Expalin why we use z?/2 when computing approximate confidence intervals around d.
What will be an ideal response?
Ans: To answer this question we need to think about the effect of z?/2 versus t?/2 on the width of a confidence interval, and the effect of ?d versus sd on the width of a confidence interval. All things being equal, a confidence interval involving t?/2 will be wider than a confidence interval involving z?/2. We need the extra width provided by t?/2 to ensure that ? is captured (1–?)100% of the time. Similarly, all things being equal, a confidence interval involving sd will be wider than a confidence interval involving ?d. This makes the confidence interval wider in a way that depends on sample size. Therefore, in this approximation, the required increase in confidence interval width when estimating ? is achieved by using sd (which depends on sample size) rather than t?/2 (which also depends on sample size).
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