How to Calculate The 90 Confidence Interval for The Mean

Calculating the 90% confidence interval for the mean is a fundamental statistical technique used to estimate the range within which the true population mean is likely to fall. This guide will walk you through the process, explain the underlying concepts, and provide practical examples to help you apply this method effectively.

What is a 90% Confidence Interval for the Mean?

A 90% confidence interval for the mean is a range of values that is likely to contain the true population mean with 90% probability. It's calculated from a sample of data and provides a measure of the uncertainty or precision of the sample mean as an estimate of the population mean.

The confidence interval is constructed using the sample mean, the standard error of the mean, and the critical value from the t-distribution (for small samples) or the standard normal distribution (for large samples). The width of the interval depends on the sample size and the variability of the data.

Key Points:

The confidence level (90% in this case) represents the probability that the interval contains the true population mean.
A higher confidence level results in a wider interval, providing more certainty but less precision.
The confidence interval is not the probability that the true mean falls within the interval; it's about the method's reliability over many samples.

When to Use a 90% Confidence Interval

You should use a 90% confidence interval when you need to estimate the range of the true population mean based on a sample of data. This is particularly useful in situations where:

You want to make inferences about a population parameter based on sample data.
You need to communicate the uncertainty associated with your sample mean.
You're comparing two or more sample means to determine if they come from the same population.
You're working with small sample sizes and need to account for additional variability.

Common applications include quality control, market research, medical studies, and social science surveys. The 90% confidence level provides a balance between precision and reliability, making it suitable for many practical scenarios.

How to Calculate the 90% Confidence Interval

Calculating the 90% confidence interval for the mean involves several steps. Here's a detailed breakdown of the process:

Collect your sample data: Gather the measurements or observations that will form your sample.
Calculate the sample mean: Sum all the values in your sample and divide by the number of observations.
Calculate the sample standard deviation: Measure how spread out the numbers in your sample are.
Determine the standard error of the mean: Divide the sample standard deviation by the square root of the sample size.
Find the critical t-value: Use the t-distribution table with degrees of freedom equal to sample size minus one and a confidence level of 90%.
Calculate the margin of error: Multiply the standard error by the critical t-value.
Determine the confidence interval: Subtract and add the margin of error to the sample mean.

Formula for 90% Confidence Interval:

Lower Bound = Sample Mean - (Critical t-value × Standard Error)

Upper Bound = Sample Mean + (Critical t-value × Standard Error)

Where Standard Error = Sample Standard Deviation / √(Sample Size)

For large samples (typically n > 30), you can use the standard normal distribution (z-distribution) instead of the t-distribution. The critical z-value for a 90% confidence interval is approximately 1.645.

Worked Example

Let's walk through a practical example to illustrate how to calculate the 90% confidence interval for the mean.

Example Scenario

Suppose you're conducting a study to estimate the average daily screen time of college students. You randomly select 25 students and record their daily screen time in hours. The sample data is as follows:

Sample Mean (x̄) = 5.2 hours

Sample Standard Deviation (s) = 1.8 hours

Sample Size (n) = 25

Step-by-Step Calculation

Calculate the standard error of the mean:
Standard Error = s / √n = 1.8 / √25 = 1.8 / 5 = 0.36 hours
Find the critical t-value:
For a 90% confidence interval with 24 degrees of freedom (n-1), the critical t-value is approximately 1.711.
Calculate the margin of error:
Margin of Error = t × Standard Error = 1.711 × 0.36 ≈ 0.616 hours
Determine the confidence interval:
Lower Bound = x̄ - Margin of Error = 5.2 - 0.616 ≈ 4.584 hours

Upper Bound = x̄ + Margin of Error = 5.2 + 0.616 ≈ 5.816 hours

The 90% confidence interval for the mean daily screen time is approximately 4.58 to 5.82 hours. This means we can be 90% confident that the true average daily screen time for all college students falls within this range.

How to Interpret the Results

Interpreting the confidence interval results correctly is crucial for making valid conclusions. Here are some key points to consider:

The confidence interval provides a range of plausible values for the population mean.
The confidence level (90%) indicates the probability that the interval contains the true mean if the sampling process were repeated many times.
A wider interval suggests more uncertainty or variability in the data.
If the interval includes zero, it suggests that the population mean might not be significantly different from zero.
Comparing confidence intervals from different samples can help assess differences between groups.

Important Note: The confidence interval does not provide information about individual observations. It only applies to the population mean.

FAQ

What does a 90% confidence interval mean?: A 90% confidence interval means that if we were to take many samples and calculate a 90% confidence interval for each, we would expect approximately 90% of these intervals to contain the true population mean.
How does sample size affect the confidence interval?: Larger sample sizes result in narrower confidence intervals because the standard error decreases as the sample size increases. This provides more precise estimates of the population mean.
Can I use a 90% confidence interval for non-normal data?: For small samples (n < 30), the t-distribution should be used even if the data is not normally distributed. For larger samples, the central limit theorem often applies, and the normal distribution can be used regardless of the underlying distribution.
What if my sample size is very small?: With very small sample sizes (typically n < 30), the confidence interval will be wider due to increased variability. In such cases, it's important to ensure the data meets the assumptions of the t-distribution.
How do I compare two confidence intervals?: To compare two confidence intervals, check if they overlap. If they don't overlap, it suggests there may be a statistically significant difference between the two population means at the chosen confidence level.