Confidence Interval

Basic Concepts

The hypothesis testing procedure described in Null and Alternative Hypothesis simply determines whether the null hypothesis should be rejected or not. Often we would like additional information.

For example, suppose the null hypothesis is that the population mean has the fixed value μ0, i.e. the null hypothesis is H0: μ = μ0. Given any sample, we would like to use the data in the sample to calculate an interval (called a confidence interval) corresponding to that sample such that 95% of such samples will produce a confidence interval that contains the population mean μ (where α = .05, and so 95% = 1 – α); i.e. we seek an interval for which we are 95% confident that a < μ < b. Here a and b are the endpoints of the confidence interval for that sample.

Caution

Note that this does not mean that there is a 95% probability that the interval contains the population mean. Instead, it means that the population mean will lie in the calculated confidence interval in 95% of the samples (assuming that we generate a large number of samples).

Observations

Furthermore, there is a link between hypothesis testing and the construction of a confidence interval. Namely, if a < μ0 < b, then we can’t reject the null hypothesis, while if μ0 ≥ b or μ0 ≤ a, then we can reject the null hypothesis.

We will show how to calculate such confidence intervals in several places on this website (see for example Confidence Interval for Sampling Distributions or Confidence Interval for ANOVA).

One-tailed Confidence Interval

In addition to the two-tailed confidence intervals described above, we can define a one-tailed confidence interval similarly. E.g. for the null hypothesis H0: μ ≤ μ0, a 95% confidence interval takes the form (a, ∞), while for the null hypothesis H0: μ ≥ μ0, a 95% confidence interval takes the form (-∞, b).

Reference

Howell, D. C. (2010) Statistical methods for psychology, 7th Ed. Wadsworth. Cengage Learning
https://labs.la.utexas.edu/gilden/files/2016/05/Statistics-Text.pdf

4 thoughts on “Confidence Interval”

  1. I need help please on how to solve the following:

    Deep-breathing exercise Progressive muscle relaxation Suggestive hypnosis
    8 10 6
    6 10 7
    7 8 8
    9 9 7
    8 9 7
    8 9 6
    7 8 5
    6 10 6
    5 9 8
    6 6 7
    7 10 7
    8 9 4
    10 10 5
    8 10 6
    9 9 6
    6 8 7
    8 8 6
    7 8 5
    9 9 8
    8 8 4
    7 10 5
    6 10 6
    8 8 7
    10 9 6
    7 9 5

    Conduct a hypothesis testing if you were comparing deep breathing exercise and progressive muscle relaxation. Show the step-by-step process of hypothesis testing:
    Null and alternative hypotheses
    Level of significance
    Test-statistic
    Critical region
    Computation of test statistic and p value
    Decision

    Conduct a hypothesis testing if you were comparing the three interventions altogether: deep breathing exercise, progressive muscle relaxation, suggestive hypnosis. Show the step-by-step process of hypothesis testing
    • Null and alternative hypotheses
    • Level of significance
    • Test-statistic
    • Critical region
    • Computation of test statistic and p value
    • Decision
    • Conclusion

    thank you

    Reply

  2. I need help to complete the following portion of an assignment:
    i)Test the hypotheses that the average household income in the township is greater than $100,000.
    (ii) Construct a 95% CI for the proportion of households with family history of heart disease, separately for each race. Would you say the proportion of households with history of heart disease differ by race?
    (i) Formulate the null and alternative hypotheses you would use to for the test this claim.
    (ii) Conduct the test and appropriately accept or reject your null hypotheses using ANOVA.
    (iii) Is there any statistically significant difference in the average household income based on race?
    Thanks,
    Ryan

    Reply

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