A confidence interval for population proportions is very similar to a confidence interval for population means. In general, a confidence interval is an estimate found by using a sample statistic and adding and subtracting an amount corresponding to how confident we are that the interval created captures the population parameter.
For a confidence interval for population proportions, the statistic is the sample proportion and population parameter is the population proportion. The following can be used to calculate the confidence interval:
To construct a confidence interval for population proportions, the following steps must be followed:
EXAMPLE
Obecalp is a popular prescription drug but is thought to cause headaches as a side effect. In a random sample of 206 patients taking Obecalp, 23 experienced headaches.Step 1: Verify the conditions necessary for inference.
Stating the conditions isn't enough, and it's not just a formality--you must verify them. Recall the conditions needed:
Condition | Description |
---|---|
Randomness | How was the sample obtained? |
Independence | Population ≥ 10n |
Normality | np ≥ 10 and nq ≥ 10 |
Step 2: Calculate the confidence interval.
To do this, we will take the point estimate, p-hat, plus or minus the z* critical value times the standard error of p-hat, which is the square root of p-hat times q hat, over n. The population proportion is not known, so you’ll use p-hat for the standard error.
First, let's find the corresponding z* critical value for a 95% confidence interval by using a z-table. For a confidence interval, we can follow the same steps as a two-sided test. If we have a 95% confidence interval, this is actually the same as a 5% significance level. However, this is split between two tails, the lower and upper part of the distribution. Each tail will have 2.5%.
We can use the upper limit to find the critical z-score. Remember, a distribution is 100%, so to find the upper limit, we can subtract 0.025 from 1, which gives us 0.975. Now, we can use a z-table.
Standard Normal Distribution Z-Table | ||||||||||
z | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |
---|---|---|---|---|---|---|---|---|---|---|
0.0 | 0.5000 | 0.5040 | 0.5080 | 0.5120 | 0.5160 | 0.5199 | 0.5239 | 0.5279 | 0.5319 | 0.5359 |
0.1 | 0.5398 | 0.5438 | 0.5478 | 0.5517 | 0.5557 | 0.5596 | 0.5636 | 0.5675 | 0.5714 | 0.5753 |
0.2 | 0.5793 | 0.5832 | 0.5871 | 0.5910 | 0.5948 | 0.5987 | 0.6026 | 0.6064 | 0.6103 | 0.6141 |
0.3 | 0.6179 | 0.6217 | 0.6255 | 0.6293 | 0.6331 | 0.6368 | 0.6406 | 0.6443 | 0.6480 | 0.6517 |
0.4 | 0.6554 | 0.6591 | 0.6628 | 0.6664 | 0.6700 | 0.6736 | 0.6772 | 0.6808 | 0.6844 | 0.6879 |
0.5 | 0.6915 | 0.6950 | 0.6985 | 0.7019 | 0.7054 | 0.7088 | 0.7123 | 0.7157 | 0.7190 | 0.7224 |
0.6 | 0.7257 | 0.7291 | 0.7324 | 0.7357 | 0.7389 | 0.7422 | 0.7454 | 0.7486 | 0.7517 | 0.7549 |
0.7 | 0.7580 | 0.7611 | 0.7642 | 0.7673 | 0.7704 | 0.7734 | 0.7764 | 0.7794 | 0.7823 | 0.7852 |
0.8 | 0.7881 | 0.7910 | 0.7939 | 0.7967 | 0.7995 | 0.8023 | 0.8051 | 0.8078 | 0.8106 | 0.8133 |
0.9 | 0.8159 | 0.8186 | 0.8212 | 0.8238 | 0.8264 | 0.8289 | 0.8315 | 0.8340 | 0.8365 | 0.8389 |
1.0 | 0.8413 | 0.8438 | 0.8461 | 0.8485 | 0.8508 | 0.8531 | 0.8554 | 0.8577 | 0.8599 | 0.8621 |
1.1 | 0.8643 | 0.8665 | 0.8686 | 0.8708 | 0.8729 | 0.8749 | 0.8770 | 0.8790 | 0.8810 | 0.8830 |
1.2 | 0.8849 | 0.8869 | 0.8888 | 0.8907 | 0.8925 | 0.8944 | 0.8962 | 0.8980 | 0.8997 | 0.9015 |
1.3 | 0.9032 | 0.9049 | 0.9066 | 0.9082 | 0.9099 | 0.9115 | 0.9131 | 0.9147 | 0.9162 | 0.9177 |
1.4 | 0.9192 | 0.9207 | 0.9222 | 0.9236 | 0.9251 | 0.9265 | 0.9279 | 0.9292 | 0.9306 | 0.9319 |
1.5 | 0.9332 | 0.9345 | 0.9357 | 0.9370 | 0.9382 | 0.9394 | 0.9406 | 0.9418 | 0.9429 | 0.9441 |
1.6 | 0.9452 | 0.9463 | 0.9474 | 0.9484 | 0.9495 | 0.9505 | 0.9515 | 0.9525 | 0.9535 | 0.9545 |
1.7 | 0.9554 | 0.9564 | 0.9573 | 0.9582 | 0.9591 | 0.9599 | 0.9608 | 0.9616 | 0.9625 | 0.9633 |
1.8 | 0.9641 | 0.9649 | 0.9656 | 0.9664 | 0.9671 | 0.9678 | 0.9686 | 0.9693 | 0.9699 | 0.9706 |
1.9 | 0.9713 | 0.9719 | 0.9726 | 0.9732 | 0.9738 | 0.9744 | 0.9750 | 0.9756 | 0.9761 | 0.9767 |
2.0 | 0.9772 | 0.9778 | 0.9783 | 0.9788 | 0.9793 | 0.9798 | 0.9803 | 0.9808 | 0.9812 | 0.9817 |
2.1 | 0.9821 | 0.9826 | 0.9830 | 0.9834 | 0.9838 | 0.9842 | 0.9846 | 0.9850 | 0.9854 | 0.9857 |
2.2 | 0.9861 | 0.9864 | 0.9868 | 0.9871 | 0.9875 | 0.9878 | 0.9881 | 0.9884 | 0.9887 | 0.9890 |
In a z-table, the value 0.975 corresponds with a 1.9 in the left column and 0.06 in the top row. This tells us that the z-score is 1.96.
Another way is to use a t-table, which you will learn more about in a later lesson but is available to view at the end of this tutorial. We don't use the t-distribution for proportions; however, we can use the last row in this table to find the confidence levels. Z confidence level, critical values, are found in the last row of this t table, under the infinity value, or ">1000". Essentially, the normal distribution is the t distribution with infinite degrees of freedom. We're going to look in this row to find the z critical value that we should use, which is the same as the 1.96 we got from before.
Now that we have the corresonding z* critical value, we need to use p-hat, which is 23 out of 206, q-hat, which is the complement of p-hat, and the sample size, n, which is 206 and put all this information in the formula:
From this formula, we obtain 0.112, which was our p-hat, plus or minus 0.043, which is the margin of error. When we evaluate the interval, it's going to be from 0.069 all the way up to 0.155.
Step 3: Interpret the confidence interval.
The confidence interval of 0.069 to 0.155 means we're 95% certain that if everyone who was taking Obecalp was in the study, the true proportion of all Obecalp users who would experience headaches is somewhere between 6.9% and 15.5%. We don't know exactly where in that range, but the true proportion is probably somewhere in this range.
Source: Adapted from Sophia tutorial by Jonathan Osters.