantages of Estimation

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Common Mistakes to Avoid

1. Using z-distribution when $\sigma$ is unknown: Always use the t-distribution for small samples ($n<30$) with unknown variance.
2. Ignoring sample size: Small samples lead to wide confidence intervals and unreliable estimates.
3. Misinterpreting confidence intervals: A 95% CI does not mean there is a 95% probability that $\mu$ lies in the interval. Instead, it means 95% of such intervals will contain $\mu$ in repeated sampling.
4. Assuming normality without checking: Confidence intervals for means assume normality (or large $n$ by CLT). Always check with a histogram or Q-Q plot.

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Exam Tip

1. Memorize Key Formulas:

   • Confidence interval for $\mu$ (known $\sigma$): $\bar{x}\pm z_{\alpha /2}\cdot \frac{\sigma }{\sqrt{n}}$
   • Confidence interval for $\mu$ (unknown $\sigma$): $\bar{x}\pm t_{\alpha /2,n-1}\cdot \frac{s}{\sqrt{n}}$
   • Confidence interval for $p$: $\hat{p}\pm z_{\alpha /2}\cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
   • Sample size for mean: $n={\left(\frac{z_{\alpha /2}\cdot \sigma }{\text{ME}}\right)}^2$

2. Understand When to Use t vs. z:

   • Use t-distribution for small samples ($n<30$) or unknown $\sigma$.
   • Use z-distribution for large samples ($n\ge 30$) or known $\sigma$.

3. Practice Calculations:

   • Always show your steps clearly in exams. For example:
     • Calculate $\bar{x}$ and $s$ from raw data.
     • Look up $t$- or $z$-values from tables.
     • Interpret the confidence interval in context.

4. Real-World Applications:

   • Expect questions linking estimation to engineering scenarios (e.g., quality control, survey sampling, or experimental design). For example:
     • "A factory tests 25 batteries and finds $\bar{x}=12$ hours, $s=1.5$ hours. Estimate the mean lifetime with 90% confidence."
     • "How many sensors must be tested to estimate the mean failure rate within 0.1% with 99% confidence?"

5. Graphical Interpretation:

   • Sketch confidence intervals on number lines or normal/t-distribution curves to visualize uncertainty. For example:

6. Sample Size Questions:

   • Always round up to the nearest whole number for sample size calculations.
   • For proportions, use $p=0.5$ if the true proportion is unknown (gives the largest sample size).

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