A sufficiently large independent random sample can support the H2 approximate mean test when the population variance is unknown. Use an unbiased estimate from the relevant sample and describe the test as approximate. Calculate \(s^2=\frac{1}{n-1}\sum(x_i-\overline{x})^2\). The estimated standard error is \(\dfrac{s}{\sqrt{n}}\), where \(s=\sqrt{s^2}\). Under \(H_0:\mu=\mu_0\), use \[Z=\frac{\overline{x}-\mu_0}{\frac{s}{\sqrt{n}}}\] with the large-sample normal approximation. Bridge the idea For \(n=100\) and an unbiased sample variance \(s^2=81\), the estimated standard error is \(\dfrac{9}{10}=0.9\). The value 81 is not a known population variance and the standard error is not \(\dfrac{81}{10}\). Needed condition Reason Sufficiently large sample Supports the approximate normal mean model and the large-sample use of estimated variance. Independent random sample from the relevant population Repeated or biased observations do not become valid simply by increasing their count. Current population and conditions After an intervention, do not automatically reuse an old variance estimate for the new process. Exam wording: “No population-normality assumption is required for the sufficiently large independent random sample. Use the unbiased estimate of the population variance in the large-sample approximate mean test.” For a small sample, do not justify this substitution by normality alone.

Large Samples with Estimated Population Variance

Junior College 2

TGM Original Questions

A sufficiently large independent random sample can support the H2 approximate mean test when the population variance is unknown. Use an unbiased estimate from the relevant sample and describe the test as approximate.

Calculate \(s^2=\frac{1}{n-1}\sum(x_i-\overline{x})^2\). The estimated standard error is \(\dfrac{s}{\sqrt{n}}\), where \(s=\sqrt{s^2}\). Under \(H_0:\mu=\mu_0\), use \[Z=\frac{\overline{x}-\mu_0}{\frac{s}{\sqrt{n}}}\] with the large-sample normal approximation.

Bridge the idea

For \(n=100\) and an unbiased sample variance \(s^2=81\), the estimated standard error is \(\dfrac{9}{10}=0.9\). The value 81 is not a known population variance and the standard error is not \(\dfrac{81}{10}\).

Needed condition	Reason
Sufficiently large sample	Supports the approximate normal mean model and the large-sample use of estimated variance.
Independent random sample from the relevant population	Repeated or biased observations do not become valid simply by increasing their count.
Current population and conditions	After an intervention, do not automatically reuse an old variance estimate for the new process.

Exam wording: “No population-normality assumption is required for the sufficiently large independent random sample. Use the unbiased estimate of the population variance in the large-sample approximate mean test.” For a small sample, do not justify this substitution by normality alone.

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