## One Sample Confidence Intervals

 Getting Started General Instructions | Introduction to Your Study | Experimental Design | Stating a Hypothesis Descriptive Statistics Histograms | Central Tendency | Standard Deviation | Confidence Intervals Comparing Two Samples Samples and Populations | Choosing a T-Test | Paired T-Test | P-Values and T-Tables Important Concepts The Normal Distribution | Z Scores | Probability Distributions Levels You are currently on Confidence Intervals at level 3. Level 1 | Level 2 | Level 3 Next Topic Standard Deviation | Samples and Populations

### Explanation

The Theory Behind Confidence Intervals
To understand how confidence limits work, you need to understand the following concepts. We provide a tutorial for each of them, so if you are unsure of any of them, you should read the appropriate sections first.
1. Any single sample mean is one of many possible sample means that might have been found for different random samples. In theory, all of those other possible sample means form a distribution called the sampling distribution of the mean;
2. The sampling distribution of the mean is normal in shape, regardless of the shape of the population distribution;
3. The mean of the sampling distribution of the mean is the same as the population mean;
4. The standard deviation of the sampling distribution of the mean can be estimated by dividing the standard deviation of the sample by the square root of N (this is known as the Standard Error of the sample and is discussed at level three of the standard deviation topic);
5. We can measure the distance between a single value (such as our sample mean) and the mean of a normal distribution using z-scores. A z-score of 1 indicates that a value is one standard deviation from the mean. Z-scores can be converted into probabilities if they are from a normal distribution;
6. In particular, remember that 95% of the values in normally distributed data lie within 1.96 standard deviations of the mean.
If the above is clear, then the following should make sense. This is how confidence intervals are derived:
• If you move 1.96 standard deviations in both directions from the mean of a normal distribution, you will cover 95% of the data in that distribution;
• We do not know the population mean but the converse of the above statement is also true: If you take each point in any sampling distribution of the mean and travel 1.96 standard deviations in both directions, then 95% of the time you will cross the population mean;
• A single standard deviation in any sampling distribution of the mean is estimated as the standard deviation of a sample divided by the square root of the sample size;
• 1.96 standard deviations in the sampling distribution of the mean is simply 1.96 times the figure in the point above. This gives us the formula for confidence intervals that you can see at level two.
• If 95 out of 100 sample means are within 1.96 standard deviations from the mean, then we can be 95% confident that any single sample mean is within that range. That is how the confidence limit is derived.
To summarise:

95% of sample means lie within 1.96 standard errors of the population mean. Any single sample mean consequently has a 95% chance of being within 1.96 standard errors from the true population mean.

You know that the standard error of a sample is its standard deviation divided by the square root of the sample size, so the following should be clear:

• Larger samples lead to lower standard errors;
• Smaller samples lead to higher standard errors;
• Larger standard deviations lead to larger standard errors;
• Smaller standard deviations lead to lower standard errors.
And, of course, lower standard errors lead to tighter confidence intervals.

If you want to use a confidence interval that is not 95%, here are the distances away from the mean for some other percentages of a normal distribution. The distances are in units of standard errors (measured from the sample).
Confidence levelStandard Errors from mean
99%2.58
95%1.96
90%1.64
80%1.28

These portions are shown in the diagram below. The scale along the bottom is number of standard errors and the coloured portions show the proportions of the data from the table above. To cover 99% of the data, you are including all coloured portions in both directions. To cover 95%, you include all the red, blue and green sections.

### Exploration

As confidence intervals are directly related to standard error and sample size, we will show you a slightly different version of the standard error game that you saw in the section on standard deviations.

The game generates 1000 random samples of numbers from a population with a known mean. Notice how the distribution is always normal, as it is the sampling distribution of the mean. For each sample, the game will calculate a confidence interval. It will then check to see whether that interval covers the population mean. The game will then report how many of the samples had a confidence interval that captured the population mean.

Choose between a low or high standard deviation for your sample distribution. Try sample sizes from 3 to 50

The program tells you the average size of the confidence intervals of all the samples. See how it is related to the spread of the sampling distribution of the mean (i.e. the standard error).
• What happens to the spread of the sampling means (standard error) as the sample size grows?

• What happens to the confidence interval as the standard error grows?

• What size of standard deviation gives the tightest confidence intervals?

( You need to enable Java to see this applet. )
This game generates 1000 samples, but you only have one. The point of the game is to show that the 95% confidence interval of any single sample really does capture the population mean around 95% of the time. So the game is a simulation of what the theory predicts will happen.

### Application

You saw at level one of this topic that a higher confidence level leads to a wider confidence interval. You now know how the standard deviation of a sample affects the width of a confidence interval. It should be clear, then, that you can use higher confidence levels with samples with very low standard deviation and still find narrow confidence intervals.

The table below shows the confidence intervals for items correctly recalled when recall interval is 100 msecs at four different confidence levels.
Confidence levelLower limitUpper limit
99%15.0112.79
95%14.7413.06
90%14.613.2
80%14.4513.35
Look at the confidence intervals for each confidence level. You can see there is a trade-off as the 99% confidence interval is wider than the 95% one, which is wider than the 90% one, and so on. Think about which might be the best to report - for example is the jump from 90% confidence to 80% confidence worth the small improvement in the width of the interval?

The standard deviation of items correctly recalled when recall interval is 100 msecs is 1.92 and there are 20 rows so the standard error is 0.43
If you doubled the size of your sample, and the sample standard deviation remained unchanged, what effect would that have on the confidence intervals?
Imagine the standard deviation of your data was smaller, what effect would that have on the confidence intervals?

 Standard Deviation | Samples and Populations