Confidence Intervals for Sampling Distributions

Suppose we take a sample of size n from a normal population N(μ, σ) and ask whether the sample mean differs significantly from the overall population mean. As we have seen in Single Sample Hypothesis Testing

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The exact point of rejection (at the right tail), zcrit, has value

Critical value right

And so
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In the two-tailed case, we have two critical values: +zcrit and –zcrit, and so we have

image420

Definition 1: The confidence interval is the interval

Confidence interval

i.e. the interval
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where zcrit is the critical value of z on the right tail, i.e. where zcrit > 0.

The term zcrit ∙ std err is the called the margin of error. This term depends on the value of zcrit which in turn depends on the value of α, and so we refer to this interval as the 1 – α % confidence interval, since we are 1 – α % confident that the population mean will occur in this interval.

Example 1: Find the confidence interval for Example 2 (two-tailed case) of Single Sample Hypothesis Testing.

As we have seen, the mean of the sample is 75 and the standard error is 2.58 (based on a sample of size 60). From Definition 1, the confidence interval is given by

image420

And so we consider the 95% confidence interval to be (75 – 5.06, 75 + 5.06) = (69.94, 80.06).

If the null hypothesis is true, we are 95% confident that the population mean will be in this interval. Since indeed the population mean, 80, is within the interval, we retain the null hypothesis.

Example 2: Find the confidence interval for Example 2 (two-tailed case) of Single Sample Hypothesis Testing.

This time the sample size is larger (n = 100) and so the standard error is smaller (s.e. = 2.0) and so the confidence interval is also smaller:

Thus, the 95% confidence interval is (75 – 3.92, 75 + 3.92) = (71.08, 78.92). Since the population mean, 80, is outside the interval, we reject the null hypothesis.

3 Responses to Confidence Intervals for Sampling Distributions

  1. hector loyola says:

    outstanding website….

  2. Afs says:

    Charles,
    Thank you for your outstanding work. I am have question for example 1.
    Isn’t the Zcri for 95% 1.64? Why it is considered 1.94?

    • Charles says:

      You need to use the two-tailed version of the critical value = NORMSINV(.975) = 1.96.
      The number 1.94 is actually a typo and should be 1.96 (I have now corrected this on the referenced webpage).
      Charles

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