Statistics Questions and Problems

Problem 1: ACT Scores and Probability

UESTIONS :
1. The scores of your state’s high school seniors on the ACT college entrance examination in a recent year had mean μ = 22.3 and standard deviationσ = 5.2. The distribution of scores is only roughly Normal.
(a) What is the approximate probability that a single student randomly chosen from all those taking the test scores 27 or higher?
(b) Now consider an SRS of 16 students who took the test. What are the mean and standard deviation of the sample mean score of these 16 students?
(c) What is the approximate probability that the mean score of these 16 students is 27 or higher?
(d) Which of your two Normal probability calculations in parts (a) and (c) is more accurate? Why?
2. A set of final exam grades in ST2500 course is normally distributed with mean 70and standard deviation of 8.
(a) What is the probability of getting a grade of A(greater or equal 80) on this exam?
(b) What is the probability of that a student scored between 65 and 79
(c) The probability is 10% that a student taking the exam scores higher than what grade?
3. The number of loaves of white bread demanded daily at a bakery is normally distributed with mean 7000 loaves and variance 84000. The company decides to produce a sufficient number of loaves so that it will fully supply demand on 90% of all days.
(a) How many loaves of bread should the company produce? (4)
(b) Based on (a), on what percentage of days will the company be left with more than 500 loaves of unsold bread?
4. The file pines.txt gives the diameter at breast height (DBH) for 40 longleaf pine trees from the Wade Tract in Thomas County, Georgia.
(a) Construct a histogram of the tree diameters. Do the credit scores appear approximately normally distributed? Explain
(b) Construct a normal probability plot (QQ-plot) of the tree diameters. Do the tree diameter appear approximately normally distributed? Explain.
(c) If your answer in (b) is yes, find the probability that a tree will have a diameter of at least 40. Use the sample mean of the data for μ, and the sample standard deviation for σ.
5. “What do you think is the ideal number of children for a family to have?” A Gallup Poll asked this question of 1020 randomly chosen adults. Over half (53%) thought that a total of two children was ideal.Suppose that p = 0.53 is exactly true for the population of all adults. Gallup announced a margin of error of ±4 percentage points for this poll. What is the probability that the sample proportion for an SRS of size n = 1020 falls between 0.49 and 0.57? (4)
6. To assess the accuracy of a laboratory scale, a standard weight known to weigh 10 grams is weighed repeatedly. The scale readings are Normally distributed with unknown mean (this mean is 10 grams if the scale has no bias). The standard deviation of the scale readings is known to be 0.0002 gram.
(a) The weight is measured five times. The mean result is 10.0023 grams. Give a 98% confidence interval for the mean of repeated measurements of the weight. (4)
(b) How many measurements must be averaged to get a margin of error of ±0.0