Final answer:
To find the probability that x is between three and nine when x ~ N(6,2), standardize the values using the formula z = (x - μ) / σ. Look up the z-scores in the standard normal distribution table or use a calculator to find the cumulative probabilities. Subtract the cumulative probability of -1.5 from the cumulative probability of 1.5 to find the probability that x is between three and nine.
Step-by-step explanation:
To find the probability that x is between three and nine when x ~ N(6,2), we need to standardize the values using the formula z = (x - μ) / σ, where μ is the mean and σ is the standard deviation. In this case, μ = 6 and σ = 2. So, z1 = (3 - 6) / 2 = -1.5 and z2 = (9 - 6) / 2 = 1.5.
Next, we look up the z-scores in the standard normal distribution table or use a calculator. The area between -1.5 and 1.5 represents the probability that x is between three and nine. This area can be found by subtracting the cumulative probability of -1.5 (which represents the area to the left of -1.5) from the cumulative probability of 1.5 (which represents the area to the left of 1.5). The cumulative probabilities can be found using the standard normal distribution table or a calculator.
Using a standard normal distribution table or a calculator, we find that the cumulative probability of -1.5 is approximately 0.0668, and the cumulative probability of 1.5 is approximately 0.9332. Therefore, the probability that x is between three and nine is approximately 0.9332 - 0.0668 = 0.8664, or 86.64%.