Help me with this math problem---**Question Stem:**
An intelligence quotient (IQ) is a measure of human intelligence.
Given that IQs are whole numbers that follow a Binomial distribution with mean 100 and standard deviation 20, use the Normal approximation to find the probability that a person selected at random has an IQ greater than 130.
Approximately what percent of people have an IQ greater than 130?
**Options:**
A 6.3%
B 7.6%
C 43.7%
D 56.3%
视频信息
答案文本
视频字幕
We have an IQ distribution problem. IQs follow a binomial distribution with mean 100 and standard deviation 20. We need to find the probability that a randomly selected person has an IQ greater than 130. We'll use normal approximation to solve this problem and choose from the given options.
We can use normal approximation because with large sample sizes, binomial distributions approximate normal distributions. Our IQ distribution becomes a normal distribution with mean 100 and standard deviation 20. The shaded area represents the probability we need to find: P of X greater than 130.
To use standard normal tables, we need to standardize using the z-score formula. We calculate z equals x minus mu divided by sigma. Substituting our values: z equals 130 minus 100 divided by 20, which equals 30 divided by 20, giving us z equals 1.5. This transforms our problem from P of X greater than 130 to P of Z greater than 1.5.
Now we calculate the probability using the standard normal distribution. P of Z greater than 1.5 equals 1 minus P of Z less than or equal to 1.5. From the standard normal table, P of Z less than or equal to 1.5 is approximately 0.9332. Therefore, P of Z greater than 1.5 equals 1 minus 0.9332, which equals 0.0668, or approximately 6.68 percent. This means about 6.68 percent of people have an IQ greater than 130.
Our calculated result is 6.68 percent. Comparing this with the given options: A is 6.3 percent, B is 7.6 percent, C is 43.7 percent, and D is 56.3 percent. Option A at 6.3 percent is closest to our calculated value of 6.68 percent. To summarize our solution: we identified the distribution parameters, applied normal approximation, standardized using the z-score, and calculated the probability using the standard normal distribution. The answer is A.