To find the critical values for each of these significance levels (α) using the standard normal (Z) table, we'll follow these steps:

• For a two-tailed test, divide the significance level by 2 to account for both tails.
• For a one-tailed test (right-tailed or left-tailed), we look for the area in only one tail.

a. α = 0.10, two-tailed test

• For a two-tailed test, the significance level is split between the two tails. Each tail has an area of 0.05 (since $\alpha/2 = 0.10/2 = 0.05$).

• Look for the Z-value that leaves 0.05 in each tail. This corresponds to ±1.645.

  Critical value: $\pm 1.645$

b. α = 0.01, right-tailed test

• For a right-tailed test, we are looking for the Z-value that leaves 0.01 in the right tail.

• The Z-value corresponding to an area of 0.99 to the left of it (since we need the right 0.01) is 2.33.

  Critical value: $2.33$

c. α = 0.005, left-tailed test

• For a left-tailed test, we need the Z-value that leaves 0.005 in the left tail.

• The Z-value corresponding to 0.005 in the left tail is −2.575.

  Critical value: $-2.575$

d. α = 0.01, left-tailed test

• For a left-tailed test, we need the Z-value that leaves 0.01 in the left tail.

• The Z-value corresponding to an area of 0.01 in the left tail is −2.33.

  Critical value: $-2.33$

e. α = 0.05, right-tailed test

• For a right-tailed test, we are looking for the Z-value that leaves 0.05 in the right tail.

• The Z-value corresponding to an area of 0.95 to the left of it is 1.645.

  Critical value: $1.645$

Summary:

• a. ±1.645
• b. 2.33
• c. -2.575
• d. -2.33
• e. 1.645

Would you like further explanation on any of these?

Here are 5 related questions:

1. What is the critical value for a two-tailed test with α = 0.05?
2. How do we interpret the critical values in hypothesis testing?
3. How does the area under the standard normal curve relate to significance levels?
4. What is the Z-value for a right-tailed test with α = 0.001?
5. How do we calculate a Z-score for a given data point?

Tip: In a two-tailed test, the critical region is split between both tails, so always divide α by 2 before finding the critical value.