(Solved): (a) The tensile strength for a type of wire is normally distributed with unknown mean \( \mu \) an ...

(a) The tensile strength for a type of wire is normally distributed with unknown mean \( \mu \) and unknown variance \( \sigma^{2} \). Six pieces of wire were randomly selected from a large roll; \( Y_{i} \), the tensile strength for portion \( i \), is measured for \( i=1,2, \cdots, 6 \). The population mean \( \mu \) and variance \( \sigma^{2} \) can be estimated by \( \bar{y} \) and \( s^{2} \), respectively. Because \( \sigma_{\bar{y}}^{2}=\sigma^{2} / n \), it follows that \( \sigma_{\bar{y}}^{2} \) can be estimated by \( s^{2} / n \). Find the appropriate probability that \( \bar{y} \) will be within \( 2 s / n \) of the true population mean \( \mu \). (b) A bottling machine can be regulated so that it discharges an average of \( \mu \) ounces per bottle. It has been observed that the amount of fill dispensed by the machine is normally distributed with \( \sigma=1.0 \) ounce. Suppose that we plan to select a random sample ten bottles and measure the amount of fill in each bottle. If these ten observations are used to calculate \( s^{2} \), it might be useful to specify an interval of values that will include \( s^{2} \) with a high probability. Find numbers \( b_{1} \) and \( b_{2} \) such that \[ P\left(b_{1} \leq s^{2} \leq b_{2}\right)=0.90 \]