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Confidence Interval Calculator

Confidence Interval Calculator

Confidence Interval Equation

Sample mean (x̅)

Standard deviation (s)

Sample size (n)

Confidence level

%
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Use this online confidence interval calculator that helps you to calculate the confidence interval with lower bound and upper bound. Also, this handy upper and lower bound calculator figure out the Standard Error, Z-score, Right Tailed P-Value, and Margin of error. Read on to know about the functionality of this confidence level calculator and how to calculate confidence intervals?

What is the Confidence Interval?

Generally, a confidence interval is the level of uncertainty in any calculations within any particular statistics. We use it with a margin of error. It tells us how confident we can be about the results from a poll or survey of the targeted population. Confidence interval is fundamentally linked to the confidence level.

Confidence interval is sometimes interpreted as meaning that the “true value” of your estimate is within the confidence interval. But in fact, it’s not. The confidence interval cannot tell you the likelihood of finding the true value of the statistics. Make an estimate because it is based on a sample rather than the entire population.

The confidence interval simply indicates what range of values ​​you can expect if you run the sample again or run the experiment again in exactly the same way. The more accurate the sample design or the more realistic the experiment, the more likely your confidence interval will contain the estimated true value. However, this accuracy is determined by your research methods, not by statistical information compiled after collecting the data. !

Confidence interval Example:

If you calculate a confidence interval with a 95% confidence level, it means that you are confident that 95 out of 100 times your estimated results will fall between the upper and lower values. However, a confidence interval calculator can make a more precise estimation as compared to manual methods.

However, an online Standard Error Calculator allows you to calculate the sample mean dispersion from the given raw data set.

Confidence Interval Formula:

Confidence interval formula is:

$$CI = x̄ ± z* σ / (\sqrt{n})$$

In this formula:

  • CI = confidence interval
  • x̄ = sample mean
  • Z = confidence level value
  • Σ = sample standard deviation
  • N = sample

The confidence interval equation can be divided into three parts:

  • sample statistic
  • a confidence level
  • and a margin of error

A sample statistic is the value of the population, and the combination of confidence level and margin of error indicates the total amount of uncertainty associated with any taken sample.

Confidence Interval equation = Point Estimate + Confidence Level * Margin of Error

How to calculate confidence interval?

If we have A group of 10-foot surgery patients with a mean weight of 240 pounds and The sample standard deviation is 25 pounds, then what will be a confidence interval?

Solution:

The confidence interval calculator provides you with a quick solution because by entering all the values ​​of a variable into the input data, you can obtain accurate results through subsequent automatic calculations. However, you can perform manual calculations by applying the confidence interval formula.

Steps for calculating confidence interval are:

  • First of all, subtract 1 from 10 to have a degree of freedom: \( 10-1 = 9 \)
  • Now subtract confidence level from 1 then divide it by 2: \( (1 – .95) / 2 = .025 \)
  • According to the distribution table 9 degrees of freedom and α = 0.025, the result is 2.262
  • Now you have to Divide sample standard deviation by the square root of sample size: \( 25 / \sqrt{10 } = 7.90 \)
  • Multiply the answers of point 3 and 4: \( 2.26 × 7.90 = 17.88 \)
  • For the calculation of the lower end of the range, you have to subtract step 5 from your sample mean:
  • \( 240 – 17.88 = 222.11 \)
  • For the calculation of the upper end of the range, you have to add step 5 to your sample mean: 240 + 17.88 = 257.88

Furthermore,  the Margin of Error Calculator helps to determine the margin of error based on the Confidence Level, Proportion Percentage, Sample Size, and Population size.

Confidence Interval Table-values:

Table representing the Z-values for some common confidence levels is given below:

Confidence Level Z- Value
70% 1.036
75% 1.150
80% 1.282
85% 1.440
90% 1.645
95% 1.960
98% 2.326
99% 2.576
99.5% 2.807
99.9% 3.291
99.99% 3.891
99.999% 4.417

It is difficult to remember the z-score used to calculate the interval, so you can use the CI calculator because you don’t have to manually enter the z-score.

How Confidence Interval Calculator Works?

This confidence level calculator for the population means, standard deviation, and sample size work as follows:

Input:

  • Enter the value of the sample mean, standard deviation, total sample size, and confidence level.
  • It displays the confidence interval equation on the top.
  • Hit the calculate button.

