Minimum Sample Size for Confidence

Solution & Explanation

To determine the minimum sample size $N$ required to ensure that the sample's conversion rate $\hat{P}$ is within $\delta$ of the actual click-through rate $P$, with a confidence level of 95%, we need to leverage the properties of the sampling distribution of the sample proportion.

Key Concepts:

1. Confidence Interval:

   • A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data.
   • For a 95% confidence interval, the critical value $Z$ is 1.96.

2. Margin of Error ($\delta$):

   • The margin of error defines how much the sample estimate $\hat{P}$ can deviate from the true population parameter $P$.
   • It is calculated using the formula: $\delta = Z \times \sqrt{\frac{\hat{P}(1 - \hat{P})}{N}}$

3. Solving for Sample Size ($N$):

   • Rearrange the margin of error formula to solve for $N$: $\delta = 1.96 \times \sqrt{\frac{\hat{P}(1 - \hat{P})}{N}}$ $\left(\frac{\delta}{1.96}\right)^2 = \frac{\hat{P}(1 - \hat{P})}{N}$ $N \geq \frac{\hat{P}(1 - \hat{P})}{\left(\frac{\delta}{1.96}\right)^2}$

Explanation:

• Binomial Distribution:

  • Since we are dealing with a click-through rate, our data follows a binomial distribution where each click is a success (click) or failure (no click).

• Normal Approximation:

  • For large $N$, the sampling distribution of $\hat{P}$ can be approximated by a normal distribution due to the Central Limit Theorem.

• Confidence Level:

  • 95% confidence level corresponds to a $Z$ value of 1.96. This means that we expect 95% of the sample proportions to fall within $\delta$ of the true population proportion.

• Plug in Values:

  • To determine $N$, you need an estimate of $\hat{P}$. If $\hat{P}$ is not known, a conservative approach is to use $\hat{P} = 0.5$ because it maximizes the product $\hat{P}(1 - \hat{P})$, leading to a larger sample size.

• Final Formula:

  • The formula to find the minimum $N$ is: $N \geq \frac{1.96^2 \times \hat{P}(1 - \hat{P})}{\delta^2}$
  • Use this formula to calculate $N$ for different values of $\hat{P}$ and $\delta$ based on your specific requirements.

This formula ensures that your sample size is sufficient to estimate the click-through rate with the desired precision and confidence.