Notation - Advanced Quantitative Methods - Exam | Exams Mathematics

December 2011 Final Exam Advanced Quantitative Methods – 201-301-RE Page 1 of 4

1. A bowl of Halloween candy contains 45 snack-sized candy bars. There are 16 Coffee Crisps, 20 Mars

bars, and 9 Snickers.

(a) You draw three candy bars (one at a time, without peeking) from the bowl.

Find the probability that all three are Mars bars. [2 marks]

(b) Repeat part (a), but find instead the probability that none are Snickers. [2 marks]

2. Assume that one person is selected randomly from the 2223 people aboard the Titanic. Let us use the

following notation to represent different events of choosing a person from each category:

S = survived, D = died, M = man, W = woman, B = boy, G = girl

Titanic Mortality

Men (M)

Women(W)

Boys (B)

Girls (G)

Total

Survived (S)

332

318

706

Died (D)

1360

104

1517

Total

1692

422

2223

Apply probability rules to determine the following:

(a) P(B) [1 mark]

(b) P(B, given S) [2 marks]

(d) P(B and G) [1 mark]

(e) Are the events B and S independent? Explain your answer. [2 marks]

3. Stuart is working too many hours and is available to attend only one class each morning. There are six

possible courses he could take on Monday-Wednesday-Friday mornings and three courses he could take

on Tuesday-Thursday mornings, as well as 2 that are offered on-line. If he takes exactly three courses

(but not more than one on-line), how many different schedules could Stuart put together?

[3 marks]

4. For one cancer-screening test, subjects with cancer will test positive 94% of the time, while subjects

without cancer will test positive 3% of the time. Suppose for a particular community, 2% of the

population has cancer.

a) What percent of the population does not have cancer? [1 mark]

b) If one person in this community tests positive, what is the probability that they actually have

cancer? [3 marks]

5. Of the world population, approximately 10% are left-handed. In a random sample of 42 people, what is

the probability of each of the following? Do not use any approximation technique.

(a) Exactly 6 are left-handed. [2 marks]

(b) At least 2 are left-handed. [3 marks]

6. Groups of five babies are randomly selected. In each group, let x = number of babies with green eyes.

Let P(x) = probability that x babies among the five have green eyes.

P(x)

0.528

0.360

0.098

0.013

0.001

0.000

(a) Compute the probability that x is more than 2. [2 marks]

(b) Compute the expected value of the x distribution. [2 marks]

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1. A bowl of Halloween candy contains 45 snack-sized candy bars. There are 16 Coffee Crisps, 20 Mars bars, and 9 Snickers. (a) You draw three candy bars (one at a time, without peeking) from the bowl. Find the probability that all three are Mars bars. [2 marks] (b) Repeat part (a), but find instead the probability that none are Snickers. [2 marks] 2. Assume that one person is selected randomly from the 2223 people aboard the Titanic. Let us use the following notation to represent different events of choosing a person from each category: S = survived, D = died, M = man, W = woman, B = boy, G = girl Titanic Mortality Men (M) Women(W) Boys (B) Girls (G) Total Survived (S) 332 318 29 27 706 Died (D) 1360 104 35 18 1517 Total 1692 422 64 45 2223 Apply probability rules to determine the following: (a) P(B) [1 mark] (b) P(B, given S) [ 2 marks] (c) P(B and S) [ 2 marks] (d) P(B and G) [1 mark] (e) Are the events B and S independent? Explain your answer. [2 marks] 3. Stuart is working too many hours and is available to attend only one class each morning. There are six possible courses he could take on Monday-Wednesday-Friday mornings and three courses he could take on Tuesday-Thursday mornings, as well as 2 that are offered on-line. If he takes exactly three courses (but not more than one on-line), how many different schedules could Stuart put together? [ 3 marks] 4. For one cancer-screening test, subjects with cancer will test positive 94% of the time, while subjects without cancer will test positive 3% of the time. Suppose for a particular community, 2% of the population has cancer. a) What percent of the population does not have cancer? [1 mark] b) If one person in this community tests positive, what is the probability that they actually have cancer? [3 marks] 5. Of the world population, approximately 10% are left-handed. In a random sample of 42 people, what is the probability of each of the following? Do not use any approximation technique. (a) Exactly 6 are left-handed. [2 marks] (b) At least 2 are left-handed. [3 marks] 6. Groups of five babies are randomly selected. In each group, let x = number of babies with green eyes. Let P( x ) = probability that x babies among the five have green eyes. x P(x) 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. (a) Compute the probability that x is more than 2. [2 marks] (b) Compute the expected value of the x distribution. [ 2 marks] (c) Compute the standard deviation of the x distribution. [3 marks]

