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The mean age of Senators in the 109th Congress was 60.35 years. A random sample of 40 senators from various state senates had an average age of 55.4 years, and the population standard deviation is 6.5 years. At

= 0.05, is there sufficient evidence that state senators are on average younger than the Senators in Washington?

Which of the following are the right hypotheses?

a)

b)

c)

d)

e)

f)

1 answer

To determine if there is sufficient evidence that state senators are on average younger than the Senators in Washington, we need to formulate our null and alternative hypotheses.

Let's denote:

  • \( \mu_s \) = mean age of state senators
  • \( \mu_w \) = mean age of Senators in Washington (109th Congress = 60.35 years)

The correct hypotheses to test whether state senators are younger than Washington senators would be:

  • Null Hypothesis (\( H_0 \)): \( \mu_s \geq 60.35 \) (The mean age of state senators is greater than or equal to the mean age of Washington senators.)
  • Alternative Hypothesis (\( H_a \)): \( \mu_s < 60.35 \) (The mean age of state senators is less than the mean age of Washington senators.)

In terms of symbols, we can write:

a) Null Hypothesis (\( H_0 \)): \( \mu_s \geq 60.35 \)
b) Alternative Hypothesis (\( H_a \)): \( \mu_s < 60.35 \)

To summarize, the appropriate hypotheses are:

a) \( H_0: \mu_s \geq 60.35 \)
b) \( H_a: \mu_s < 60.35 \)

You would need to check through the options (c, d, e, f) you have for the correct matching hypotheses. However, the essential formulation would be as stated above.