Homework 3 - Alyssa Taylor sharma

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Indiana University, Purdue University, Indianapolis *

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Apr 3, 2024

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PBHL-B300/B301 Homework #3 1. If you flip a fair coin 10 times, what is the probability of: (a) getting all tails? – 0.098% (b) getting all heads? – 0.098% (c) getting at least one tail? – 99.9% 2. The 2010 American Community Survey estimates that 14.6% of Americans live below the poverty line, 20.7% speak a language other than English (foreign language) at home, and 4.2% fall into both categories. (a) Are living below the poverty line and speaking a foreign language at home disjoint? – No, they are not disjoint events because they have elements in common. (b) What percent of Americans live below the poverty line and only speak English at home? - 10.4% (c) What percent of Americans live below the poverty line or speak a foreign language at home? - 31.1% ( 14.6 + 20.7) - 4.2 = 31.1 (d) What percent of Americans live above the poverty line and only speak English at home? – 68.9% 100% - 31.1% = 68.9% (e) Is the event that someone lives below the poverty line independent of the event that the person speaks a foreign language at home? No, they are not independent. 3. A Pew Research poll asked 1,306 Americans, “From what you’ve read and heard, is there solid evidence that the average temperature on earth has been getting warmer over the past few decades, or not?”. The table below shows the distribution of responses by party and ideology, where the counts have been replaced with relative frequencies. Response Earth is warming Not warming Don’t Know Refuse Total Conservative Republican 0.11 0.20 0.02 0.33 Party and Mod/Lib Republican 0.06 0.06 0.01 0.13 Ideology Mod/Cons Democrat 0.25 0.07 0.02 0.34 Liberal Democrat 0.18 0.01 0.01 0.20 Total 0.60 0.34 0.06 1.00 (a) Are believing that the earth is warming and being a liberal Democrat mutually exclusive? No because there are some conservative republicans, mod/lib republicans, and mod/cons democrats that also believe the earth is warming. There are also a few liberal democrats that do not believe it is warming. (b) What is the probability that a randomly chosen respondent believes the earth is warming or is a liberal Democrat? – 62% (0.6 + 0.2 - 0.18= 0.62) (c) What is the probability that a randomly chosen respondent believes the earth is warming given that he is a liberal Democrat? 90% ( 0.18/0.2 = 0.9)

(d) What is the probability that a randomly chosen respondent believes the earth is warming given that he is a conservative Republican? 33.3% (0.11/0.33 = 0.33) (e) Does it appear that whether or not a respondent believes the earth is warming is independent of their party and ideology? Explain your reasoning. – It is not independent on their party and ideology. 90% of liberal democrats believe the earth is warming as well as 33% of conservative republicans. (f) What is the probability that a randomly chosen respondent is a moderate/liberal Republican given that he does not believe that the earth is warming? – 17.65% (0.06/0.34 = .1764706) 4. An airline charges the following baggage fees: $25 for the first bag and $35 for the second. Suppose 54% of passengers have no checked luggage, 34% have one piece of checked luggage and 12% have two pieces. We suppose a negligible portion of people check more than tw o bags. (a) Build a probability model, compute the average revenue per passenger. Event X P(X) X*P(X) 0 checked bags $0 .54 0 1 checked bag $25 .34 8.5 2 checked bags $60 .12 7.2 Average rev. - $15.70 (b) About how much revenue should the airline expect for a flight of 120 passengers? $ 1 8 8 4 ( 1 5 . 7 0 * 1 2 0 ) 5. In triathlons, it is common for racers to be placed into age and gender groups. Friends Leo and Mary both completed the Triathlon, where Leo competed in the Men, Ages 30-34 group while Mary competed in the Women, Ages 25-29 group. Leo completed the race in 1:22:28 (4948 seconds), while Mary completed the race in 1 :31:53 (5513 seconds). Obviously Leo finished faster, but they are curious about how they did within their respective groups. Can you help them? Here is some information on the performance of their groups: The finishing times of the Men, Ages 30-34 group has a mean of 4313 seconds with a standard deviation of 583 seconds. The finishing times of the Women, Ages 25-29 group has a mean of 5261 seconds with a standard deviation of 807 seconds. The distributions of finishing times for both groups are approximately Normal. Remember: a better performance corresponds to a faster finish (and a shorter time). (a) Write down the short-hand for these two normal distributions. - Men’s finishing times: M ∼ N(4313,583) Women’s finishing times: W ∼ N(5261,807 (b) What are the Z-scores for Leo’s and Mary’s finishing times? What do these Z-scores tell you ? Mary -0.312 Leo – 1.089 (c) Did Leo or Mary rank better in their respective groups? Explain your reasoning. – No. they were still in the lower end of the finishing times. (d) What percent of the triathletes did Leo finish faster than in his group? - 13.8%

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