Final answer:
The question is about finding the probability of a binomial proportion, and it is solved using a normal approximation to find the probability of fewer than 136 people in the sample being born in their state of residence.
Step-by-step explanation:
The student is asking for the probability that fewer than 136 people in a random sample of 222 were born in their state of residence, given that 67.5% of the U.S. population were born in their state of residence. This is a binomial probability problem that can be approximated using the normal distribution since the sample size is large enough.
To find the probability P(x < 136), we first need to calculate the mean (μ) and standard deviation (σ) of the binomial distribution:
- μ = n * p = 222 * 0.675
- σ = √(n * p * (1 - p)) = √(222 * 0.675 * (1 - 0.675))
Then we calculate the z-score for 135.5 (we use 135.5 instead of 136 due to continuity correction) using the formula:
z = (x - μ) / σ
Finally, we look up this z-score in the standard normal distribution table, or use a calculator, to find the probability that corresponds to this z-score. The calculated probability will be the answer to the student's question