Topic 4.18: t & z tests, errors (HL only) Questions

Write down what a Type I error is.

You are told the population standard deviation. Which test is appropriate?

In a hypothesis test, H₀ usually represents:

H₁: μ ≠ 50 corresponds to a test that is:

The probability of a Type I error is equal to:

Rejecting H₀ when it is actually true is called a:

A textbook states a species of beetle has mean wing length 11.0 mm. A biologist suspects the local population is larger and measures a sample, obtaining p = 0.032 from a one-tailed t-test at the 5% level. State the hypotheses and the conclusion in context.

Two delivery firms are compared. A two-sample t-test of H₀: μ_A = μ_B against H₁: μ_A ≠ μ_B gives p = 0.27 at the 5% significance level. State the conclusion and explain whether firm A is faster.

A teacher records each of 8 students' typing speed before and after a course. A paired t-test of H₀: μ_d = 0 (d = after − before) against H₁: μ_d > 0 gives p = 0.09 at the 5% level. State the conclusion in context.

To compute the probability of a Type II error you must be given:

Explain (i) why an AI HL question almost always asks for a t-test rather than a z-test, and (ii) how the significance level α relates to the conclusion.

At the 5% level a test gives p = 0.046. The correct decision is:

The same group of patients has blood pressure measured before and after a drug. The appropriate test for a mean change is a:

Find the probability of a Type I error, and state what would reduce the chance of a Type II error.

Explain why making a test stricter (lowering the significance level from 5% to 1%) reduces the chance of a Type I error but increases the chance of a Type II error.

A clinic's blood-test machine normally reports a mean reading of 100 units with known sd σ = 8. A sample of n = 16 is taken and H₀: μ = 100 is tested against H₁: μ > 100 at the 5% level (reject H₀ for large sample means). Find the critical value of the sample mean.

A discrete (e.g. binomial) test rejects H₀ for X in a critical region. P(Type I error) is found by computing:

A spam filter is tuned to flag emails. Let H₀ be 'the email is genuine'. Describe, in context, what a Type I error and a Type II error mean for this filter, and which is usually worse for the user.

A café claims its lattes contain 180 mg of caffeine on average. A sample of 15 lattes has mean 188 mg and sample sd 14 mg (σ unknown). Test at the 5% level whether the mean differs from 180 mg.

Making a test stricter by lowering α from 5% to 1% will: