Question

Designs for comparing means using College Board data on SAT scores. Please write the null and alternative hypotheses such that it is clear we are testing which of the following: 1) a single mean versus a claim in null hypothesis, 2) Means from two independent samples, or 3) Means from matched pairs. Also state the degrees of freedom associated with the t-test. (2 points for correct hypotheses, 1 point for df)

a.) The college board aims to achieve a mean of 500 on the Math section of the SAT each year. A researcher randomly selects 50 Math SAT scores from Fort Collins college-bound seniors to see if the Fort Collins mean is different than 500.

b.) A researcher randomly selects 23 male Math SAT scores and 29 female Math SAT scores from Fort Collins college-bound seniors to see if their means are different.

c.) A researcher randomly selects 26 Math SAT scores from one Fort Collins high-school and 21 scores from another Fort Collins high-school and tests for a difference between means.

d.) A researcher randomly selects 25 SAT scores and tests for a difference between the mean of the Math SAT and the mean of the Critical Reading SAT score.

e.) A researcher randomly selects 20 high-schools across Colorado and then randomly selects 2 male Math SAT scores from each school to test for a difference in the overall mean with the national average for males.

Answer 1

Solution;-

a)

Test

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u = 500
Alternative hypothesis: u 500(Claim)

Note that these hypotheses constitute a two-tailed test.

D.F = n - 1

D.F = 49

b)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u₁ = u ₂
Alternative hypothesis: u₁ u ₂ (Claim)

Note that these hypotheses constitute a two-tailed test.

D.F = n₁ + n₂ - 2

D.F = 23 + 29 - 2

D.F = 50

c)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u₁ = u ₂
Alternative hypothesis: u₁ u ₂

Note that these hypotheses constitute a two-tailed test.

D.F = n₁ + n₂ - 2

D.F = 26 + 21 - 2

D.F = 45

d) matched pair test

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u_d = 0

Alternative hypothesis: u_d ≠ 0

Note that these hypotheses constitute a two-tailed test.

D.F = n - 1

D.F = 25 - 1

D.F = 24