Question

Explain in detail about the test of significance of a single mean.

Answer 1

A random sample of size n (n ≥ 30) is drawn from a population. We want to test the population mean has a specified value µ₀ . 

Procedure for testing: (For two-tail test) 

The null hypothesis is H₀: µ = µ₀ . 

The alternative hypothesis is H₁ : µ ≠ µ₂ 

Since n is large the sampling distribution of \(\overline{x}\) (the sample mean) is approximately normal. 

The test statistic

For a significance level α = 0.05 (5% level) 

If |Z| < 1.96, H₀ is accepted at 5% level. If |Z| > 1.96, H₀ is rejected at 5% level 

For α = 0.01 (1% level), if |Z| < 2.58, H₀ is accepted. If |Z| > 2.58, H is rejected.

Procedure for the one-tail test: (left tail)

H₀: µ ≥ µ₀ 

H₁ : µ < µ₁ 

At α = 0.05, |Z| = 1.645 

If Z < -1.645, H₀ is rejected If Z > -1.645, H₀ is accepted 

One tail test: (right tail) 

If Z < 1.645, H₀ is accepted If Z > 1.645, H₀ is rejected at 5% level of significance.