Question 15.1: Refer to the GPA’s in Example 22 of Chapter 1, where we assu......
Refer to the GPA’s in Example 22 of Chapter 1, where we assume that the given GPA scores are observed values of r.v.’s X_{i},i=1,\ldots, 34 with (unknown) mean μ and (unknown) variance \sigma^{2}, both finite. Construct a confidence interval for μ with confidence coefficient approximately 95%.
In the discussion of Example 1 in Chapter 13, we saw that: \sum\limits_{i}x_{i}=100.73\operatorname{and}\sum_{i}x_{i}^{2}=304.7885, so that:
\bar{x}=\frac{100.73}{34}\simeq2.963,\quad s_{n}^{2}=\frac{304.7885}{34}-\frac{(100.73)^{2}}{34^{2}}\simeq0.187,\quad\mathrm{and}\quad s_{n}\simeq0.432.
Since z_{0.025}=1.96, formula (5) gives:
\left[{\bar{X}}_{n}-z_{\frac{\alpha}{2}}{\frac{S_{n}}{\sqrt{n}}},\quad{\bar{X}}_{n}+z_{\frac{\alpha}{2}}{\frac{S_{n}}{\sqrt{n}}}\right] (5)