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# Confidence Interval Calculator

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Confidence interval measures the accuracy of an estimated population parameter. Plug in the appropriate values into the confidence interval calculator. The calculator will determine the confidence interval of the mean by using the formula below.

Confidence level formula :
If n \geq 30, the confidence interval is given as $x\pm$ $z_{\frac{\alpha }{2}} \times \frac{\sigma }{\sqrt{n}}$.

If n < 30, the confidence interval is given as $x\pm$ $t_{\frac{\alpha }{2}} \times \frac{\sigma }{\sqrt{n}}$.

For 95% confidence limit, the confidence interval is $x\pm 1.96 \times$ $\frac{\sigma }{\sqrt{n}}$.
And for 99% confidence limit, the confidence interval is $x\pm 2.58 \times$ $\frac{\sigma }{\sqrt{n}}$.

Where,
x = Sample mean ,
$\sigma$ = Standard deviation,
$\alpha$ = level of significance = 1 - $\frac{Confidence \ level}{100}$.

## Steps

Step 1 : Note down the given parameter from the problem.

Step 2 : Using the formula above, calculate the confidence interval according to the problem at the given confidence level.

## Problems

Based on Confidence interval, some of the problems are solved below.

### Solved Examples

Question 1: If 7.4 kg is the mean of a sample of 100 ad standard deviation of 1.2 kg. Determine 95% confidence limits for the population mean.
Solution:

Given :
Sample size, n = 100
Sample mean, x = 7.4 kg
Standard deviation, $\sigma$ = 1.2 kg
Confidence interval = 95%

As n > 30,
confidence interval = $x\pm 1.96 \times$ $\frac{\sigma }{\sqrt{n}}$

= $7.4 \pm 1.96 \times$ $\frac{1.2 }{\sqrt{100}}$

= $7.4 \pm 0.2352$

95% of confidence interval is from 7.1648 to 7.6352.

Question 2: The mean and the standard deviation of a sample of 64 observations are 160 and 10. Determine 95% confidence limit for the population mean.

Solution:

Given :
Sample size, n = 64
Sample mean, x = 160
Standard deviation, $\sigma$ = 10
Confidence interval = 95%

As n > 30,
confidence interval = $x\pm 1.96 \times$ $\frac{\sigma }{\sqrt{n}}$

= $160 \pm 1.96 \times$ $\frac{10}{\sqrt{64}}$

= $160 \pm 2.45$

95% of confidence interval is from 157.55 to 162.45.