Hypothesis testing; chi-squared (independence, goodness of fit); t-test practice questions.

A χ² goodness-of-fit test checks whether a six-sided die is fair using the 6 observed frequencies. State the number of degrees of freedom.

• 4
• 5
• 6
• 30

Answer: 5

For a goodness-of-fit test, degrees of freedom = number of categories − 1 = 6 − 1 = 5.

A χ² test of independence is carried out at the 5% level. The calculated χ²_calc = 7.81 and the critical value is 9.49. State the correct conclusion.

• Reject H₀; the variables are dependent
• The test is invalid
• H₀ must be that the variables are dependent
• Do not reject H₀; no association shown

Answer: Do not reject H₀; no association shown

Since χ²_calc (7.81) < critical value (9.49), we do not reject H₀, so there is insufficient evidence that the variables are associated.

A researcher uses a two-sample t-test to compare the mean lifetimes of two brands of battery. Which is the appropriate null hypothesis H₀?

• The two population means are equal
• The two population means are different
• Brand A has a longer mean lifetime
• The two samples have equal size

Answer: The two population means are equal

The null hypothesis for a two-sample t-test states that the two population means are equal (μ₁ = μ₂); a difference is the alternative.

Which condition should hold for a χ² goodness-of-fit test to be valid?

• Every observed frequency must be exactly equal
• The sample size must be below 30
• The data must be normally distributed
• Every expected frequency should be at least 5

Answer: Every expected frequency should be at least 5

A standard validity requirement for χ² tests is that all expected frequencies are at least 5; otherwise categories are usually combined.

A one-tailed t-test is used to test whether a new teaching method gives a HIGHER mean score than the old method. Which is the correct alternative hypothesis H₁?

• μ_new = μ_old
• μ_new < μ_old
• μ_new > μ_old
• μ_new ≠ μ_old

Answer: μ_new > μ_old

A one-tailed test for a higher mean uses H₁: μ_new > μ_old, matching the directional claim being investigated.

A χ² test of independence gives χ²_calc = 12.6 with a p-value of 0.002, tested at the 5% level. State the correct conclusion.

• Do not reject H₀; the variables are independent
• Reject H₀; significant evidence of association
• The test proves one variable causes the other
• The degrees of freedom must be zero

Answer: Reject H₀; significant evidence of association

Since p = 0.002 < 0.05, we reject H₀ and conclude there is significant evidence of an association (though not causation).