Calculate 90% Confidence Interval

How to Calculate a 90% Confidence Interval

Essential Tools and Resources

To calculate a 90% confidence interval accurately, you need a calculator for basic arithmetic and a statistical table to find the z-score. These tools help in performing calculations and referencing the necessary statistical values efficiently.

Steps for Calculation

The process comprises seven critical steps: identifying the sample size (n), calculating the sample mean (x̄), determining the standard deviation (s), selecting the confidence interval, finding the appropriate z-value from the statistical table (z-score for a 90% confidence interval is 1.645), calculating the confidence interval using the given formulas, and finally, drawing conclusions from the results.

Formulas Used in the Calculation

The specific calculations involve several key formulas. The standard error (SE) is calculated using SE = s / \sqrt{n}. To find the margin of error (ME), multiply the standard error by the z-score: ME = SE \times 1.645. The confidence interval's upper and lower bounds are found by adding and subtracting the margin of error from the mean, respectively: upper\ bound = x̄ + ME and lower\ bound = x̄ - ME.

Example

For practical application, consider calculating a 90% confidence interval for the proportion of mobile-phone owners in a certain age group. With a proportion of successes (p̂) calculated from your sample data, apply the formulas to obtain the margin of error, and thereby the confidence interval. For instance, with a sample proportion of 0.734 and a z-score of 1.64, calculate ME = 1.64 \times \sqrt{0.734(1 - 0.734)/1250} = 0.0205. The interval extends from 71.4% to 75.5% (or 0.714 to 0.755).

Following these steps and using these tools and formulas will lead to accurate and reliable results when calculating a 90% confidence interval.

How to Calculate a 90% Confidence Interval

Understanding the Basics

A 90% confidence interval provides a range of values which is likely to contain the population parameter with 90% certainty. It's commonly employed in statistics to indicate the reliability of an estimate.

Step-by-Step Calculation

The calculation of a 90% confidence interval can be achieved by following a systematic process. First, determine the number of samples, n, and calculate the sample mean, x̄. Then, determine the standard deviation of the sample, s.

With these values, calculate the standard error using SE = s / sqrt(n). The next step involves determining the Z-value for 90% confidence which typically is 1.645. Apply this value to calculate the margin of error (ME) as ME = SE * Z(0.90).

Finally, establish the confidence interval. The upper bound is calculated as upper bound = x̄ + ME, and the lower bound as lower bound = x̄ - ME. This provides the range in which the true mean likely falls with 90% confidence.

Conclusion Drawing

Once the interval is calculated, interpret the results to understand the range within which the true population parameter lies with 90% confidence. This assists in making informed decisions or hypotheses about the studied population.

This precise approach not only ensures the calculation is accurate but also meaningful in practical applications, enhancing the reliability of statistical analysis.

Examples of Calculating a 90% Confidence Interval

Example 1: Mean of a Normally Distributed Population

If we know the population standard deviation (σ), use the Z-distribution for the sample mean. For a random sample of size n = 30, with sample mean \bar{x} = 65, and σ = 8, calculate the 90% CI. First, find the Z-score corresponding to 90%, which is 1.645. Then, calculate CI: (\bar{x} - Z \times \frac{\sigma}{\sqrt{n}}, \bar{x} + Z \times \frac{\sigma}{\sqrt{n}}). Plugging in the values, (62.59, 67.41).

Example 2: Mean with Unknown Population Standard Deviation

When σ is unknown, use the t-distribution. Suppose n = 25, \bar{x} = 48, and sample standard deviation s = 12. The t-score for 24 degrees of freedom and a 90% level is 1.711. Compute the CI as (\bar{x} - t \times \frac{s}{\sqrt{n}}, \bar{x} + t \times \frac{s}{\sqrt{n}}). The interval is (43.46, 52.54).

Example 3: Proportion in Binomial Distribution

For proportions, use a normal approximation if np and n(1-p) are both > 5. Consider a sample with n = 150, and 60 successes. The sample proportion p = 0.4. Calculate the 90% CI using the formula (\hat{p} - Z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \hat{p} + Z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}). This gives (0.34, 0.46).

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