Final answer:
Calculating E(e⁻ᵣ²+³ᵣ) involves integration with respect to the standard normal distribution and may require numerical methods. Standard normal distribution is used for constructing confidence intervals, with the central limit theorem allowing the treat sums of large samples as normally distributed.
Step-by-step explanation:
Let z be a standard normal random variable. To find E(e⁻ᵣ²⁼³ᵣ), or the expected value of the given function of z, we would need to integrate the function e⁻ᵣ²+³ᵣ weighted by the standard normal probability density function. However, this particular computation is quite complex and typically not computed by hand, possibly requiring numerical methods or software.
The question seems to focus on understanding of the standard normal distribution, expectation, and how to compute probabilities and related values for normally distributed random variables.
When we know the population standard deviation, σ, and we wish to calculate an error bound EBM or construct a confidence interval, we use a standard normal distribution Z~ N(0, 1) to find the appropriate z-value. For instance, to find the value of z for a 95 percent confidence level, we can use a standard normal probability table or the invNorm function on a calculator such as the TI-83, 83+, or 84+, where the command invNorm(0.975,0,1) would return Z0.025, which is approximately 1.96.
The central limit theorem is also relevant here, as it allows us to treat sums of large samples as normally distributed even when the original distribution is not normal, provided certain conditions are met. For such cases, the mean and standard deviation of the sampling distribution can be calculated as multiples of the original mean and standard deviation, adjusted by the sample size.