Question

An alien empire is considering taking over planet Earth, but they will only do so if the portion of rebellious humans is less than 10 % 10%10, percent. They abducted a random sample of 400 400400 humans, performed special psychological tests, and found that 14 % 14%14, percent of the sample are rebellious. Let's test the hypothesis that the actual percentage of rebellious humans is 10 % 10%10, percent versus the alternative that the actual percentage is higher than that. The table below sums up the results of 1000 10001000 simulations, each simulating a sample of 400 400400 humans, assuming there are 10 % 10%10, percent rebellious humans. According to the simulations, what is the probability of getting a sample with 14 % 14%14, percent rebellious humans or more?'

Answer 1

It's 0.7% (make sure to include the % sign)

Explanation:

Khan Academy

Answer 2

Probability of getting a sample with 14% rebellious humans or more is 0.0104.

Explanation:

We are given that an alien empire is considering taking over planet Earth, but they will only do so if the portion of rebellious humans is less than 10%.

They abducted a random sample of 400 humans, performed special psychological tests, and found that 14% of the sample are rebellious.

Let p = population proportion rebellious humans

The z score probability distribution for sample proportion is given by;

Z =
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }` ~ N(0,1)

where,
`\hat`
`\hat p` = sample proportion rebellious humans = 14%

n = sample of humans = 400

p = population proportion rebellious humans = 10%

Now, probability of getting a sample with 14% rebellious humans or more is given by = P(
`\hat p`
`\geq` 14%)

P(
`\hat p`
`\geq` 0.14) = P(
`\frac{\hat p-p}{\sqrt{(\hat p(1-\hat p))/(n) } }`
`\geq`
`\frac{0.14-0.10}{\sqrt{(0.14(1-0.14))/(400) } }` ) = P(Z
`\geq` 2.31)

= 1 - P(Z < 2.31) = 1 - 0.9896 = 0.0104

The above probability is calculated by looking at the value of x = 2.31 in the z table which has an area of 0.9896.

Hence, the required probability is 0.0104.