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.