stats la2

Confidence Interval (CI)

INTRODUCTION

In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated confidence level; the 95% confidence level is most common, but other levels, such as 90% or 99%, are sometimes used.^[1][2] The confidence level represents the long-run proportion of corresponding CIs that contain the true value of the parameter. For example, out of all intervals computed at the 95% level, 95% of them should contain the parameter’s true value.^[3]

Example

Suppose a student measuring the boiling temperature of a certain liquid observes the readings (in degrees Celsius) 102.5, 101.7, 103.1, 100.9, 100.5, and 102.2 on 6 different samples of the liquid. He calculates the sample mean to be 101.82. If he knows that the standard deviation for this procedure is 1.2 degrees, what is the confidence interval for the population mean at a 95% confidence level?

In other words, the student wishes to estimate the true mean boiling temperature of the liquid using the results of his measurements. If the measurements follow a normal distribution, then the sample mean will have the distribution N(,). Since the sample size is 6, the standard deviation of the sample mean is equal to 1.2/sqrt(6) = 0.49.

The selection of a confidence level for an interval determines the probability that the confidence interval produced will contain the true parameter value. Common choices for the confidence level C are 0.90, 0.95, and 0.99. These levels correspond to percentages of the area of the normal density curve. For example, a 95% confidence interval covers 95% of the normal curve -- the probability of observing a value outside of this area is less than 0.05. Because the normal curve is symmetric, half of the area is in the left tail of the curve, and the other half of the area is in the right tail of the curve. As shown in the diagram to the right, for a confidence interval with level C, the area in each tail of the curve is equal to (1-C)/2. For a 95% confidence interval, the area in each tail is equal to 0.05/2 = 0.025.

The value z^* representing the point on the standard normal density curve such that the probability of observing a value greater than z^* is equal to p is known as the upper p critical value of the standard normal distribution. For example, if p = 0.025, the value z^* such that P(Z > z^*) = 0.025, or P(Z < z^*) = 0.975, is equal to 1.96. For a confidence interval with level C, the value p is equal to (1-C)/2. A 95% confidence interval for the standard normal distribution, then, is the interval (-1.96, 1.96), since 95% of the area under the curve falls within this interval.

Confidence Intervals for Unknown Mean and Known Standard Deviation

For a population with unknown mean and known standard deviation , a confidence interval for the population mean, based on a simple random sample (SRS) of size n, is +z^*, where z^* is the upper (1-C)/2 critical value for the standard normal distribution.

Note: This interval is only exact when the population distribution is normal. For large samples from other population distributions, the interval is approximately correct by the Central Limit Theorem.

In the example above, the student calculated the sample mean of the boiling temperatures to be 101.82, with standard deviation 0.49. The critical value for a 95% confidence interval is 1.96, where (1-0.95)/2 = 0.025. A 95% confidence interval for the unknown mean is ((101.82 - (1.96*0.49)), (101.82 + (1.96*0.49))) = (101.82 - 0.96, 101.82 + 0.96) = (100.86, 102.78).

As the level of confidence decreases, the size of the corresponding interval will decrease. Suppose the student was interested in a 90% confidence interval for the boiling temperature. In this case, C = 0.90, and (1-C)/2 = 0.05. The critical value z^* for this level is equal to 1.645, so the 90% confidence interval is ((101.82 - (1.645*0.49)), (101.82 + (1.645*0.49))) = (101.82 - 0.81, 101.82 + 0.81) = (101.01, 102.63)

Confidence Intervals for Unknown Mean and Unknown Standard Deviation

In most practical research, the standard deviation for the population of interest is not known. In this case, the standard deviation is replaced by the estimated standard deviation s, also known as the standard error. Since the standard error is an estimate for the true value of the standard deviation, the distribution of the sample mean is no longer normal with mean and standard deviation . Instead, the sample mean follows the t distribution with mean and standard deviation . The t distribution is also described by its degrees of freedom. For a sample of size n, the t distribution will have n-1 degrees of freedom. The notation for a t distribution with k degrees of freedom is t(k). As the sample size n increases, the t distribution becomes closer to the normal distribution, since the standard error approaches the true standard deviation for large n.

For a population with unknown mean and unknown standard deviation, a confidence interval for the population mean, based on a simple random sample (SRS) of size n, is +t^*, where t^* is the upper (1-C)/2 critical value for the t distribution with n-1 degrees of freedom, t(n-1).

Confidence Intervals for Unknown Mean and Unknown Standard Deviation

Real life Example

The dataset “Normal Body Temperature, Gender, and Heart Rate” contains 130 observations of body temperature, along with the gender of each individual and his or her heart rate. Using the MINITAB “DESCRIBE” command provides the following information:

Descriptive Statistics

Variable        N     Mean   Median  Tr Mean    StDev  SE Mean
TEMP            130   98.249   98.300   98.253    0.733    0.064

Variable      Min      Max       Q1       Q3
TEMP        96.300  100.800   97.800   98.700

To find a 95% confidence interval for the mean based on the sample mean 98.249 and sample standard deviation 0.733, first find the 0.025 critical value t^* for 129 degrees of freedom. This value is approximately 1.962, the critical value for 100 degrees of freedom (found in Table E in Moore and McCabe). The estimated standard deviation for the sample mean is 0.733/sqrt(130) = 0.064, the value provided in the SE MEAN column of the MINITAB descriptive statistics. A 95% confidence interval, then, is approximately ((98.249 - 1.962*0.064), (98.249 + 1.962*0.064)) = (98.249 - 0.126, 98.249+ 0.126) = (98.123, 98.375).

For a more precise (and more simply achieved) result, the MINITAB “TINTERVAL” command, written as follows, gives an exact 95% confidence interval for 129 degrees of freedom:

MTB > tinterval 95 c1

Confidence Intervals

Variable     N      Mean    StDev  SE Mean       95.0 % CI
TEMP        130   98.2492   0.7332   0.0643  ( 98.1220, 98.3765)

According to these results, the usual assumed normal body temperature of 98.6 degrees Fahrenheit is not within a 95% confidence interval for the mean.

Conclusion

This document focused on the formulas for estimating different unknown population parameters. In each application, a random sample or two independent random samples were selected from the target population and sample statistics (e.g., sample sizes, means, and standard deviations or sample sizes and proportions) were generated. Point estimates are the best single-valued estimates of an unknown population parameter. Because these can vary from sample to sample, most investigations start with a point estimate and build in a margin of error. The margin of error quantifies sampling variability and includes a value from the Z or t distribution reflecting the selected confidence level as well as the standard error of the point estimate. It is important to remember that the confidence interval contains a range of likely values for the unknown population parameter; a range of values for the population parameter consistent with the data. It is also possible, although the likelihood is small, that the confidence interval does not contain the true population parameter. This is important to remember in interpreting intervals. Confidence intervals are also very useful for comparing means or proportions and can be used to assess whether there is a statistically meaningful difference. This is based on whether the confidence interval includes the null value (e.g., 0 for the difference in means, mean difference and risk difference or 1 for the relative risk and odds ratio).

References

https://www.khanacademy.org/math/statistics-probability/confidence-intervals-one-sample

https://www.simplilearn.com/tutorials/data-analytics-tutorial/confidence-intervals-in-statistics