You live in City A, and your friend lives in City B. Your friend believes that his city has significantly more sunny days each year than your city.

Question

What are the null hypothesis and alternative hypothesis your friend would use to test his claim? p_1 refers to City A, and p_2 refers to City B.

A) null: p_2 - p_1 > 0; alternative: p_2 - p_1 < 0
B) null: p_1 - p_2 ≠ 0; alternative: p_2 - p_1 ≠ 0
C) null: p_1 - p_2 > 0; alternative: p_1 - p_2 < 0
D) null: p_2 - p_1 = 0; alternative: p_2 - p_1 > 0

Answer 1

To determine the appropriate null and alternative hypotheses given your friend's belief that his city (City B) has significantly more sunny days each year than your city (City A), we need to clarify the meanings of the variables:

• Let \( p_1 \) represent the proportion of sunny days in City A.
• Let \( p_2 \) represent the proportion of sunny days in City B.

Your friend believes that City B has significantly more sunny days, which suggests that \( p_2 \) is greater than \( p_1 \).

In hypothesis testing:

• The null hypothesis typically represents no effect or no difference.
• The alternative hypothesis represents the effect or difference that we are testing for.

Given this information, the correct hypotheses would be:

• Null hypothesis (\( H_0 \)): The proportions are equal (no significant difference in sunny days), which can be expressed as \( p_2 - p_1 = 0 \).
• Alternative hypothesis (\( H_a \)): City B has significantly more sunny days than City A, expressed as \( p_2 - p_1 > 0 \).

Thus, the correct choice is:

D) null: \( p_2 - p_1 = 0 \); alternative: \( p_2 - p_1 > 0 \).