Output:

This confidence level calculator gives you:

  • Values of confidence intervals with lower bound and upper bound.
  • Tells you The population mean (μ) enclosed by confidence interval \( x̅ ± E \) that contains the percentage of samples.
  • Standard Error, Z-score, Right Tailed P-Value, Separate value of Lower and upper bound, and Margin of error (E).

How to Construct a Confidence Interval?

An online confidence interval calculator helps you to construct a confidence interval instant but if  you want these calculations manually you need to follow these steps to construct a confidence interval are:

  • First of all, you have to Identify the sample statistic. For this purpose, select the statistic for example sample mean, sample proportion for the estimation of a population parameter.
  • Now Select a confidence level. It will describe the uncertainty of a sampling method.
  • Calculate the margin of error to construct a confidence interval. For Margin of error calculation = Critical value * Standard deviation of a statistic.
  • State the confidence interval and Confidence interval = sample statistic + Margin of error

Example: 

We draw a random sample of 230 men from a population of 1,000 men and weigh them. We find that the average in our sample weight is 150 pounds, and the standard deviation of the sample is 40 pounds. What is the 95% confidence interval?

  • \( 150 + 1.86 \)
  • \( 150 + 40 \)
  • None of the above

Solution:

Find standard error. The standard error (SE) of the mean is:

$$ SE = s / sqrt( n ) $$

$$ SE = 40 / sqrt(230) = 40/15.17 = 2.6367 $$

Now, find critical value. Compute alpha (α):

$$ Alpha α = 1 – (confidence level / 100) = 0.05 $$

Then, Find the critical probability:

$$ p* = 1 – α/2 = 1 – 0.05/2 = 0.975 $$$

So, find the degrees of freedom:

$$ df = n – 1 = 230 – 1 = 229 $$

The critical value is the t statistic having 229 degrees of freedom, also cumulative probability equal to 0.975 from the confidence interval calculator, the critical value is 2.6367.

Important Factors that Affect Confidence Intervals:

Confidence interval statics that is responsible for the values of confidence intervals are:

Confidence level:

When we draw a random sample from any population multiple times, a certain percentage of the confidence intervals will comprise the mean of that population. This percentage is known as the confidence level.

Sample standard deviation:

It is the usual or typical difference among the data point in any population.

Sample mean:

It is an average of a set of data. You can use it to compute the:

  • central tendency,
  • standard deviation
  • variance
  • confidence interval

Sample size:

It is the Total number of participants that are included in any study. It also represents the number of variables or observations.

Population size:

It is the overall set of data forms which you have taken out your sample size. For example, if your total population is 100 then your sample size might be 20 or 50.

However, a confidence interval calculator will find all these factors that affect confidence interval.

FAQs:

What is the Significance of the Confidence Interval?

It gives us the probability that any selected parameter will fall between an estimated pair of values around the mean. It will measure the uncertainty or certainty in any sampling method. They are usually assembled on the basis of confidence levels of \( 95% or 99% \).

What is a Good Confidence Interval?

A good confidence interval depends on Sample Size and Variability. If the sample size is small and the variability is high, then the confidence interval level will be wider confidence with a larger margin of error.

How do you know if a Confidence Interval is Significant?

When the level of significance is 0.05, then the corresponding confidence level will be 95%.  If there is no null hypothesis value is associated with the confidence interval, then the results are statistically significant.

What is the relationship between P-value and Confidence Interval?

If the confidence interval is narrower then the p-value will be smaller. However, the confidence interval provides valuable facts and figures about the extent of the impact studied and the reliability of the estimate.

Conclusion:

This confidence interval calculator helps you to calculate the values of upper and lower bound to assess the level of certainty and uncertainty in any estimated results. It is functioned to give you fast and easy calculations therefore students and educators may put their faith in this upper and lower bound calculator for learning and educating purposes.

Reference: 

From the source of Wikipedia: confidence interval, Philosophical issues, Statistical hypothesis testing, Confidence region, Confidence band, Significance of t-tables and z-tables.

From the source of Investopedia: Confidence Interval, Calculating a Confidence Interval, Special Considerations.

From the source of Yale: Confidence Intervals for Unknown Mean and Known Standard Deviation, Confidence Intervals for Unknown Mean and Unknown Standard Deviation.