Stat-Hypothesis Testing-1

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In a big city 325 men out of 600 were found to be smokers.
Does this information support the conclusion that the majority
of men in this city are smokers? State the hypothesis clearly.

Solution: Null hypothesis is that the proportion of smokers
in the whole city is 50%, i.e. 50 / 100 = 0.5

H₀ : P = 0.5

We are interested to see if the proportion of smokers is more than 50%,
i.e. alternative hypotheisi is

H₁ : P > 0.05

The proportion of smokers in the sample is 325 out of n = 600

Using the sample proportion,

Observed value (p) = 325 / 600 = 0.542

If the null hypothesis H0 is true,

Expected Value (P₀) = 0.5

and S.E. of p = √ {0.5 * (1 – 0.5) / 600} = 0.0204

The test statistic is:

z = (Observed Value – Expected Value) / S.E.

= (0.542 – 0.5) / 0.0204

= 2.1

Since the alternative hypothesis H₁ : P > 0.5 is one-sided,
the critical region of the test is one-tailed.

At 5% level of significance

Critical Region is z ≥ 1.645

The value of test statistic z viz. 2.1 lies in the critical region,
and hence is significant.

We therefore reject the null hypothesis that majority of men in the city are smokers.