Questions: A scientist claims that 4% of viruses are airborne. If the scientist is accurate, what is the probability that the proportion of airborne viruses in a sample of 662 viruses would be greater than 6%? Round your answer to four decimal places.

A scientist claims that 4% of viruses are airborne.

If the scientist is accurate, what is the probability that the proportion of airborne viruses in a sample of 662 viruses would be greater than 6%? Round your answer to four decimal places.

Transcript text: A scientist claims that $4 \%$ of viruses are airborne. If the scientist is accurate, what is the probability that the proportion of airborne viruses in a sample of 662 viruses would be greater than $6 \%$ ? Round your answer to four decimal places.

Solution

Solution Steps

Step 1: State the Hypotheses

We want to test the claim that $4\%$ of viruses are airborne. We set up our hypotheses as follows:

Null Hypothesis ($H_0$): $p = 0.04$
Alternative Hypothesis ($H_a$): $p > 0.04$

Step 2: Calculate the Test Statistic

The test statistic $Z$ is calculated using the formula: \[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] where:

$\hat{p} = 0.06$ (sample proportion)
$p_0 = 0.04$ (hypothesized population proportion)
$n = 662$ (sample size)

Substituting the values, we find: \[ Z = \frac{0.06 - 0.04}{\sqrt{\frac{0.04(1 - 0.04)}{662}}} = 2.626 \]

Step 3: Determine the P-value

The P-value is calculated based on the test statistic. For $Z = 2.626$, the P-value is found to be: \[ \text{P-value} = 0.0043 \]

Step 4: Identify the Critical Region

For a significance level of $\alpha = 0.05$ in a one-tailed test, the critical value of $Z$ is: \[ Z_{critical} = 1.6449 \] Thus, the critical region is defined as: \[ Z > 1.6449 \]

Step 5: Make a Decision

We compare the test statistic to the critical value: