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Question 1 (12 marks) In a market survey a sample of 255 people from different age groups were asked about their preferred flavour of ice cream. Their responses are in the accompanying table.
Age Group Preferred Flavour of Ice Cream Chocolate (C) Vanilla (V) Strawberry (S) Under 21 years of age (U) 40 25 20 Between 21 and 40 (B) 35 20 30 Over 40 years of age (O) 20 30 35
For a person selected at random from the sample:
a) Find the probability that the person is between 21 and 40 years of age. b) Find the probability that the person does not have a preference for chocolate ice cream. c) Find the probability that the person is between 21 and 40 years of age given that the person has preference for strawberry ice cream. d) Find the probability that the person is between 21 and 40 years of age and has preference for strawberry ice cream.
Let B denotes the event that the person is between 21 and 40 years of age, O denotes the event that the person is over 40 years of age and S denote the event that the person has a preference for strawberry ice cream.
e) Are the events B and O independent? Explain your answer. f) Are the events B and S mutually exclusive? Explain your answer.
Question 2 (4 marks) A bag contains 125 jelly beans. There are 36 red, 30 pink, 28 green and 31 black. You grab a handful of 5 jelly beans from the bag.
a) Find the probability that all five are red. b) Find the probability that at least one of your five jelly beans is not red.
Question 3 (4 marks) Waldo works as a waiter and must wear black pants and a white collared shirt as his uniform for each shift. He does have the freedom to change his tie, belt and shoes, and in fact is encouraged to select those that are colourful in order to make his uniform more interesting. In his closet, he has 8 crazy ties, 4 different- coloured belts, and 5 different pairs of shoes to select from whenever he dresses for work. How many shifts can he work before he must repeat his choice of uniform?
Question 4 (4 marks) How many musical arrangements can you make if you are to select 4 songs from a catalogue of 20 and no repetitions are allowed?
Question 5 (4 marks) Justin Bieber has to select three teenage girls to receive back-stage passes at an upcoming concert. If he makes his selection from the 25 girls who won recent radio contests, how many possible trios does he have to choose from?
Question 6 (4 marks) For one pregnancy test, subjects who are pregnant will test positive 80% of the time, while subjects who are not pregnant will test positive 4% of the time. Suppose for a particular group of women using this pregnancy test, 85% are pregnant. If one of these women receives a positive test result, what is the probability that she is actually pregnant?
Question 7 (4 marks) One airline has a policy of booking as many as 15 persons on an airplane that can seat only 14. (Past studies have revealed that only 85% of the booked passengers actually show up for their flight.) Find the probability that if this airline books 15 persons, not enough seats will be available.
Question 8 (6 marks) Let X denotes the number of prior sentences for a randomly selected inmate at the Bordeau prison. Assume that X has the following probability distribution:
X P(X)
a) Given that the above table does represent a probability distribution, find the missing value in the P(X) column. b) Compute the expected value of the X distribution. c) Compute the standard deviation of the X distribution.
Question 9 (4 marks) A multiple-choice test consists of 25 questions with possible answers of a, b, c, d, and e. Use the Normal distribution with continuity correction to estimate the probability that with random guessing, the number of correct answers is between 3 and 10 inclusive.
Question 10 (5 marks) Men have hip breadths that are normally distributed with a mean of 14.4 inches and a standard deviation of 1 inch.
a) In a group of 18 college football players, what is the probability that their average hip breadth would be more than 15 inches? b) Suppose you want to build a bench to seat 18 male college football players. What is the minimum length of the bench if you want a 0.975 probability that it will fit the combined hip breadths of 18 randomly selected men? Round your answer to the nearest inch.
Question 11 (3 marks) A study is proposed to determine how much students are spending on textbooks. The population standard
deviation is assumed to be σ = $32. How many students should be included in the sample to be 95%
confident that the sample mean x is within $10 of the population mean μ for all students at this college?
Question 12 (4 marks) Collette is self-employed, selling cosmetics at home parties. She wants to estimate the average amount a
client spends at each party. A random sample of 45 receipts had a mean of x = $32.72 with a sample
standard deviation of s = $4.85. Find a 95% confidence interval for the mean amount spent by her clients.
Question 13 (6 marks) A random sample of 74 male athletes showed that 21 were wearing contact lenses during performances. Let p be the proportion of male athletes who wear contact lenses.
a) Find a 90% confidence interval for p. b) If you have no preliminary estimate for p, how many male athletes should you include in a random sample to be 99% sure that the point estimate p ˆ^ will be within a distance of 0.04 from p.
Question 18 (6 marks) John Abbott College is studying alternative course formats. One course format is fast track (course completed in five weeks) and the other course format is the traditional 15-week semester. One of the studies involves grade distributions. Two sections of a single course were scheduled. Both sections were taught by the same instructor. One section used the fast track format, the other the 15-week format. The grade distribution for each section is given in the table below. Use a 5% significance level to determine if the course format and the grade distribution are dependent.
Course Format Grade Fast track Traditional Above 80% 26 30 Between 60% and 80% 6 12 Less than 60% 18 8
a) State the null and the alternate hypotheses. b) What is the value of the sample test statistic? c) Find (or estimate) the P-value. d) State your conclusions in the context of the application.
Question 19 (6 marks) Has the length of telephone calls changed over the past ten years? Ten years ago, 10% of all phone calls lasted less than 1 minute, 60% lasted between 1 and 10 minutes and 30% lasted over 10 minutes. This year a random sample of 1000 calls showed that 125 lasted less than 1 minute, 585 lasted between 1 and 10 minutes, and 290 lasted longer than 10 minutes. Use a 1% significance level to test the claim that the distribution of phone call lengths has changed.
a) State the null and the alternate hypotheses. b) What is the value of the sample test statistic? c) Find (or estimate) the P-value. d) State your conclusions in the context of the application.
Answers
- a) 85/255 b) 160/255 c) 30/85 d) 30/255 e) No, because P(B|O) ≠ P(B) f) No, because P(B and S) ≠ 0.
- a) 0.00161 b) 0.
- 160
- 116,
- 2300
- a) 0.301 b) 0.73 c) 0.
- a) 0.0054 b) 268 inches
- 40 students
- Interval from $31.26 to $34.
- a) Interval from 0.1976 to 0.3700 b) 1041 male athletes
- a) Ho: μ = 4.3; H1: μ < 4.3 b) t = - 1.6197 c) 0.050 < P-value < 0.075. d) Do not reject Ho. We cannot
conclude that the mean recovery time for outpatients is less than the time for those recovering in the
hospital.
- a) Ho: p = 0.68; H1: p > 0.68 b) z = 2.48 c) P-value: 0.0066 d) Reject Ho. The realtor’s estimate is too low.
16. a) Ho: μ d = 0; H1: μ d ≠ 0 b) t = - 0.930; ( d = - 0.8; s = 1.9235) c) P-value between 0.250 and 0.
d) Do not reject Ho. We cannot conclude that the ratings before and after the debate are different.
- a) Ho: μ1= μ2; H1: μ1> μ 2 b) t = 3.365 c) 0.0005 < P-value < 0.
d) Reject HO. The sodium content is higher in the smoked ham.
- a) Ho: The course format and the grade distribution are independent. H1: The course format and the
grade distribution are dependent. b) χ2 = 6.1319 c) 0.025 < P-value < 0.05 d) Reject Ho. The course format
and the grade distribution are dependent.
- a) Ho: The distribution is the same as last year. H1: The distribution has changed. b) χ2 = 6.
c) 0.025 < p-value < 0.050 d) Do not reject Ho. We can not conclude that the distribution has changed.
