Question

Let X₁,X₂ , … , X_n(n ≥ 3) be a random sample from a N(μ,σ²) distribution, where μ ∈ \(\R\) and σ > 0 are both unknown. Then which one of the following is a simple null hypothesis ?

• A. H₀ : μ < 5, σ² = 3
• B. H₀ : μ = 5, σ² > 3
• C. H₀ : μ = 5, σ² = 3
• D. H₀ : μ = 5

Answer 1

The Correct Option is C

To determine which of the given options is a simple null hypothesis in the context of a normally distributed random sample, let's first understand the definition of a simple null hypothesis.

A null hypothesis, denoted by \(H_0\), is considered simple when it specifies a single value for all the parameters of the distribution. In the case of a normal distribution \(N(\mu, \sigma^2)\), a simple null hypothesis will specify both the mean \(\mu\) and the variance \(\sigma^2\).

Now, let's evaluate each of the options:

• \(H_0 : \mu < 5, \sigma^2 = 3\) - Here, the hypothesis specifies a variance \(\sigma^2 = 3\), but the mean \(\mu\) is not a single value; it is specified as less than 5. This is not a simple hypothesis because the mean is not fixed.
• \(H_0 : \mu = 5, \sigma^2 > 3\) - This hypothesis specifies a mean \(\mu = 5\), but the variance is not a single fixed value; it is greater than 3. Thus, this is not a simple hypothesis.
• \(H_0 : \mu = 5, \sigma^2 = 3\) - Here, both the mean and variance are specified as single values: \(\mu = 5\) and \(\sigma^2 = 3\). This qualifies as a simple hypothesis.
• \(H_0 : \mu = 5\) - Although the mean is specified, the variance \(\sigma^2\) is not mentioned, meaning this is not a simple hypothesis.

Based on the analysis above, the correct option that represents a simple null hypothesis is \(H_0 : \mu = 5, \sigma^2 = 3\).