## One Sample Confidence Intervals

 Getting Started General Instructions | Introduction to Your Study Descriptive Statistics Histograms | Scatter Plots | Central Tendency | Standard Deviation | Confidence Intervals Relating Variables Correlation Important Concepts The Normal Distribution | Z Scores | Probability Distributions Levels You are currently on Confidence Intervals at level 2. Level 1 | Level 2 | Level 3 Next Topic Standard Deviation | Samples and Populations

### Explanation

Calculating a Confidence Interval
Calculating the confidence interval is pretty straight forward. The only complication is that it includes a constant, Z, which is related to the required confidence level. Z does not take the value of the confidence level itself (95 for example), but takes a constant that relates to the number of standard deviations that cover the chosen percentage (95% for example) of a standard normal distribution. You can look these values up in a table of z-scores, or see the most useful values at level three of this topic. For now, this is all you need to know:

The z value you need for a 95% confidence interval is 1.96.

Understanding where the confidence intervals come from requires you to follow level three of this tutorial, so at this level we will just show you how to calculate them.

### Exploration

The formula for confidence intervals is shown below. Hover over or click on any part of it for a description of that part.

To read, it is 'The sample mean plus and minus z times the sample standard deviation over the square root of the sample size, where z is the z value for the chosen confidence interval.'

• x is the mean of the sample, and is read 'x bar';
• The + (plus and minus) part means that you subtract this value from your mean to get the lower confidence limit and you add it to your mean to get the upper confidence limit. Between those limits is the confidence interval. That means that you use the formula twice: once with addition and once with subtraction;
• Z is the constant described above. For 95% confidence levels, Z = 1.96;
• S is the standard deviation of the sample;
• The √ symbol means square root. There is an introduction to square roots on the page describing how to calculate standard deviations.

### Application

Let us look at your variable, cost.

The sample mean of cost is 517.53.
There are 30 entries in your data, so N=30.
The sample standard deviation is 298.47
We will work to the 95% confidence level, so z = 1.96

 You will be entering answers to 2 decimal places, but working to more decimal places with the calculations. If you can round numbers down in your head, do so. If not, use the [RD] button on the calculator, but remember to put the full number into memory [M=] first so that you can get it back for further calculations. You can clear the calculator screen with the [CA] button.What is the value of N for your data?   Now use the calculator to find the square root of N with the [Sqrt] button and put it into memory using the [M=] buttonRound that number to 2 decimal places and enter it here   Now divide the sample standard deviation by the value you stored in memory. Use the [MR] to retrieve that value from memory.Put this new number into memory with [M=].Now round this new number to 2 places and enter it here   Now recall the number from step 5 from memory [MR] and multiply it by 1.96. You will need this number twice, so put it into memory with [M=]. When you have safely stored the number, round it down to 2 decimal places and enter the rounded value here.   This number must now be used to produce the lower and upper confidence limits. You will make two calculations, recalling the number from step 6 [MR] for each one. Use [CA] to clear the screen between steps 8 and 9.Subtract this number from the mean to produce the lower confidence limit   Add this number to the mean to produce the upper confidence limit Calculator( You need to enable Java to see this applet. )Help
Finally, we report the confidence intervals like this: The population mean of cost has a 95% confidence interval of between 410.72 and 624.34.Now try it on your own with data from no. apps.

The sample mean of no. apps is 56.73.
There are 30 entries in your data, so N=30.
The sample standard deviation is 30.64
Repeat the steps above with these new values (but work on paper or a spreadsheet - don't enter anything other than your final answer here).
What is the lower confidence limit for no. apps?
What is the upper confidence limit for no. apps?

 Standard Deviation | Samples and Populations