# How to calculate 90 confidence interval

**Introduction**

Confidence intervals are an essential statistical tool used to express the degree of uncertainty associated with a sample estimate. They provide a range within which the true population parameter is likely to fall, with a specified level of confidence. In this article, we will explore how to calculate the 90% confidence interval for a sample mean and proportion.

**1. Confidence Interval for a Sample Mean**

To calculate a 90% confidence interval for the mean, follow these steps:

**Step 1: Gather your data**

Obtain a sample from your population and calculate the sample mean (x̄) and the standard deviation (s).

**Step 2: Determine the desired confidence level**

In this case, we want to find the 90% confidence interval.

**Step 3: Find the appropriate Z-score**

Look up the Z-score in standard normal distribution Z-table that corresponds to a 90% confidence level. As it is symmetric around zero, you will find two values – one positive (Zα/2) and one negative (-Zα/2) – with 0.95 in each tail (the area between them denotes 90%). For a 90% confidence interval, Zα/2 ≈ 1.645.

**Step 4: Calculate the standard error**

Standard error (SE) is derived by dividing the standard deviation (s) by the square root of the sample size (n):

SE = s / √n

**Step 5: Multiply the Z-score by the standard error**

Compute this value:

Margin of Error = Zα/2 × SE

**Step 6: Constructing the confidence interval**

Finally, add and subtract this margin of error from your sample mean:

**Lower limit =** x̄ – Margin of Error

**Upper limit** = x̄ + Margin of Error

This range represents your estimated 90% confidence interval for the population mean.

**2. Confidence Interval for a Sample Proportion**

Calculating a 90% confidence interval for a sample proportion is similar, with some variation in mathematical formulas.

**Step 1: Gather your data**

Collect the sample size (n) and determine the proportion of success (p̂), which is the number of successful outcomes divided by the total number of observations.

**Step 2: Determine the desired confidence level**

Again, we are aiming to find a 90% confidence interval.

**Step 3: Find the appropriate Z-score**

The Z-score corresponding to a 90% confidence level remains Zα/2 ≈ 1.645 for proportions as well.

**Step 4: Calculate the standard error**

For proportions, SE = √(p̂ × (1 – p̂) / n)

**Step 5: Multiply the Z-score by the standard error**

Margin of Error = Zα/2 × SE

**Step 6: Constructing the confidence interval**

**Lower limit** = p̂ – Margin of Error

**Upper limit =** p̂ + Margin of Error

This range now represents your estimated 90% confidence interval for the population proportion.

**Conclusion**

Confidence intervals are critical for gauging uncertainties in our estimates and making informed decisions. This article demonstrated how to calculate both a mean and proportion 90% confidence intervals, allowing you greater insight into your data and its implications.