Question

suppose that you randomly selected 24 adults. if 11% of the population smoke,
a) using the range rule of thumb, what is the minimum number of usual smokers we can expect to get out of 24 adults?
note: round all answers to two decimal places.
b) using the range rule of thumb, what is the maximum number of usual smokers we can expect to get out of 24 adults?
note: round all answers to two decimal places.
c) would it be unusual to randomly select 24 smokers and get 13 smokers? explain!
yes, since 13 is not between the maximum and minimum usual values.
no, since 13 is between the maximum and minimum usual values.
d) if we randomly selected 24 adults and found 13 smokers in the group, what could we conclude?
we cannot conclude that the 11% of smokers is probably incorrect.
we can conclude that 11% of smokers is probably incorrect, and that the percentage of smokers is greater than 11%.

Explanation:

Step1: Calculate mean and standard deviation

The mean $\mu = np$, where $n = 24$ and $p=0.11$. So $\mu=24\times0.11 = 2.64$. The standard deviation $\sigma=\sqrt{np(1 - p)}=\sqrt{24\times0.11\times(1 - 0.11)}=\sqrt{24\times0.11\times0.89}=\sqrt{2.3496}\approx1.53$.

Step2: Find minimum using range - rule - of - thumb

The range - rule - of - thumb for minimum usual value is $\mu - 2\sigma$. So $2.64-2\times1.53=2.64 - 3.06=- 0.42\approx0.00$.

Step3: Find maximum using range - rule - of - thumb

The range - rule - of - thumb for maximum usual value is $\mu + 2\sigma$. So $2.64+2\times1.53=2.64 + 3.06 = 5.70$.

Step4: Answer part c

Since $13$ is not between $0.00$ and $5.70$, it is unusual.

Step5: Answer part d

If we get $13$ smokers out of $24$ when the expected proportion is $11\%$, it is very unlikely. So we can conclude that $11\%$ of smokers is probably incorrect, and that the percentage of smokers is greater than $11\%$.

Answer:

a) $0.00$
b) $5.70$
c) Yes, since 13 is not between the maximum and minimum usual values.
d) We can conclude that 11% of smokers is probably incorrect, and that the percentage of smokers is greater than 11%.