Question

Explain in detail 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 `bar(x)` (the sample mean) is approximately normal.

The test statistic Z = `(bar(x) - mu)/(sigma/sqrt("n")` ∼  (0, 1)

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, H₀ is rejected.

Procedure for one tail test: (left tail)

H₀: µ ≥ µ₀

H₁: µ < µ₀

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.