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MAE102 Statistics and probability

Questions:
Q1) The distribution of scores of students heading to college in the math section of the SAT follows a normal distribution with mean 520520 and standard deviation σ=115σ=115
X=X=The score of a student heading to college in the math section of the SAT
X¯=X¯=The mean score of 100100 students heading to college in the math section of the SAT
∑X=∑X=The total score of 100100 students heading to college in the math section of the SAT

(Please show all formulas, calculations and the curves)
(a) What math SAT score is 0.50.5 standard deviations above the mean?
(b) What is the probability that the score of a student is between 518518 and 530530?
(c) What is the distribution of X¯X¯?
(d) What is the probability that the mean scores of 100100 students is greater than 480480?
(e) What is the distribution of ∑X∑X?
(f) What is the probability that the total scores of 100100 students is less than 5150051500?
(g) What is the 15th15th percentile for the total scores of 100100 students?

Q2)  A camp director is interested in the mean number of letters each child sends during his or her camp session. A survey of 1616 campers is taken. The mean from the sample is 88 with a sample standard deviation of 2.82.8. Suppose that the population standard deviation of 3.13.1 is given. (Please show all formulas, calculations and the curve)
(a) Find a 9090% confidence interval for the mean number of letters each child sends during the camp session. Interpret this confidence intervals.
(b) If you wanted to estimate the mean number of letters each child sends during the camp session with 9595% confidence, and a margin of error of 0.30.3, how large a sample would you need?

Q3)  In a lake pollution study, the concentration of lead in the upper sedimentary layer of a lake bottom is measured from 2525 sediment samples. The sample mean and the sample standard deviation of the measurements are found to be 0.380.38 and 0.060.06, respectively. Suppose:
H0:μ=0.35H0:μ=0.35
H1:μ≠0.35H1:μ≠0.35
(a) State the Type I and Type II errors.
(b) Conduct a hypothesis test at 0.050.05 level of significance by doing the seven-step p-value approach. (please show all seven steps, formulas and calculations)