Question

If a null hypothesis is rejected at a significance level of 1%, then we should say that it was rejected at 1%. Reporting that the null was also rejected at the 5% level of significance is unnecessary and unwise.

True False

The p-value equals alpha, the level of significance of the hypothesis test.

True False

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
Let X₁, X₂, X₃, and X₄ be a random sample of observations from a population with mean μ and variance σ². The observations are independent because they were randomly drawn. Consider the following two point estimators of the population mean μ:

₁ = 0.10 X₁ + 0.40 X₂ + 0.40 X₃ + 0.10 X₄ and
₂ = 0.20 X₁ + 0.30 X₂ + 0.30 X₃ + 0.20 X₄

Which of the following statements is true?

HINT: Use the definition of an unbiased estimator and the expected value rules to figure this question out. See example in Review notes.

• A. Both ₁ and ₂ are unbiased estimators of μ.
• B. Both ₁ and ₂ are biased estimators of μ.
• C. ₁ is unbiased, but ₂ is a biased estimator of μ.
• D. ₁ is biased, but ₂ is an unbiased estimator of μ.
• Let X₁, X₂, X₃, and X₄ be a random sample of observations from a population with mean μ and variance σ². Consider the following estimator of μ: ₁ = 0.15 X₁ + 0.35 X₂ + 0.20 X₃ + 0.30 X₄. Using the linear combination of random variables rule and the fact that X₁, ..., X₄ are independently drawn from the population, calculate the variance of ₁?
• A. 0.20 σ²
• B. 0.55 σ²
• C. 0.275 σ²
• D. 0.125 σ²

Answer 1

1. If you reject the null hypothesis at 1 per cent significance level that means the experimental results are 99 per cent certain/confident of the rejection of null hypothesis or the rejection is supported with 99 per cent certainty. With this logic, we need not test for 5 per cent confidence level because it will also be rejected with 95 per cent certainty as it has already been rejected with higher (99 per cent) certainty. Thus, the statement is TRUE.

2. p-value is used to test the significance of the hypothesis. If p-value is equal or less than the alpha or the level of significance we conclude that the null hypothesis is being rejected. But if p-value turns out to be greater than alpha, we fail to reject the null hypothesis. Thus, p-value is not equal to alpha in all cases and the statement is thus FALSE.