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16.10.2020 •
Mathematics
A scientist claims that 7% 7 % of viruses are airborne. If the scientist is accurate, what is the probability that the proportion of airborne viruses in a sample of 600 600 viruses would differ from the population proportion by greater than 3% 3 % ? Round your answer to four decimal places.
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Ответ:
The probability is![P(|p-\^{p}| 0.03) = 0.0040](/tpl/images/0809/1269/64ad1.png)
Step-by-step explanation:
From the question we are told that
The population proportion is![p = 0.07](/tpl/images/0809/1269/db64e.png)
The mean of the sampling distribution is![\mu_p = 0.07](/tpl/images/0809/1269/ad616.png)
The sample size is n = 600
Generally the standard deviation is mathematically represented as
=>
=>
Generally the probability that the proportion of airborne viruses in a sample of 600 viruses would differ from the population proportion by greater than 3% is mathematically represented as
=>![P(|p-\^{p}| 0.03) = 1 - P(-0.03 \le p -\^{p} \le 0.03 )](/tpl/images/0809/1269/5cd0c.png)
Now add p to both side of the inequality
=>![P(|p-\^{p}| 0.03) = 1 - P( 0.07-0.03 \le \^{p} \le 0.03+ 0.07 )](/tpl/images/0809/1269/00407.png)
=>![P(|p-\^{p}| 0.03) = 1 - P(0.04 \le \^{p} \le 0.10 )](/tpl/images/0809/1269/5f548.png)
Now converting the probabilities to their respective standardized score
=>![P(|p-\^{p}| 0.03) = 1 - P(\frac{0.04 - 0.07}{0.010416} \le Z \le \frac{0.10 -0.07}{0.010416} )](/tpl/images/0809/1269/249bd.png)
=>![P(|p-\^{p}| 0.03) = 1 - P(-2.88 \le Z \le 2.88 )](/tpl/images/0809/1269/01803.png)
=>![P(|p-\^{p}| 0.03) = 1 - [P(Z \le 2.88) - P(Z \le -2.88)]](/tpl/images/0809/1269/173a1.png)
From the z-table
and
So
=>![P(|p-\^{p}| 0.03) = 0.0040](/tpl/images/0809/1269/64ad1.png)
Ответ: