Confidence interval provides a range of values that would likely contain the true population parameter, which is unknown. It represents the long-term success rate of capturing the parameter of interest.

The confidence level (e.g. 95%) is the percentage of times you expect to get close to the same estimate if you run your experiment again or resample the population in the same way.

Instead of estimating an unknown parameter by a single value, the confidence interval provides a range that is likely to contain the unknown value and a confidence that the unknown value lays within that range.

We are 95% confident that the true parameter lies within the calculated interval. In other words, we are confident that 95 out of 100 times, the estimate will fall within the upper and lower bounds of the confidence interval.

The 95% confidence level does not mean that there is a 95% probability that the true parameter lies within the specific interval calculated from a particular sample. The true parameter is either within the interval or not; the confidence level refers to the long-term behavior of intervals in repeated sampling.

$\bar{x} \pm z*\frac{s}{\sqrt{n}}$

• $\bar{x}$: The sample mean.
• $z$: The critical value.
• $s$: The sample standard deviation.
• $n$: The sample size.

Definition: $MOE$ represents the difference, in percentage points, between a point estimate and the true population value, considering a specified confidence level.

Formula: $MOE = z*\frac{s}{\sqrt{n}}$

• $z$: The critical value based on the sample distribution.
• $s$: The sample standard deviation.
• $n$: The sample size.

1. Decide the probability distribution (For this example, the data is normally distributed).

2. Choose alpha value $(a)$, the probability threshold for statistical significance.

   • $a=1-CL$ (where $a$ is alpha and $CL$ is the confidence interval)
   • $a =0.05$ (For this example, the confidence level will be $0.95$)

3. Decide one-tail or two-tail (For this example, select the confidence interval).

4. Find the critical value in the table.

   • $0.05 / 2 = 0.025$ (Split the $a$ into 2 because it is two-tailed)
   • From the $z$-table the critical value based on the two-tail 95% confidence level is $1.96$.

• The critical value defines how many SE away from the mean in order to reach the desired confidence level 95%.
• The standard error is a measure of the variability of the estimates.
• Multiply both critical value and the standard error together to establish the range of values that the true population parameter is likely to fall within at the specified level of confidence.

An increase in the confidence level will increase the confidence interval. Intuitively, we need a higher confidence interval to be more confident that the true population parameter is within the interval.

The confidence interval decreases as the number of sample increases.

An increase in standard error implies that we are more uncertain we are regarding the estimate and thus the less confident we are about our estimate.

Increase the sample size to maintain the confidence interval.