Introduction to Confidence Intervals

I. What are Confidence Intervals?

If you and your friends all toss a coin one hundred times, you will probably each get a slightly different proportion of heads. One person might get 0.55, while another friend might get 0.42, etc. Between the two extremes lies the population parameter of 0.5. Rather than estimating this parameter with one sample statistic, it might be more helpful to look at the range.

This is where confidence intervals come into play. When you make a confidence interval, you create a range that can be used to estimate the population parameter. For example, if we created a 95% confidence interval, we would be 95% confident that the true population parameter was within that calculated interval. Another way to think about it is if we took many samples and created confidence intervals for each one, we would expect the population parameter to lie within approximately 95% of the intervals. 

For more information on what these terms mean, proceed below ↓ 

II. Definitions of Key Terms & Other Important Facts

Point Estimate

This is sample statistic you have gotten from your data. It might be a sample mean (\(\bar{x}\)), sample proportion (p̂), etc.

Confidence Level

You can choose how much confidence you want to have. This level ranges from anywhere around 50% to 99%. 

Critical Value

The critical value (typically z* or t*) is a number found on a table. The value is determined by the confidence level you have chosen. For example, the z* value for an 80% confidence level is 1.28 and the z* value for a 99% confidence level is 2.58. 

Standard Error

The standard error is the standard deviation OF THE STATISTIC. Make sure to do this calculation and NOT just use the standard deviation given! For example, let's say you are constructing a 1 sample mean confidence interval and you know that the standard deviation of a population is 5 and the sample size is 36. The standard error is NOT 5. Recall that the standard deviation of the \(\bar{x}\) sampling distribution is  \({\sigma}\over \sqrt {n}\). Therefore, the standard deviation of the statistic (i.e. the standard error) is  \({5}\over \sqrt {36}\) = \(5\over6\). 

Margin of Error

The margin of error is found by multiplying the standard error by the critical value. Continuing with the prior example, if you want to construct an 80% confidence interval and you know that the standard error is \(5\over6\), then the margin of error would be \((1.28) \times ({5\over 6})\) = 1.067.

Facts About Confidence Intervals:

• The point estimate always lies in the middle of the confidence interval. 
• Higher confidence levels (i.e. 99% compared to 70%) have a wider range. 
• To make the confidence interval more narrow...
  • Increase the sample size.
  • Decrease the confidence level.  

III. Steps for Constructing Confidence Intervals

Step 1: Name the Confidence Interval 

• One Proportion, One Sample Mean Z, One Sample Mean T, Matched Pairs, etc.

Step 2: Check the Conditions

• These conditions vary depending on the type of confidence interval you are constructing. 

Step 3: Construct the Interval (Apply the Formula) 

Step 4: State the Conclusion

• Based on the data, I am ___ % confident that the population parameter is between ___ and ___ . 

IV. Example

Eleanor wants to find out how many hours teenagers spend each day watching tv. She surveys a random sample of 42 teenagers and finds the sample mean to be 3.5 hours. The population standard deviation is 0.8 hours and is normally distributed. Construct a 95% confidence interval for the average number of hours teenagers spend each day watching tv.  

Step 1: Name the Confidence Interval

In this case, we are dealing with a 1 Sample Mean Z Interval. 

Step 2: Check the Conditions

In this case, the conditions are all met. For more on the conditions of this test, see the page on 1 Sample Mean Z Confidence Intervals. 

Step 3: Construct the Interval (Apply the Formula) 

point estimate +/-  (critical value) x (standard error)  

→   (\(\bar{x}\)) +/-  (z*)(\({\sigma}\over \sqrt {n}\))  

→  (3.5) +/- (1.96)(\({0.8}\over \sqrt {42}\))  

→ (3.5) +/- (0.242) 

3.5 - 0.242 = 3.258

3.5 + 0.242 = 3.742

Interval: (3.258, 3.742)

Note: The critical value was found using a z-table. For z*, the sample size is not needed to pick a critical value, but for t*, the critical value changes depending on the sample size. A portion of the z-table is listed below with the part needed for our problem highlighted:

Step 4: State the Conclusion

Based on the data, I am 95% confident that the average number of hours teenagers spend each day watching tv is between 3.258 and 3.742 .

V. Recap

Remember:

• Confidence intervals are a core topic in statistical inference. The main idea is to give up the accuracy of a point estimate in return for the confidence of a range. 
• Always NAME the interval, check the CONDITIONS, calculate the INTERVAL, and then form a CONCLUSION. 

Links to Specific Confidence Interval Pages: 

• One Sample Mean (z)
• One Sample Mean (t)
• One Proportion
• Two Proportions / Difference of Proportions
• Matched Pairs
• Two Sample Mean / Difference of Means
• Inference for Linear Regression