Example
A company wants to know what proportion of the bell pepper seeds it sells will germinate. One hundred seeds are randomly selected from the company's inventory. They are placed in ideal conditions for germination. After two weeks, 78 of the seeds had germinated. Set a 90% confidence interval for the proportion of seeds in inventory that would germinate under ideal conditions. Interpret the interval in the context of the problem.
Solution
First, we need to determine whether the conditions are satisfied for us to use a normal approximation to find the interval.
n = 100(0.78) = 78 > 10 and n( 1 – ) = 100(1 – 0.78) = 12 > 10
Because both n and n( 1 – ) are greater than 10, we can use a largesample confidence interval based on the normal distribution.
Second, we know that the standard error of is . This would have been true even if the sample size had not been large enough to use a largesample confidence interval.
Next, we need to find z* such that P( –z* < z < z*) = 0.90. From the table of common z* values provided earlier, z* = 1.645 is the multiplier for a 90% confidence interval.
The limits of the confidence interval are then 0.78 1.645 or 0.78 0.22 The confidence interval is (0.56,1.00).
Interpretation: We are 90% confident that between 56% and 100% of the bell pepper seeds in this company's inventory would germinate under ideal conditions.
Sample Sizes, Confidence Level, and Length of Confidence Intervals
Recall that the general form of a confidence interval is point estimate multiplier × standard error. The length of the confidence interval is 2 × multiplier × standard error; the halflength of the confidence interval is multiplier × standard error. From Lesson 13, we know that as the sample size increases, the standard error decreases, so the length of the confidence interval decreases. Notice from the table of z* values that the value of the multiplier increases as the confidence level increases, making the confidence interval longer. The fact that the confidence interval gets longer as the confidence level increases holds for other forms of intervals as well.
Example
In the last example, we set a 90% confidence interval on the proportion of seeds in inventory that would germinate under ideal conditions. Now set a 95% confidence interval on this same proportion. Compare the two intervals.
Solution
Changing the confidence level has no effect on whether or not the conditions for inference are satisfied. We still have n = 100(0.78) = 78 > 10 and n( 1 – ) = 100(1 – 0.78) = 12 > 10 so we can use the normal distribution to approximate the sampling distribution of . Because the sample size has not changed, the standard error is the same. However, the multiplier z* is different. We must have 0.025 probability in each tail instead of the 0.05 in each tail that corresponded to a 90% confidence interval. From the table, we have z* = 1.96. Thus, the 95% confidence interval is 0.78 1.96 or 0.78 0.26 compared to the 90% confidence interval of 0.78 0.22 found earlier. Clearly, the 95% confidence interval is wider than the 90% confidence interval. It makes sense that, as the interval increases in length, we become more confident that the interval will capture the true population proportion.
We should also note that the 95% confidence interval ranges from 0.52 to 1.04. However, p cannot be greater than 1. Therefore, rounding the upper limit to the largest admissible value, we would say that we are 95% confident that the proportion of bell pepper seeds in inventory that would sprout under optimal conditions is between 56 and 100%. Alternately, we might say with 95% confidence that at least 56% of the bell pepper seeds in inventory will sprout under optimal conditions.
Confidence Intervals for Proportions In Short
Confidence intervals provide a plausible set of values for the unknown population parameter of interest. Appealing either to normality or the Central Limit Theorem, confidence intervals have the form point estimate multiplier × standard error.
For proportions, the point estimate is , and the standard error is . The multiplier z* is chosen so that the interval has the desired level of confidence. Relationships exist among the length of the confidence interval, the sample size, and the level of confidence.
Find practice problems and solutions for these concepts at Confidence Intervals for Proportions Practice Questions.
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