90 Confidence Interval Calculator Given X and N

This calculator computes the 90% confidence interval for a proportion given x successes and n trials. It provides the exact formula, assumptions, and interpretation guide for statistical analysis.

How to Use This Calculator

To calculate the 90% confidence interval for a proportion:

Enter the number of successes (x) in the first input field.
Enter the total number of trials (n) in the second input field.
Click the "Calculate" button to compute the confidence interval.
Review the results and interpretation guidance below.

The calculator will display the lower and upper bounds of the 90% confidence interval, along with a visual representation of the result.

Formula Explained

The 90% confidence interval for a proportion is calculated using the following formula:

CI = p ± z*(√(p*(1-p)/n)) where: p = x/n (sample proportion) z = 1.645 (z-score for 90% confidence) n = total number of trials

This formula accounts for the standard error of the proportion and the appropriate z-score for 90% confidence. The result provides a range that is likely to contain the true population proportion 90% of the time.

Interpreting the Results

The 90% confidence interval provides a range of values that is likely to contain the true population proportion. Here's how to interpret the results:

The lower bound represents the smallest proportion that could be true with 90% confidence.
The upper bound represents the largest proportion that could be true with 90% confidence.
If the interval does not include 0.5 (or another value of interest), you can be 90% confident that the true proportion differs from that value.
A narrower interval indicates more precise estimates, while a wider interval suggests more uncertainty.

Remember that confidence intervals are based on assumptions about the data. The results are only valid if the sample is representative and the sample size is sufficient.

Worked Example

Suppose you conducted a survey and found that 60 out of 100 people supported a particular policy. Let's calculate the 90% confidence interval for this proportion.

Input	Value
Number of successes (x)	60
Total number of trials (n)	100

Using the formula:

p = 60/100 = 0.6 CI = 0.6 ± 1.645*(√(0.6*0.4/100)) CI = 0.6 ± 1.645*0.04899 CI = 0.6 ± 0.0808 Lower bound = 0.6 - 0.0808 = 0.5192 Upper bound = 0.6 + 0.0808 = 0.6808

The 90% confidence interval for this proportion is approximately 51.9% to 68.1%. This means we are 90% confident that the true population proportion falls within this range.

Frequently Asked Questions

What is a 90% confidence interval?: A 90% confidence interval is a range of values that is likely to contain the true population proportion 90% of the time. It provides a measure of the precision of the estimate.
How do I know if my sample size is sufficient?: For the confidence interval to be reliable, your sample size should be large enough. A common rule of thumb is to have at least 30 successes and 30 failures in your sample.
What does it mean if the confidence interval includes 0.5?: If the confidence interval includes 0.5, it suggests that the true proportion could be equal to 0.5. This means there is no statistically significant difference from 50% at the 90% confidence level.
Can I use this calculator for other confidence levels?: This calculator specifically calculates the 90% confidence interval. For other confidence levels, you would need to adjust the z-score in the formula accordingly.
What are the assumptions for this calculation?: The calculation assumes that the sample is representative of the population and that the sample size is sufficient. It also assumes that the data is normally distributed.