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Answer:
There is a 68% chance that between 17% and 30% are smokers.
There is a 95% chance that between 10% and 37% are smokers.
There is a 99.7% chance that between 4% and 44% are smokers.
Step-by-step explanation:
According to the Central limit theorem, if from an unknown population large samples of sizes <em>n</em> > 30, are selected and the sample proportion for each sample is computed then the sampling distribution of sample proportion follows a Normal distribution.
The mean of the sampling distribution of sample proportion is:
The standard deviation of the sampling distribution of sample proportion is:
Given:
<em>n</em> = 40
<em>p</em> = 0.236
Compute the mean and standard deviation of this sampling distribution of sample proportion as follows:
The Empirical Rule states that in a normal distribution with mean <em>µ</em> and standard deviation <em>σ</em>, nearly all the data will fall within 3 standard deviations of the mean. The empirical rule can be divided into three parts:
- 68% data falls within 1 standard-deviation of the mean.
That is P (µ - σ ≤ X ≤ µ + σ) = 0.68.
- 95% data falls within 2 standard-deviations of the mean.
That is P (µ - 2σ ≤ X ≤ µ + 2σ) = 0.95.
- 99.7% data falls within 3 standard-deviations of the mean.
That is P (µ - 3σ ≤ X ≤ µ + 3σ) = 0.997.
Compute the range of values that has a probability of 68% as follows:
Thus, there is a 68% chance that between 17% and 30% are smokers.
Compute the range of values that has a probability of 95% as follows:
Thus, there is a 95% chance that between 10% and 37% are smokers.
Compute the range of values that has a probability of 99.7% as follows:
Thus, there is a 99.7% chance that between 4% and 44% are smokers.