7.4 Two-Tail Test Notes

7.4 Two-Tail Test

Key concepts and facts from section 7.4 of the Pearson A Level Maths Applied Master One textbook, focusing on two-tail tests.

Key Facts

• Two-Tail Test Form:

  • Null Hypothesis: H_0: P =
  • Alternative Hypothesis: H_1: P \neq
  • Used to check for a change in the proportion P.

• Keywords:

  • change
  • altered
  • different

• Binomial Distribution:

  • If X ~ Bin(n, p), where n is the fixed number of trials and p is the fixed probability, and x is the observed value:
    • If x < E(X) = n \times p, find P(X \leq x).
    • If x > E(X) = n \times p, find P(X \geq x).

• Significance Level:

  • Divide the significance level by 2 for a two-tail test.
    • If P > (significance level / 2), accept H0 and reject H1.
    • If P < (significance level / 2), reject H0 and accept H1.

• Critical Region:

  • If the observed value x falls inside the critical region, reject H0 and accept H1.
  • If the observed value x does not fall inside the critical region, accept H0 and reject H1.
  • Actual Significance Level = Probability of the critical region.

Exam-Style Question 1

• Scenario: AGA's carrots have a 25% chance of being longer than 7 cm. A new fertilizer is tested on 30 carrots, and 13 are longer than 7 cm. Test at a 5% significance level whether the fertilizer has changed the proportion.

• Solution:

  1. Define the test statistic:
     • Let X be the number of carrots with a length more than 7 cm.
  2. Distribution:
     • X ~ Bin(30, 0.25)
  3. Hypotheses:
     • H_0: p = 0.25
     • H_1: p \neq 0.25
  4. Observed value vs. Expected value:
     • Observed value: x = 13
     • Expected value: E(X) = 30 \times 0.25 = 7.5
     • Since 13 > 7.5, calculate P(X \geq 13).
  5. Calculate Probability:
     • P(X \geq 13) = 1 - P(X < 13) = 1 - P(X \leq 12)
     • Using binomial CDF: 1 - 0.9784 = 0.0216
  6. Compare with Significance Level:
     • Significance level / 2 = 0.05 / 2 = 0.025
     • Since 0.0216 < 0.025, reject H0 and accept H1.
  7. Conclusion:
     • There is sufficient evidence at the 5% significance level that the new fertilizer has changed the probability of a carrot being longer than 7 cm.

• Variation: If the observed value was less than the expected value, use P(X \leq x) instead of P(X \geq x).

Exam-Style Question 2

• Scenario: 10% of customers arrive late. A new manager believes the proportion has changed. A random sample of 50 customers is taken.

• Part A: Write down the suitable null and alternative hypotheses.

  • Solution:
    • Let X be the number of patients who arrive late for their appointment.
    • X ~ Bin(50, 0.1)
    • H_0: p = 0.1
    • H_1: p \neq 0.1

• Part B: Using a 5% level of significance, find the critical region.

  • Solution:
    • Lower Critical Region (LCR): (p < 0.1)
      • Let C_1 be the lower critical value.
      • Find the largest value of C such that P(X \leq C_1) < 0.025
      • Using binomial CDF, x = 0, so C_1 = 0
      • LCR: X = 0
    • Upper Critical Region (UCR): (p > 0.1)
      • Let C_2 be the upper critical value.
      • Find the smallest value C2 such that P(X \geq C2) < 0.025
      • Rewrite: P(X \leq C_2 - 1) > 0.975
      • Using the binomial CDF table, x = 9, so C2 - 1 = 9, thus C2 = 10
      • UCR: X \geq 10
    • Critical Region: X = 0 or X \geq 10

• Part C: Find the actual significance level of the test.

  • Solution:
    • Actual Significance Level (ASL) = P(X = 0) + P(X \geq 10)
    • P(X \geq 10) = 1 - P(X < 10) = 1 - P(X \leq 9)
    • Using binomial CDF, P(X \leq 9) = 0.9755
    • Using binomial PD with N = 50, p = 0.1, x = 0, P(X = 0) = 0.0052
    • ASL = 0.0052 + (1 - 0.9755) = 0.0297 = 2.97\%

• Part D: The manager observes that 15 of the 50 customers arrive late. Comment on the manager's belief with reference to your answer in Part B.

  • Solution:
    • Observed value: 15
    • Since 15 > 10, the observed value falls inside the upper critical region.
    • Reject H0 and accept H1.
    • Conclusion: There is sufficient evidence at the 5% significance level that the proportion of patients that arrive late has indeed changed.