Welcome to the Lab for Module 6. In this lab we'll be getting more experience calculating probabilities for both Normal and t distributed random variables. Some of the functions you may need are: pnormCx, mean = mu, sd = sigma, lower.tail = TRUE) This is the function to calculate the area under a normal curve where X ~ N(,u., or). Recall that \"areas\" of probability distributions correSpond to probabilities. Therefore, if you Specify lower.tail=TRUE'., you are nding the P(X x]. pt(x, df = n-1, lower.tail = TRUE) This is the function to calculate the area under a t curve where X ~ tdfztnnl). 1. Find the following probabilities and sketch the corresponding areas under the t distribution curve. A. X ~ t24, nd P(X > 2). B. X ~ t74, nd P(X 1.6). 2. We can also use our knowledge of Normal, Z, and t, distributions and R to calculate values that cut-off specic probabilities. Using the functions quorm(probability, mean = MU, sd = SIGMA, lower.tail = TRUE.) and qt(probability, df = n-1, lower.tail = TRUE.) A. For a sample size of 40, what value cuts off a probability of 0.025 in the lower tail? B. For a sample size of 25, what value cuts off a probability of 0.025 in the lower tail? C. For a sample size of 28, what value cuts off a probability of 0.1 in the upper tail? D. For a sample size of 65, what value cuts off a probability of 0.01 in the upper tail? 3. A. For the population of women between the ages of 3 and 74 who participated in the National Health Interview Survey, the distribution of hemoglobin levels is normally distributed and has mean ,u = 13.3 g/100ml and standard deviation 0 = 1.12 g / 100ml. What is the probability that a woman between the ages of 3 and 74 participating in the NHIS has a hemoglobin level that is considered \"low,\" which is less than 12 g/100ml? B. Describe the parameters and distribution of the sampling distribution of the means of samples of size 5. C. Using the same population distribution of hemoglobin levels in Part A, what is the probability of observing a sample of 5 women having a mean hemoglobin level less than 12 g / 100ml? D. Comment on the differences between the probabilities in Parts A and B. 4. Suppose IQ Scores are distributed Normally with mean ,LL = 100 and standard deviation 0 = 15. A. What is the probability that a randomly selected person will have an IQ over 114? B. What is the probability that a randomly selected person will have an IQ between 86 and 106? C. Describe the parameters and distribution of the sampling distribution of the means of samples of size 16. D. What is the probability that a sample of size 16 will have a mean IQ score over 114? E. What is the probability that a sample of size 16 will have a mean IQ score between 86 and 106