The problem involves identifying which confidence interval formula is used in estimating the mean weight of a certain type of pizza, based on a sample of 65 pizzas. Let’s analyze the provided options:

1. Key details from the question:

   • A sample of 65 pizzas is randomly selected.
   • The goal is to estimate the average weight with a confidence interval.
   • The sample size $n = 65$ is sufficiently large, so we expect the use of either the $z$-statistic (if the population standard deviation is known) or the $t$-statistic (if the population standard deviation is unknown, and the sample standard deviation $S$ is used).

2. Analysis of the options:

   • (a) $\bar{Y} \pm t_{1-\alpha/2}(n-1) \times \frac{S}{\sqrt{n}}$: This formula uses the $t$-statistic for constructing a confidence interval for the mean, relying on the sample standard deviation $S$. It’s appropriate when the population standard deviation is unknown.
   • (b) $\hat{\theta} \pm z_{1-\alpha/2} \times SE(\hat{\theta})$: This is a general confidence interval formula, often used in parameter estimation with the standard error of $\hat{\theta}$. It’s less specific to the mean.
   • (c) $\hat{\delta} \pm z_{1-\alpha/2} \times SE(\hat{\delta})$: This is for estimating a difference in parameters, combining two standard errors.
   • (d) Similar to (c), but with $t$-statistics and specific to comparing means ($\bar{Y}_1 - \bar{Y}_2$).
   • (e) $\bar{Y} \pm z_{1-\alpha/2} \times \frac{\sigma}{\sqrt{n}}$: This formula assumes the population standard deviation $\sigma$ is known, which is less likely in the given scenario.
   • (f) Involves a more complex variance formula, not applicable to single mean estimates.

3. Conclusion: The correct choice is (a), as it corresponds to estimating a single mean using the $t$-distribution, the appropriate method when the sample size is given and the population standard deviation is unknown.

Would you like a more detailed explanation or clarification about the other options?

Related questions:

1. What is the difference between $z$- and $t$-distributions in confidence intervals?
2. Why is the $t$-distribution used when the population standard deviation is unknown?
3. What assumptions must hold for using this formula?
4. How does increasing the sample size impact the confidence interval width?
5. What happens to the confidence interval if we increase the confidence level (e.g., from 95% to 99%)?

Tip: When deciding between $t$- and $z$-statistics, check if the population standard deviation is known. If not, use $t$-statistics!