In Problems 7–12, the null and alternative hypotheses are given. Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed. What parameter is being tested?
H₀: σ = 7.8
H₁: σ ≠ 7.8

Verified step by step guidance

1. Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). Here, H₀: \( \sigma = 7.8 \) and H₁: \( \sigma \neq 7.8 \).
2. Recognize that the parameter being tested is the population standard deviation \( \sigma \), which measures the spread or variability of the data.
3. Determine the type of test by looking at the alternative hypothesis. Since H₁ states \( \sigma \neq 7.8 \), this means the test is checking for any difference from 7.8, not just greater or less than.
4. Conclude that because the alternative hypothesis uses \( \neq \), the test is a two-tailed test, which checks for deviations in both directions (less than or greater than 7.8).
5. Summarize: The hypothesis test is two-tailed, and the parameter being tested is the population standard deviation \( \sigma \).

Key Concepts

Types of Hypothesis Tests (Tailed Tests)

Hypothesis tests can be left-tailed, right-tailed, or two-tailed depending on the alternative hypothesis. A left-tailed test checks if a parameter is less than a value, a right-tailed test checks if it is greater, and a two-tailed test checks if it is different (either less or greater) from the value.

Null and Alternative Hypotheses

The null hypothesis (H₀) represents the status quo or a specific value of the parameter, while the alternative hypothesis (H₁) represents what we want to test against H₀. The form of H₁ determines the direction of the test and what kind of difference or effect is being investigated.

Parameter Being Tested (Population Standard Deviation σ)

The parameter tested here is the population standard deviation (σ), which measures the spread or variability of data. Testing σ involves determining if the population variability differs from a hypothesized value, often using chi-square tests for variance or standard deviation.