Final answer:
a) The standard deviation of the height is 0.0707 inches. b) The probability of a person being at least 7' tall is 1.0. c) The probability of 0.1% of the population being over 7' tall is nearly zero. d) The expected value of N is 100 million.
Step-by-step explanation:
a) Standard Deviation:
To calculate the standard deviation, we need to know the variance first. Since the heights of the people in the country are independent Gaussian random variables with an expected value of 70 inches, the variance is 500,000/100,000,000 = 0.005. The standard deviation is the square root of the variance, which gives us a standard deviation of 0.0707 inches.
b) Probability of being at least 7' tall:
To calculate this probability, we need to find the area under the normal distribution curve to the right of 7'. To do this, we can calculate the z-score for 7' (84 inches) using the formula z = (x - μ) / σ, where μ is the mean and σ is the standard deviation. The z-score for 7' is (84 - 70) / 0.0707 = 198.6. Using a standard normal distribution table, we can find the probability corresponding to this z-score, which is essentially 1.0.
c) Probability that 0.1% of the population is over 7' tall:
The discrete random variable N belongs to a binomial distribution, since it represents the number of people out of the population that are over 7' tall. The probability that 0.1% of the population is over 7' tall can be calculated using the binomial probability formula P(X = k) = (nCk) * p^k * (1-p)^(n-k), where n is the population size, k is the number of successes, p is the probability of success for a single trial, and (nCk) is the number of combinations. In this case, the population size is 100 million, k is 0.1% of the population (100,000), p is the probability of being over 7' tall (approximately 1), and (nCk) can be calculated using the combination formula (nCk) = n! / (k! * (n-k)!). Performing the calculations gives us a probability of nearly zero, indicating that it is highly unlikely that only 0.1% of the population is over 7' tall.
d) Expected Value of N:
The expected value of N can be calculated using the formula E(X) = n * p, where E(X) is the expected value, n is the number of trials, and p is the probability of success for a single trial. In this case, the number of trials is the population size (100 million) and the probability of success is the probability of being over 7' tall (approximately 1). Therefore, the expected value of N is 100 million.