Sample test statistic

Question

You are testing the claim that the proportion of men who own cats is smaller than the proportion of women who own cats.

You sample 100 men, and 80% own cats.

You sample 200 women, and 75% own cats.

Find the test statistic

Answer 1

Rejection of a null hypothesis that is true is a Type I Error.

Acceptance of a null hypothesis that is false is a Type II Error.

In testing a hypothesis, the maximum probability with which

one is willing to risk a Type I Error is called the Level Of

Significance of the test.

Levels Of Significance of 0.05 or 0.01 are most common.

Choosing a 0.01 Significance Level to create a hypothesis

test translates to there being 1 chance out of 100 that a true

hypothesis will be rejected (Type I Error). In other words,

for a true Null Hypothesis, there is 99% confidence that

the right decision would be made. It is then said that the

probability of rejection of a true Null Hypothesis at the

0.01 Significance Level is equal to 0.01.

Analyze this claim at Level Of Significance α = 0.01:

p₁ = Male Cat Owners or 80 out of 100 Men Sampled;

p₂ = Female Cat Owners or 150 out of 200 Women Sampled.

Write the null and alternative hypotheses:

H₀ is p₁ − p₂ ≤ 0;

H₁ is p₁ − p₂ > 0.

A one-tailed test on the right side of the distribution with α = 0.01

has a critical z-score of z_c = 2.33. Rejection of H₀ (or p₁ − p₂ ≤ 0)

demands that z_{(p-bar1 − p-bar2)} be greater than 2.33.

Determine the sample proportions and the estimated overall proportion:

p-bar₁ = x₁/n₁ or 80/100 or 0.80;

p-bar₂ = x₂/n₂ or 150/200 or 0.75;

p-cap = (x₁+ x₂)/(n₁ + n₂) or (80 + 150)/(100 + 200) or 23/30.

Estimated standard error of the difference between the two proportions is

σ-cap_{(p-bar1 − p-bar2)} equal to [(p-cap)(1 − p-cap)(1/n₁ + 1/n₂)]^0.5 which here

comes to [(23/30)(1 − 23/30)(1/100 + 1/200)]^0.5  or 0.0518009009.

Now take z_{(p-bar1 − p-bar2)} as [(p-bar₁ − p-bar₂) − (p₁ − p₂)*] divided by σ-cap_{(p-bar1 − p-bar2),}

equal here to [(0.80 − 0.75) − (0)*] ÷ 0.0518009009 which reduces to 0.965234178.

0.965234178 is the statistic to be tested against z_c = 2.33. Since 0.965234178 is less than 2.33,

H₀ is not rejected. There is not enough evidence to support the alternative hypothesis that the proportion

of Male Cat Owners is larger than the proportion of Female Cat Owners at the 0.01 Significance Level.

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*Note that (p₁ − p₂) in this set of circumstances is set equal to zero. Predicting that 1 proportion is

bigger by a specific amount means that (p₁ − p₂) is that specific amount. Merely predicting

that 1 proportion is bigger (or that p₁ ≠ p₂) sets (p₁ − p₂) to 0 here.