6. Parameter Estimation and Confidence Intervals: Estimation
Confidence Interval for the Population Proportion
A confidence interval for the population proportion is a range of values, based on sample data, which are highly plausible candidates for the true value of the population proportion.
To construct a confidence interval for the population proportion , we will need to make use of the sampling distribution of the sample proportion.
Remember that the sample proportion (approximately) follows the distribution if both of the following conditions are satisfied:
- There are at least 10 positive cases:
- There are at least 10 negative cases:
The problem, however, is that since the value of is unknown, we cannot use it to check the conditions for normality.
The solution is to use the sample proportion as an estimate for the population proportion and check the conditions for normality using instead.
Likewise, without knowing , we cannot compute the standard error of the proportion . Instead, we will use the estimated standard error of the proportion in the calculation of a confidence interval for the population proportion :
The width of a confidence interval is determined by the margin of error.
Margin of Error
The margin of error of a confidence interval for the population proportion is the distance from the center of the interval to either the lower bound or the upper bound .
To calculate the margin of error of a confidence interval for the population proportion , use the following formula:
Where is the critical value of the Standard Normal Distribution such that .
Calculating z* with Statistical Software
Let be the confidence level in .
To calculate the critical value in Excel, make use of the function NORM.INV():
To calculate the critical value in R, make use of the function qnorm():
Factors that Influence the Margin of Error
The margin of error of a confidence interval for the population proportion is dependent on factors: the confidence level, the sample proportion, and the sample size.
- As the confidence level increases, the margin of error increases and the confidence interval becomes wider.
- As the sample proportion approaches a value of (from either side), the margin of error increases and the confidence interval becomes wider.
- As the sample size increases, the margin of error decreases and the confidence interval becomes narrower.
He randomly selects a sample of from this population and finds that of them are vegetarian/vegan.
Calculate the margin of error of the confidence interval for the population proportion . Round your answer to decimal places.
There are a number of different ways we can calculate the margin of error. Click on one of the panels to toggle a specific solution.
The margin of error of a confidence interval for the population proportion is calculated with the following formula:
Calculate the sample proportion :
Investigate whether the sampling distribution of the sample proportion may be considered approximately normal:
Since both conditions are satisfied, the sampling distribution of the sample proportion is approximately normally distributed with parameters and .
However, because the population proportion is unknown, the standard error of the proportion cannot be calculated.
Instead, we will use the sample proportion to calculate the estimated standard error of the proportion :
For a given confidence level , the critical value of the standard normal distribution is the value such that .
To calculate this critical value in Excel, make use of the following function:
NORM.INV(probability, mean, standard_dev)
- probability: A probability corresponding to the normal distribution.
- mean: The mean of the distribution.
- standard_dev: The standard deviation of the distribution.
Here, we have . Thus, to calculate such that , run the following command:
This gives:
With this information, the margin of error can be calculated:
The margin of error of a confidence interval for the population proportion is calculated with the following formula:
Calculate the sample proportion :
Investigate whether the sampling distribution of the sample proportion may be considered approximately normal:
Since both conditions are satisfied, the sampling distribution of the sample proportion is approximately normally distributed with parameters and .
However, because the population proportion is unknown, the standard error of the proportion cannot be calculated.
Instead, we will use the sample proportion to calculate the estimated standard error of the proportion :
For a given confidence level , the critical value of the standard normal distribution is the value such that .
To calculate this critical value in R, make use of the following function:
qnorm(p, mean, sd, lower.tail)
- p: A probability corresponding to the normal distribution.
- mean: The mean of the distribution.
- sd: The standard deviation of the distribution.
- lower.tail: If TRUE (default), probabilities are , otherwise, .
Here, we have . Thus, to calculate such that , run the following command:
This gives:
With this information, the margin of error can be calculated:
General Formula for a Confidence Interval for the Population Proportion
Assuming the sampling distribution of the sample proportion is (approximately) normal, the general formula for computing a for the population proportion , based on a random sample of size , is:
Of these, cultures showed some resistance to penicillin.
Construct a confidence interval for the proportion of strep cultures among all Florida patients that are penicillin-resistant. Round your answers to decimal places.
There are a number of different ways we can compute the confidence interval. Click on one of the panels to toggle a specific solution.
Calculate the sample proportion :
Investigate whether the sampling distribution of the sample proportion may be considered approximately normal:
Since both conditions are satisfied, the sampling distribution of the sample proportion is approximately normal.
Assuming the sampling distribution of the sample proportion is (approximately) normal, the general formula for computing a for the population proportion , based on a random sample of size , is:
For a given confidence level , the critical value of the standard normal distribution is the value such that .
To calculate this critical value in Excel, make use of the following function:
NORM.INV(probability, mean, standard_dev)
- probability: A probability corresponding to the normal distribution.
- mean: The mean of the distribution.
- standard_dev: The standard deviation of the distribution.
Here, we have . Thus, to calculate such that , run the following command:
This gives:
Calculate the lower bound of the confidence interval:
Calculate the lower bound of the confidence interval:
Thus, the confidence interval for the population proportion is:
Calculate the sample proportion :
Investigate whether the sampling distribution of the sample proportion may be considered approximately normal:
Since both conditions are satisfied, the sampling distribution of the sample proportion is approximately normal.
Assuming the sampling distribution of the sample proportion is (approximately) normal, the general formula for computing a for the population proportion , based on a random sample of size , is:
For a given confidence level , the critical value of the standard normal distribution is the value such that .
To calculate this critical value in R, make use of the following function:
qnorm(p, mean, sd, lower.tail)
- p: A probability corresponding to the normal distribution.
- mean: The mean of the distribution.
- sd: The standard deviation of the distribution.
- lower.tail: If TRUE (default), probabilities are , otherwise, .
Here, we have . Thus, to calculate such that , run the following command:
This gives:
Calculate the lower bound of the confidence interval:
Calculate the lower bound of the confidence interval:
Thus, the confidence interval for the population proportion is:
Controlling the Margin of Error
Suppose you would like the margin of error for a confidence interval for the population proportion to be no larger than .
Then the minimum sample size required is
rounded up to the next whole number.
If the researcher wants the margin of error of the confidence interval for the population proportion to be no larger than , what is the minimum sample size she needs to achieve this goal?
There are a number of different ways we can calculate the minimum sample size. Click on one of the panels to toggle a specific solution.
For a given confidence level , the critical value of the standard normal distribution is the value such that .
To calculate this critical value in Excel, make use of the following function:
NORM.INV(probability, mean, standard_dev)
- probability: A probability corresponding to the normal distribution.
- mean: The mean of the distribution.
- standard_dev: The standard deviation of the distribution.
Here, we have . Thus, to calculate such that , run the following command:
This gives:
With this information, the minimum sample size can be calculated:
Rounding this value up gives .
Thus, for the margin of error to be no larger than , you need a sample size of at least .
For a given confidence level , the critical value of the standard normal distribution is the value such that .
To calculate this critical value in R, make use of the following function:
qnorm(p, mean, sd, lower.tail)
- p: A probability corresponding to the normal distribution.
- mean: The mean of the distribution.
- sd: The standard deviation of the distribution.
- lower.tail: If TRUE (default), probabilities are , otherwise, .
Here, we have . Thus, to calculate such that , run the following command:
This gives:
With this information, the minimum sample size can be calculated:
Rounding this value up gives .
Thus, for the margin of error to be no larger than , you need a sample size of at least .