CI and Sample Size Determination

 Let’s delve into confidence intervals and sample size determination with a numerical example. As an undergraduate civil engineering student, understanding these concepts is crucial for making informed decisions based on data. 

Confidence Interval Formula

The confidence interval is based on the mean and standard deviation. Thus, the formula to find CI is

X̄ ± Zα/2 × [ σ / √n ]

Where
X̄ = Mean
Z = Confidence coefficient
α = Confidence level
σ = Standard deviation
N = sample space
The value after the ± symbol is known as the margin of error.

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Calculating the Sample Size for a Population Mean

The margin of error $E$ for a confidence interval for a population mean is

$E=\frac{z\times \sigma }{\sqrt{n}}$

where $z$ is the $z$-score so that the area under the standard normal distribution in between $-z$ and $z$ is the confidence level $C$.

Rearranging this formula for $n$ we get a formula for the sample size $n$:

$n={\left(\frac{z\times \sigma }{E}\right)}^2$

In order to use this formula, we need values for $z$, $E$ and $\sigma$:

• The value for $z$ is determined by the confidence level of the interval, calculated the same way we calculate the $z$-score for a confidence interval.
• The value for the margin of error $E$ is set as the predetermined acceptable error, or tolerance, for the difference between the sample mean $\overline{x}$ and the population mean $\mu$.  In other words, $E$ is set to the maximum allowable width of the confidence interval.
• An estimate for the population standard deviation $\sigma$ can be found by one of the following methods:
  • Conduct a small pilot study and use the sample standard deviation from the pilot study.
  • Use the sample standard deviation from previously collected data.  Although crude, this method of estimating the standard deviation may help reduce costs significantly.
  • Use $\frac{\text{Range}}{4}$ where $\text{Range}$ is the difference between the maximum and minimum values of the population under study.

Confidence Intervals:

A confidence interval provides a range of values within which we expect an estimate to fall if we were to repeat our experiment or resample the population. It quantifies the uncertainty around our estimate. Here’s how it works:

1. What Is a Confidence Interval?

   • A confidence interval consists of the mean of your estimate plus and minus the variation in that estimate.
   • Think of it as a range of values where you are confident your estimate will fall if you repeat the experiment.
   • The confidence level represents the percentage of times you expect your estimate to fall within this range.

2. Calculating Confidence Intervals:

   • Suppose you survey both British and American people about their weekly television-watching habits.
   • You find that both groups watch an average of 35 hours of television per week.
   • However, the British group has more variation in their viewing hours compared to the Americans.
   • Even though both groups have the same point estimate (average hours watched), the British estimate will have a wider confidence interval due to the greater variation in their data.

3. Example: Variation Around an Estimate:

   • Let’s say you construct a 95% confidence interval for the average hours of television watched by both groups.
   • This means you are confident that 95 out of 100 times, the estimate will fall between the upper and lower bounds specified by the confidence interval.

4. Numerical Example:

   • Sample mean for both groups: 35 hours
   • Sample standard deviation for British group: 10 hours
   • Sample size for both groups: 100
   • Confidence level: 95%

   Using the formula for the confidence interval: [ \text{Confidence Interval} = \text{Sample Mean} \pm Z \cdot \frac{\text{Sample Standard Deviation}}{\sqrt{\text{Sample Size}}} ]

   • The critical value (Z) for a 95% confidence interval is approximately 1.96 (you can find this value from statistical tables).
   • Plugging in the numbers: 
   • [ \text{Confidence Interval} = 35 \pm 1.96 \cdot \frac{10}{\sqrt{100}} ]
   • [ \text{Confidence Interval} = 35 \pm 1.96 \cdot 1 = (33.04, 36.96) ]

   Therefore, we are 95% confident that the true average hours of television watched by both groups fall within the range of 33.04 to 36.96 hours.

Sample Size Determination:

Determining an appropriate sample size is essential for accurate statistical analysis. It affects the precision of your estimates and the width of your confidence intervals. Here’s a simplified approach:

1. Factors Influencing Sample Size:

   • Desired confidence level (e.g., 95%, 99%)
   • Margin of error (how much variation you can tolerate)
   • Standard deviation (variability in the population)
   • Z-score (critical value based on confidence level)

2. Formula for Sample Size: [ n = \left(\frac{Z \cdot \sigma}{E}\right)^2 ]

   • (n) = required sample size
   • (Z) = critical value (e.g., 1.96 for 95% confidence)
   • (\sigma) = population standard deviation (if unknown, use a conservative estimate)
   • (E) = desired margin of error

3. Example: Determining Sample Size:

   • Suppose you want to estimate the average strength of a certain material with a 95% confidence level and a margin of error of ±5 MPa.
   • Assume a conservative estimate of the population standard deviation ((\sigma)) as 20 MPa.
   • Using the formula: [ n = \left(\frac{1.96 \cdot 20}{5}\right)^2 = 15.36 ]

   Round up to the nearest whole number: (n = 16).

Therefore, you would need a sample size of 16 specimens to estimate the material strength with the desired confidence level and margin of error.

Remember, these examples are simplified, but they illustrate the concepts. In practice, consider the specific context and consult statistical software or textbooks for precise calculations. 📊🔍