Chapter 7.2, Problem 24E

Applet Exercise Refer to Example 7.6. Suppose that T has a t distribution with 5 df.

1. a Use the applet Student’s t Probabilities and Quantiles to find the exact probability that T is greater than 2.
2. b Use the applet Student’s t Probabilities and Quantiles to find the exact probability that T is less than −2.
3. c Use the applet Student’s t Probabilities and Quantiles to find the exact probability that T is between −2 and 2.
4. d Your answer to part (c) is considerably less than 0.9544 = P(−2 ≤ Z ≤ 2). Refer to Figure 7.3 and explain why this is as expected.

7.6 The tensile strength for a type of wire is normally distributed with unknown mean μ and unknown variance σ². 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,…, 6. The population mean μ and variance σ² can be estimated by $\bar{Y}$ and S², respectively. Because ${\sigma }_{\bar{Y}}^2={\sigma }^2\text{/}n$, it follows that ${\sigma }_{\bar{Y}}^2$ can be estimated by S²/n. Find the approximate probability that $\bar{Y}$ will be within $2S\text{/}\sqrt{n}$ of the true population mean μ.