7. Membership in Mensa requires an IQ score above 131.5. Nine candidates take IQ tests, and their summary results indicated that their mean IQ score is 133. IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. (a) If one person is randomly selected from the general population, find the probability of getting someone with an IQ score of at least 133. [ 2 marks] (b) If 9 people are randomly selected, find the probability that their mean IQ score is at least 133. [ 2 marks] 8. Six percent of typical people have blood that is group O and type Rh –. These people are considered to be universal donors, because they can give blood to anyone. One hospital is conducting a blood drive and 200 volunteers show up to donate blood. Use the Normal distribution with continuity correction to estimate the probability that at least 10 of the volunteers are universal donors. [ 5 marks] 9. Randomly selected students participated in an experiment to test their ability to determine when one minute (or 60 seconds) had passed. Forty students yielded a sample mean of (^) x = 58.3 seconds with a sample standard deviation of s = 9.5 seconds. Find a 9 5 % confidence interval for the population mean of all students. [5 marks] 10. One poll of 1501 randomly selected U.S. adults showed that 70% of the respondents believe in global warming. Find a 9 0 % confidence interval for the proportion of all U.S. adults who believe in global warming. [ 5 marks] 11. Suppose that we want to estimate the mean IQ score for the population of statistics students. The population standard deviation is assumed to be σ = 15. How many students should be included in the sample to be 9 5 % confident that the sample mean x is within 3 points of the population mean μ for all statistics students? [5 marks] 12. Ruth is concerned about the spending habits of teens. She read a report that the national weekly spending average for teens in the age group 12 to 15 years is $42. She took a random sample of 60 teens who live in a rural area and found that they spent an average of $39 per week with sample standard deviation $7.50. Test the claim that rural teens from this area spend less than the national average. Use a 1% significance level. (a) State the null and alternate hypotheses. [2 marks] (b) What is the value of the sample test statistic? [2 marks] (c) Find (or estimate) the P - value. [2 marks] (d) Based on your answers for parts (a) through (c), will you reject or fail to reject the null hypothesis? [1 mark] 13. A particular type of avalanche studied in Canada had an average thickness of μ = 67 cm. This type of avalanche was studied in a region of the southwest United States. A random sample of 16 such avalanche thicknesses had a mean of 70 cm and a known standard deviation of σ = 11.3. Assume this thickness has an approximately normal distribution. Use a 5% level of significance to test the claim that this mean avalanche thickness in this US region is more than that in Canada. (a) State the null and alternate hypotheses. [2 marks] (b) What is the value of the sample test statistic? [2 marks] (c) Find (or estimate) the P - value. [2 marks] (d) Based on your answers for parts (a) through (c), will you reject or fail to reject the null hypothesis? [1 mark] 14. A random sample of 8 years of Denver, Colorado weather records gave a sample average number of sunny days per year of 263 sunny days with known standard deviation σ 1 = 24 days. An independent random sample of 6 years of weather records from Phoenix, Arizona had an average of 296 sunny days per year with known standard deviation σ 2 = 18.3 days. Test to see if the population mean numbers of sunny days are different for the two cities. Use a 1% significance level. (a) State the null and alternate hypotheses. [2 marks] (b) What is the value of the sample test statistic? [2 marks] (c) Find (or estimate) the P - value. [ 3 marks] (d) Based on your answers for parts (a) through (c), will you reject or fail to reject the null hypothesis? [1 mark]

ANSWERS:

a) 0.0803 b) 0.
a) 0. 0288 b) 0.0411 c) 0.01305 d) 0 e) No, since P(B) ≠ P(B|S)
36
a) 98% b) 0.
a) 0.118 b) 0.
a) 0.014 b) 0.599 c) 0.
a) 0.0139 b) 0
(55.25, 61.35)
(0.681, 0.719)
97
a) H 0 : μ = 42, H 1 : μ < 42 b) t = - 3.098 c) 0.0005 < p < 0.005 d) Reject H 0.
a) H 0 : μ = 67, H 1 : μ > 67 b) z = 1.06 c) p = 0.1446 d) Fail to Reject H 0.
a) H 0 : μ 1 = μ 2 , H 1 : μ 1 ≠ μ 2 b) z = - 2.92 c) p = 0.0036 d) Reject H 0.
a) (0.02, 0.30) Yes. b) No, since the new interval would now contain 0.
a) H 0 : Distributions are the same, H 1 : Distributions are different b) Χ^2 = 70. c) p < 0.005 d) Reject H 0.
a) H 0 : Getting an infection is independent of treatment group H 1 : Getting an infection is dependent of treatment group b) Χ^2 = 2.925 c) 0.1 < p < 0.9 d) Fail to Reject H 0.

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