Respuesta :
Answer:
The probability that at least 280 of these students are smokers is 0.9664.
Step-by-step explanation:
Let the random variable X be defined as the number of students at a particular college who are smokers
The random variable X follows a Binomial distribution with parameters n = 500 and p = 0.60.
But the sample selected is too large and the probability of success is close to 0.50.
So a Normal approximation to binomial can be applied to approximate the distribution of X if the following conditions are satisfied:
1. np ≥ 10
2. n(1 - p) ≥ 10
Check the conditions as follows:
[tex]np=500\times 0.60=300>10\\n(1-p)=500\times(1-0.60)=200>10[/tex]
Thus, a Normal approximation to binomial can be applied.
So,
[tex]X\sim N(\mu=600, \sigma=\sqrt{120})[/tex]
Compute the probability that at least 280 of these students are smokers as follows:
Apply continuity correction:
P (X ≥ 280) = P (X > 280 + 0.50)
= P (X > 280.50)
[tex]=P(\frac{X-\mu}{\sigma}>\frac{280-300}{\sqrt{120}}\\=P(Z>-1.83)\\=P(Z<1.83)\\=0.96638\\\approx 0.9664[/tex]
*Use a z-table for the probability.
Thus, the probability that at least 280 of these students are smokers is 0.9664.
The probability of a minimum of 280 students being radio listeners would be:
0.9664
Standard deviation:
The standard deviation is a statistic that measures the dispersion of a dataset relative to its mean and is calculated as the square root of the variance in statistics.
So, the formula is,
[tex]SD=\sqrt{\sigma}[/tex]
Suppose it is known that 60% of students at a college are smokers.
A sample of 500 students from the college is selected at random.
[tex]mean=np\\=500\times 0.6\\=300[/tex]
The standard deviation is,
[tex]S=\sqrt{np(1-p)}\\ =\sqrt{500\times 0.6\times 0.4}\\ =10.95445[/tex]
So, the probability is,
[tex]P(X > 280)=\frac{P(x-mean)}{s} \\=\frac{280-300}{10.95445} \\=0.9664[/tex]
Learn more about the topic standard deviation:
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