Let's go through each part to confirm and understand the answers:

Part (a)

• Hypotheses: $H_0: p = 0.3$ vs. $H_A: p \neq 0.3$
• Alpha level ($\alpha$) = 0.05
• Test type: Two-tailed

For a two-tailed test at $\alpha = 0.05$, the critical z-values are at $\pm 1.96$.

• Answer: $z^* = \pm 1.96$

Part (b)

• Hypotheses: $H_0: p = 0.5$ vs. $H_A: p > 0.5$
• Alpha level ($\alpha$) = 0.01
• Test type: One-tailed (right-tailed)

For a right-tailed test at $\alpha = 0.01$, the critical z-value is $2.33$.

• Answer: $z^* = 2.33$

Part (c)

• Hypotheses: $H_0: \mu = 40$ vs. $H_A: \mu \neq 40$
• Alpha level ($\alpha$) = 0.01
• Sample size ($n$) = 48
• Test type: Two-tailed, using the $t$-distribution since the sample size is less than 50

With $\alpha = 0.01$ for a two-tailed test and $n - 1 = 47$ degrees of freedom, the critical $t$-value is approximately $\pm 2.68$.

• Answer: $t^* = \pm 2.68$

Part (d)

• Hypotheses: $H_0: p = 0.3$ vs. $H_A: p > 0.3$
• Alpha level ($\alpha$) = 0.10
• Sample size ($n$) = 330
• Test type: One-tailed (right-tailed), using the $z$-distribution

For a right-tailed test at $\alpha = 0.10$, the critical z-value is $1.28$.

• Answer: $z^* = 1.28$

Part (e)

• Hypotheses: $H_0: \mu = 50$ vs. $H_A: \mu < 50$
• Alpha level ($\alpha$) = 0.01
• Sample size ($n$) = 1000
• Test type: One-tailed (left-tailed), using the $z$-distribution due to large sample size

For a left-tailed test at $\alpha = 0.01$, the critical z-value is $-2.33$.

• Answer: $z^* = -2.33$

Summary of Answers

• (a) $z^* = \pm 1.96$
• (b) $z^* = 2.33$
• (c) $t^* = \pm 2.68$
• (d) $z^* = 1.28$
• (e) $z^* = -2.33$

Would you like further details on the calculations or concepts involved in finding these critical values?

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1. How is the choice between $z$- and $t$-values made in hypothesis testing?
2. What determines whether a test is one-tailed or two-tailed?
3. How do critical values change with different significance levels?
4. Why is the sample size relevant when deciding between $z$- and $t$-values?
5. What is the effect of increasing the sample size on the $t$-distribution?

Tip: In hypothesis testing, always clarify the direction of the alternative hypothesis, as it affects whether a test is one-tailed or two-tailed.