Download Math 131 Final Exam

Math 131 Final Exam
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1. Suppose X is a discrete random variable with probability distribution function
x
p(x)
0
0.2
5
0.15
10
0.3
15
0.35
Calculate the following.
(a) P (5 ≤ X < 15)
(b) E(X)
(c) The variance and standard deviation of X
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2. Calculate the following quantities using the sample 7, -2, 10, 7, 4, 1, 0.
P
(a)
x
(b)
P
x2
(c) The sample mean
(d) The sample variance
(e) The sample standard deviation
(f) The median
(g) The mode
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3. A random sample of 12 students is taken from a large university, where 24% of the students
are business majors.
(a) Calculate the probability that exactly 4 students in the sample are business majors.
(b) Find the probability that at least 2 students in the sample are business majors.
(c) Let X be the number of students in the sample who are business majors. Find the
expected value, variance, and standard deviation of X.
4. In a random sample of 25 apples from an orchard, the average weight is 152.34 grams,
and the standard deviation is 32.98 grams. Assuming that apple weights are normally distributed, find a 95% confidence interval for the average weight of apples at this orchard.
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5. At a doctor’s office, the time a patient waits between arriving and seeing a doctor is uniformly distributed between 5 minutes and 25 minutes.
(a) Find the expected value of a patient’s waiting time.
(b) If a patient arrives at the doctor’s office at 4:02, find the probability that the patient sees
a doctor before 4:19.
6. Assume that men have heights that are normally distributed with a mean of 70 inches and a
standard deviation of 2 inches.
(a) What is the probability that a randomly selected man’s height is between 67 and 74
inches?
(b) For a random sample of 50 men, find the probability that the sample mean of their
heights is over 70.5 inches.
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7. In a random sample of 268 people, 23 were left-handed. Based on this information, find a
90% confidence interval for the proportion of people in the population who are left-handed.
8. A computer retailer has a policy of rejecting any shipment of computer processors in which
more than 3% of the processors are defective. A random sample of 600 processors is taken
from a large shipment, and 26 of them are defective.
(a) Use this data to perform the following hypothesis test at the 5% significance level.
H0 : p = 0.03 vs. Ha : p > 0.03
(b) Does this data provide strong evidence that the shipment should be rejected?
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9. According to the packaging, a box of Rice Krispies contains 283 grams of cereal. In a random
sample of 653 boxes of Rice Krispies, the average mass of the contents was 282.33 grams, and
the standard deviation was 12.48 grams.
(a) Use this data to perform the following hypothesis test at the 1% significance level.
H0 : µ = 283 vs. Ha : µ < 283
(b) Does this data provide strong evidence that the average mass of cereal in a box of Rice
Krispies is less than 283 grams?
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