Statistical Inference Quiz

Grade 12 · Math · Quiz

Test your understanding of hypothesis testing, confidence intervals, error types, and the Central Limit Theorem.

Questions

  1. A sample of n = 64 has x̄ = 50 and σ = 8. What is the 95% confidence interval for the population mean?

    1. (48.04, 51.96)
    2. (47.5, 52.5)
    3. (49.5, 50.5)
    4. (46.08, 53.92)
    Show answer

    Answer: (48.04, 51.96)

    CI = x̄ ± z*(σ/√n) = 50 ± 1.96(8/8) = 50 ± 1.96 = (48.04, 51.96).

  2. A researcher sets α = 0.01. How does this affect the probability of a Type I error compared to α = 0.05?

    1. Increases probability of Type I error
    2. Decreases probability of Type I error
    3. Has no effect
    4. Eliminates Type I error
    Show answer

    Answer: Decreases probability of Type I error

    The significance level α IS the probability of Type I error. Lowering α from 0.05 to 0.01 directly reduces the chance of falsely rejecting a true H₀.

  3. A two-tailed z-test gives p = 0.08 with α = 0.05. The conclusion is:

    1. Reject H₀; statistically significant
    2. Fail to reject H₀; insufficient evidence
    3. Accept H₀; it is true
    4. Reject H₀; p > α
    Show answer

    Answer: Fail to reject H₀; insufficient evidence

    p = 0.08 > α = 0.05, so we fail to reject H₀. There is insufficient evidence at the 5% level.

  4. The Central Limit Theorem states that for large n, the sampling distribution of x̄ is approximately:

    1. Uniform
    2. The same shape as the population
    3. Normal, regardless of the population's distribution
    4. Skewed right
    Show answer

    Answer: Normal, regardless of the population's distribution

    The CLT states the sampling distribution of x̄ approaches N(μ, σ²/n) as n increases, regardless of the population's shape (typically n ≥ 30 is sufficient).

  5. A t-test for a mean has 24 degrees of freedom. This test was conducted with a sample of size:

    1. n = 23
    2. n = 24
    3. n = 25
    4. n = 26
    Show answer

    Answer: n = 25

    Degrees of freedom for a one-sample t-test = n - 1. If df = 24, then n - 1 = 24, so n = 25.

  6. Which action will most directly increase the power of a hypothesis test?

    1. Decreasing the sample size
    2. Increasing the significance level α
    3. Using a two-tailed test instead of one-tailed
    4. Decreasing α from 0.05 to 0.01
    Show answer

    Answer: Increasing the significance level α

    Power = 1 - β. Increasing α makes it easier to reject H₀, increasing power (though it also raises Type I error risk). Larger sample size also increases power.

  7. Increasing sample size reduces both the margin of error and the probability of a Type II error.

    Show answer

    Answer: True

    Larger n reduces σ/√n (narrowing CI, reducing margin of error) and increases test power (reducing Type II error probability β).

  8. A p-value of 0.001 provides ___ evidence against the null hypothesis than a p-value of 0.04.

    Show answer

    Answer: stronger

    A smaller p-value means the data would be even less likely under H₀. p = 0.001 provides much stronger evidence against H₀ than p = 0.04.

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