## probability and statistics HWprobability and statistics HW

Problem 1 (1.5 points)

A plan for an executive travelers’ club has been developed by an airline on the
premise that 5% of its current customers would qualify for membership. A random
sample of 500 customers yielded 30 who would qualify.
(a) Using this data, test at level 0.01 the null hypothesis that the company’s premise
is correct against the alternative that it is not correct. (0.5 point)
(b) What is the probability that when the test of part (a) is used, the company’s
premise will be judged correct when in fact 15% of all current customers qualify?
(1 point)
Note: To get full points, include major intermediate steps.
Problem 2 (1.5 points)
Suppose a sample of 16 wires is selected and each is tested to determine tensile
strength (N/mm2

). The resulting sample mean and standard deviation are 2150
and 40, respectively.
(a) The mean tensile strength for springs made using spinner straightening is 2100
N/mm2
. What hypotheses should be tested to determine whether the mean tensile
strength for the roller method exceeds 2100? (0.5 point)
(b) Assuming that the tensile strength distribution is approximately normal, what
test statistic would you use to test the hypotheses in part (a)? What is the p-value in
this case, and what would you conclude at significance level α = 0.05? (1 point)
Note: To get full points, include major intermediate steps.
Problem 3 (1.5 points)
A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at 40 mph under
specified conditions is known to be 120 ft. It is proposed that the new design be
1
implemented only if sample data strongly indicates a reduction in true average
braking distance for the new design.
(a) Define the parameter of interest and state the relevant hypotheses. (0.5 point)
(b) Suppose braking distance for the new system is normally distributed with
σ
2 = 100. The sample average braking distance for a random sample of 36 observations is 116.5. What is the p-value in this case, and what would you conclude at
significance level α = 0.05? (0.5 point)

(c) What is the probability that the new design is not implemented when its true
average braking distance is actually 110 ft and the test from part (b) is used? (0.5
point)
Note: To get full points, include major intermediate steps.
Problem 4 (1.5 points)
Considering the following Highway Runoff data in Table 1 for a particular location, where x = rainfall volume (m
3
) and y = runoff volume (m
3
). A simple linear
regression model is built based on such data using least squares estimation.
Table 1: The accompanying data in Problem 4
x 17 23 30 40 47 55 67 72 81
y 15 15 25 27 46 38 46 53 70
(a) Calculate point estimates of the slope and intercept. (1 point)
(b) Calculate a point estimate of the standard deviation σ. (0.5 point)
Note: To get full points, include major intermediate steps.