Research suggests that the pressure of being timed may interfere with performance on tests that involve mathematical problems. A fictional study was conducted with 30 6th graders. First, the 6th graders were given a math test that contained 50 problems and were told that they had only one hour to complete it (Timed Condition). The same 6th graders were later given a math test that contained 50 problems and were told that they could have as much time, as needed, to complete the test (Unlimited Time Condition). The total number of correct answers for each 6th grader was then calculated for each condition. Then, for each student, the difference between the two scores (Timed-Untimed) was calculated. The researchers hypothesized that the 6th graders would get fewer correct answers, when they took the test with a time limit, as compared to when they had unlimited time.

Tramo MJ, Loftus WC, Green RL, Stukel TA, Weaver JB, Gazzaniga MS. Brain Size, Head Size, and IQ in Monozygotic Twins. Neurology 1998; 50:1246-1252.

1. If μ₁ and μ₂ represent the number of correct answers during the Timed Condition and the Unlimited Time Condition, respectively, and let μd be the mean of the differences in the number of correct answers (Timed-Untimed) of all 6th graders. Which are the appropriate null and alternative hypotheses?
   
   
   • (A)

     H₀: μ_d = 0

     H_a: μ_d > 0

   • (B)

     H₀: μ_d = 0

     H_a: μ_d < 0

   • (C)

     H₀: μ_d < 0

     H_a: μ_d = 0

   • (D)

     H₀: μ₁- μ₂ = 0

     H_a: μ₁ - μ₂ < 0

   • Both (B) and (D) are correct.

2. The researchers analyzed the data and obtained the following output:
   
   
   

   • The data provide sufficient evidence to reject H₀. Thus, the researchers conclude that 6^th grade students get, on average, fewer correct answers and, thus, lower scores on math tests when taking them under timed conditions as compared to untimed conditions.
   • The data provide sufficient evidence to reject H₀. Thus, based on the data the researchers cannot conclude that mean number of correct  answers on the math test for the Timed Condition is lower compared to the Untimed Condition.
   • The data do not provide sufficient evidence to reject H₀.  Thus, the researchers conclude that 6^th grade students get, on average, fewer correct answers and, thus, lower scores, on math tests, when taking them under timed conditions as compared to untimed conditions.
   • The data do not provide sufficient evidence to reject H₀. Thus, based on the data the researchers cannot conclude that mean number of correct  answers on the math test for the Timed Condition is lower compared to the Untimed Condition.