STATS help?? dont understand how to do this question

6% of the American population are at least 75 years of age.
What is the probability that a random sample of 150 respondents would contain less than 4% of individuals at least 75 years of age?

i want to learn how to do this question. please!

This is a binomial distribution

P(person is at least 75) = 0.06
P(person is less than 75) = 0.94

P(x = n) = Prob(exactly n people out of 150 are at least 75 years of age)
. . . . . . . = (150 C n) * 0.06^n * 0.94^(150-n)

4% of 150 = 6
If LESS than 4% of individuals are at least 75, then number of individuals that are at least 75 is 0, 1, 2, 3, 4, or 5

P(less than 4% of respondents are at least 75)
= P(x < 6)
= P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5)
.. 5
= ∑ (150 C n) * 0.06^n * 0.94^(150-n)
.. 0
= 0.0000931 + 0.0008919 + 0.0042410 + 0.0133548 + 0.0313269 + 0.0583879
= 0.1082956

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Alternatively, you could find an approximation using normal distribution

We find mean and variance for expected number of individuals at least 75 years of age in a sample of 150 people

Mean = n * p = 150 * 0.06 = 9
Variance = n * p * q = 150 * 0.06 * 0.94 = 8.46
Standard deviation = √8.46 = 2.9086

So we need to find P(x < 6)
Find z-score, and use z-score chart to find probability
z = (6 - 9) / 2.9086 = −1.03
P(x < 6) = P(z < −1.03) = 0.1515

This is not a great approximation. Since we are approximating probability of discrete distribution using continuous distribution, we use a continuity correction -- subtract 1/2 from 6. This should give a better result.

Pr(x < 6) = Pr(y < 5.5)

We find z-score for y = 5.5 and use z-score chart:
z = (5.5 - 9) / 2.9086 = −1.203

P(y < 5.5) = P(z < −1.203) = 0.1151 <---- much better approximation.