Question

the mean percent of childhood asthma prevalence in 43 cities is 2.14%. a random sample of 32 of these cities is selected. what is the probability that the mean childhood asthma prevalence for the sample is greater than 2.5%? interpret this probability. assume that $sigma = 1.30%$. the probability is 0.0582. (round to four decimal places as needed.) interpret this probability. select the correct choice below and fill in the answer box to complete your choice. (round to two decimal places as needed.) a. about % of samples of 43 cities will have a mean childhood asthma prevalence greater than 2.5%. b. about % of samples of 32 cities will have a mean childhood asthma prevalence greater than 2.5%. c. about % of samples of 32 cities will have a mean childhood asthma prevalence greater than 2.14%.

Explanation:

Step1: Identify the sample - related values

We are given that the population mean $\mu = 2.14\%$, the population standard deviation $\sigma=1.30\%$, the sample size $n = 32$, and we want to find $P(\bar{X}>2.5\%)$.

Step2: Calculate the standard error of the mean

The standard error of the mean $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}=\frac{1.30}{\sqrt{32}}\approx0.2298$.

Step3: Calculate the z - score

The z - score is calculated as $z=\frac{\bar{X}-\mu}{\sigma_{\bar{X}}}=\frac{2.5 - 2.14}{0.2298}\approx1.57$.

Step4: Find the probability

We want $P(Z > 1.57)=1 - P(Z\leq1.57)$. From the standard normal table, $P(Z\leq1.57) = 0.9418$, so $P(Z>1.57)=1 - 0.9418=0.0582$.
The probability that the mean childhood asthma prevalence for the sample of 32 cities is greater than 2.5% is 0.0582. This means that about 5.82% of samples of 32 cities will have a mean childhood asthma prevalence greater than 2.5%.

Answer:

B. About 5.82% of samples of 32 cities will have a mean childhood asthma prevalence greater than 2.5%.