Answer to Question #346594 in Statistics and Probability for John Lloyd

Question

A medical expert claims that 80% of recovered COVID-19 patients have produced antibodies against the virus. To verify this, 1,000 recovered patients were tested and found that 823 of them have antibodies for the corona virus. Is this enough evidence to support the claim? Use 𝑎 = 0.01.

Answer 1

The following null and alternative hypotheses for the population proportion needs to be tested:

$H_0:p=0.8$

$H_a:p\not=0.8$

This corresponds to a two-tailed test, for which a z-test for one population proportion will be used.

Evidence:

Based on the information provided, the significance level is $\alpha = 0.01 ,$ and the critical value for a two-tailed test is $z_c = 2.5758.$

The rejection region for this two-tailed test is $R = \{z: |z|>2.5758\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{823/1000-0.8}{\sqrt{\dfrac{0.8(1-0.8)}{1000}}}=1.8183$$

Since it is observed that $|z| =1.8183<2.5758= z_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is $p=2P(Z>1.8183)= 0.069018,$ and since $p= 0.069018>0.01=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population proportion $p$ is different than 0.8, at the $\alpha = 0.01$ significance level.