Question

Consider the claim that majority(p>0.5) of people over 50 are more affected by corona virus. But a survey of 10000 people over 50 suggest that 76% of them died due to corona virus rest recovered. Use a hypothesis test to test this claim. Use both P value and Critical value method. Write statements describing a Type I and a Type II error for the Null and Alternative Hypothesis that you obtained.

Answer 1

Solution:

Here, we have to use one sample z test for the population proportion.

The null and alternative hypotheses for this test are given as below:

Null hypothesis: H₀: The proportion of people over 50 are more affected by corona virus is less than or equal to 50%.

Alternative hypothesis: H_a: The proportion of people over 50 is more affected by corona virus is greater than 50%.

H₀: p ≤ 0.5 versus H_a: p > 0.5

This is an upper tailed test.

We assume

Level of significance = α = 0.05

Test statistic formula for this test is given as below:

Z = (p̂ - p)/sqrt(pq/n)

Where, p̂ = Sample proportion, p is population proportion, q = 1 - p, and n is sample size

n = sample size = 10000

p̂ = x/n = 0.76

p = 0.5

q = 1 - p = 0.5

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.76 - 0.5)/sqrt(0.5*0.5/10000)

Z = 52.000

Test statistic = 52.00

P-value = 0.0000

(by using z-table)

P-value < α = 0.05

So, we reject the null hypothesis

There is sufficient evidence to conclude that the proportion of people over 50 is more affected by corona virus is greater than 50%.

Type I error occurred when we conclude that the proportion of people over 50 is more affected by corona virus is greater than 50%, however, in fact it is less than or equal to 50%.

Type II error occurred when we conclude that the proportion of people over 50 is more affected by corona virus is less than or equal to 50%, however, in fact it is greater than 50%.