CALCULATE

CALCULATE

** Confidence Interval calculator**, formulas, step by step calculation, real world and practice problems to learn how to find the statistical accuracy of a survey-based estimate by using the input values of confidence level, sample size, population and percentage.

It is necessary to follow the next steps:

- Choose confidence level $90\%$, $95\%$, or $99\%$. Enter sample size, population and proportion percentage in the box. The sample proportion percentage needs to be a number between 0 and 100. The sample size needs to be a positive integer.
- Press the
**"GENERATE WORK"**button to make the computation. - Confidence interval calculator will find the statistical accuracy of a survey-based estimate for a population proportion.

Confidence interval for a proportion is determined by the formula

$$(\hat p-z_c\sqrt{\frac{\hat p(1-\hat p)}{n}},\hat p+z_c\sqrt{\frac{\hat p(1-\hat p)}{n}})$$

where $\hat p$ is a sample proportion, $z_c$ is the $z$-value for confidence interval and
$n$ is the sample size.
A point estimate is a value estimate for a population parameter. The most unbiased point estimate of the population mean $\mu$ is the sample mean $\bar X$.
__The point estimate__ for the population proportion of successes, is determined by the proportion of successes in a sample i.e.
$$\hat p = \frac{x}{n}$$
where $x$ is the number of successes in the sample and $n$ is the sample size. __The interval estimate__ is an interval used to estimate a population parameter. __The confidence interval__ is an interval of numbers obtained from a point estimate of a parameter.

__The confidence level__, $c$, of interval estimate of a parameter is the probability that the interval estimate contains the population parameter.
A $c\%$ confidence interval depends on the sample. A confidence interval may or may not contain the population mean.
A $c\%$ confidence interval contains the population mean for $c\%$ of samples whose elements are chosen randomly and independently. The most common used confidence intervals are: $90\%$, $95\%$, and $99\%$.

The Central Limit Theorem states that for $n \geq 30$, the sampling distribution of sample means is a normal distribution. The level of confidence $c$ is the area under the curve between the critical values $-z_c$ and $z_c$.

A $c\%$ confidence interval for a population proportion is

$$(\hat p-z_c\sqrt{\frac{\hat p(1-\hat p)}{n}}, \hat p+z_c\sqrt{\frac{\hat p(1-\hat p)}{n}})$$

where the critical value correspond to critical values associated to the normal distribution. The critical values for $\alpha$
is
$$z_c= z_{1 - \alpha/2}$$
A binomial distribution can be approximated
by a normal distribution when $$np \geq5$$ and $nq \geq5$. If $n\hat p\geq5$, and $n(1-\hat p)\geq 5$,
then the sampling distribution of $\hat p$ is approximately normal.So, the confidence interval is defined by its lower and upper bounds. In constructing a confidence interval for a population proportion we need to follow the next steps:

- Find the point estimate $\hat p$;
- Check that the sampling distribution of $\hat p$ can be approximated by a normal distribution, i.e. $n\hat p\geq5$, and $n(1-\hat p)\geq 5$;
- Find the critical value $z_c$ in the table which corresponds to the given level of confidence $c$;
- Find the margin of error $z_c\sqrt{\frac{\hat p(1-\hat p)}{n}}$;
- Find the left and right endpoints of the confidence interval.

The concept of the confidence interval is very important in hypothesis testing since it is used as a measure of uncertainty. In other words, it can be used in considering some risk. It gives us an estimate of the range in which the real answer lies. The confidence interval can be used also to check whether or not enough simulations are done. The confidence interval decreases when the number of simulations increases. When we construct a confidence interval, it should be interpreted the meaning of the confidence level that was obtained. In the following, we give a few examples that demonstrate how to interpret confidence intervals.

**Practice Problem 1:**

In a survey of $1000$ student, $168$ said that they played football in free time. Find
a point estimate for the population proportion of students who play football. Construct a 95\% confidence interval for the population proportion of students who play football.

**Practice Problem 2:**

In a sample of $500$ adults, $241$ use public transportation. Find a $99\%$ confidence interval for the proportion of all adults use public transportation.

**Practice Problem 3:**

In a sample of $1000$ school students, $156$ smoke on a regular basis. Find the $90\%$ confidence interval estimating the population percentage for smokers at this school.

The confidence interval calculator, formula, work with steps and practice problems would be very useful for grade school students (K-12 education) to learn what is confidence interval in statistics and probability, and how to find it. Students can use this concept to determine whether a test is significant and to explain why one should not accept the null hypothesis.