confidence level

(noun)

The probability that a measured quantity will fall within a given confidence interval.

Examples of confidence level in the following topics:

  • Level of Confidence

    • The proportion of confidence intervals that contain the true value of a parameter will match the confidence level.
    • The desired level of confidence is set by the researcher (not determined by data).
    • If a corresponding hypothesis test is performed, the confidence level is the complement of respective level of significance (i.e., a 95% confidence interval reflects a significance level of 0.05).
    • In applied practice, confidence intervals are typically stated at the 95% confidence level.
    • However, when presented graphically, confidence intervals can be shown at several confidence levels (for example, 50%, 95% and 99%).

  • Changing the Confidence Level or Sample Size

    • Suppose we change the original problem by using a 95% confidence level.
    • σ = 3 ; n = 36 ; The confidence level is 95% (CL=0.95)
    • Use the original 90% confidence level.
    • x = 68• σ = 3 ; The confidence level is 90% (CL=0.90) ; = z = 1.645
    • σ = 3 ; The confidence level is 90% (CL=0.90) ; = z .05 = 1.645

  • Changing the confidence level

    • Suppose we want to consider confidence intervals where the confidence level is somewhat higher than 95%: perhaps we would like a confidence level of 99%.
    • The 95% confidence interval structure provides guidance in how to make intervals with new confidence levels.
    • The choice of 1.96 corresponds to a 95% confidence level.
    • point estimate ± z* SE (where z* corresponds to the confidence level selected)
    • Figure 4.10 provides a picture of how to identify z* based on a confidence level.We select z* so that the area between -z* and z* in the normal model corresponds to the confidence level.

  • Interpreting confidence intervals

    • A careful eye might have observed the somewhat awkward language used to describe confidence intervals.
    • Incorrect language might try to describe the confidence interval as capturing the population parameter with a certain probability.
    • This is one of the most common errors: while it might be useful to think of it as a probability, the confidence level only quantifies how plausible it is that the parameter is in the interval.
    • Another especially important consideration of confidence intervals is that they only try to capture the population parameter.
    • Confidence intervals only attempt to capture population parameters.

  • Calculating the Confidence Interval

    • The margin of error depends on the confidence level (abbreviated CL).
    • The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter.
    • If the confidence level (CL) is 95%, then we say that "We estimate with 95% confidence that the true value of the population mean is between 4.5 and 9.5.
    • The confidence level, CL, is the area in the middle of the standard normal distribution.
    • σ = 3 ; n = 36 ; The confidence level is 90% (CL=0.90)

  • Lab 2: Confidence Interval (Place of Birth)

    • Calculate the confidence interval and the error bound. i.
    • Confidence Interval: ii.
    • Using the above information, construct a confidence interval for each given confidence level given.
    • What happens to the EBP as the confidence level increases?
    • Does the width of the confidence interval increase or decrease?

  • Lab 1: Confidence Interval (Home Costs)

    • Calculate the confidence interval and the error bound. i.
    • Confidence Interval: ii.
    • Using the above information, construct a confidence interval for each confidence level given.
    • What happens to the EBM as the confidence level increases?
    • Does the width of the confidence interval increase or decrease?

  • Hypothesis Tests or Confidence Intervals?

    • What is the difference between hypothesis testing and confidence intervals?
    • Confidence intervals are closely related to statistical significance testing.
    • More generally, given the availability of a hypothesis testing procedure that can test the null hypothesis $\theta = \theta_0$ against the alternative that $\theta \neq \theta_0$ for any value of $\theta_0$, then a confidence interval with confidence level $\gamma = 1-\alpha$ can be defined as containing any number $\theta_0$ for which the corresponding null hypothesis is not rejected at significance level $\alpha$.
    • While not all confidence intervals are constructed in this way, one general purpose approach is to define a $100(1-\alpha)$% confidence interval to consist of all those values $\theta_0$ for which a test of the hypothesis $\theta = \theta_0$ is not rejected at a significance level of $100 \alpha$%.
    • For the same reason, the confidence level is not the same as the complementary probability of the level of significance.

  • Practice 2: Confidence Intervals for Means, Unknown Population Standard Deviation

    • The student will calculate confidence intervals for means when the population standard deviation is unknown.
    • We are interested in finding a confidence interval for the true mean number of colors on a national flag.
    • Construct a 95% Confidence Interval for the true mean number of colors on national flags.
    • Using the same $\bar{x}$ , sx , and level of confidence, suppose that n were 69 instead of 39.
    • Using the same $\bar{x}$ , sx , and n = 39, how would the error bound change if the confidence level were reduced to 90%?

  • Lab 3: Confidence Interval (Womens' Heights)

    • The student will calculate a 90% confidence interval using the given data.
    • The student will determine the relationship between the confidence level and the percent of constructed intervals that contain the population mean.
    • Now write your confidence interval on the board.
    • Suppose we had generated 100 confidence intervals.
    • When we construct a 90% confidence interval, we say that we are 90% confident that the true population mean lies within the confidence interval.