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The Census Bureau reports that 82% of Americans over the age of 25 are high school graduates. A survey of randomly selected residents of certain county included 1200 who were over the age of 25, and 1010 of them were high school graduates.

(a) Find the mean and standard deviation for the number of high school graduates in groups of 1200 Americans over the age of 25.
Mean = 984
Standard deviation =?

1 Answer

2 votes

Answer:

Mean = 984

Standard deviation = 13.31 (2 d.p.)

Explanation:

To find the mean and the standard deviation for the number of high school graduates in groups of 1200 Americans over the age of 25, we can model the given scenario as a binomial distribution.

Binomial distribution


\large\boxed{X\sim \text{B}(n,p)}

where:

  • X is the random variable that represents the number of successes.
  • n is the fixed number of independent trials.
  • p is the probability of success in each trial.

Given that 82% of Americans over the age of 25 are high school graduates, and the number of trials is 1200:

  • n = 1200
  • p = 0.82

Therefore:


\large\boxed{X\sim \text{B}(1200,0.82)}

where the random variable X represents the number of Americans over the age of 25 who are high school graduates.

The mean (μ) for a binomial distribution is given by:


\large\boxed{\mu=np}

Substituting n = 1200 and p = 0.82 into the formula gives:


\mu = 1200 \cdot 0.82 = 984

Therefore, the mean for the number of high school graduates in groups of 1200 Americans over the age of 25 is 984.

The standard deviation (σ) for a binomial distribution is given by:


\large\boxed{\sigma = √(n p (1 - p))}

Substituting n = 1200 and p = 0.82 into the formula gives:


\sigma = √(1200\cdot 0.82(1 - 0.82))


\sigma = √(1200\cdot 0.82(0.18))


\sigma = √(177.12)


\sigma =13.3086438...


\sigma =13.31\; \sf (2\;d.p.)

Therefore, the standard deviation for the number of high school graduates in groups of 1200 Americans over the age of 25 is 13.31 (rounded to 2 decimal places).

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