Math 1240 Activity #4 Fall 2024 PDF

## SOLUTIONS ### MATH 1240 - Activity #4 **Fall 2024** 0. Write down the name of everyone in your group, along with their favorite musician/band. Matt McMullen - Built to Spill 1. Frequentist probability. What is the probability that a thumb tack dropped on a hard surface lands point up? Use the...

## SOLUTIONS ### MATH 1240 - Activity #4 **Fall 2024** 0. Write down the name of everyone in your group, along with their favorite musician/band. Matt McMullen - Built to Spill 1. Frequentist probability. What is the probability that a thumb tack dropped on a hard surface lands point up? Use the frequentist approach to try and estimate this probability. Specifically, conduct this experiment 30 times and record your results (point up or on its side) below. What percentage of your drops landed point up? $P(point \ up) = \frac{ \# \ of \ times \ it \ was \ point \ up}{ \# \ of \ times \ it \ was \ dropped}$ 2. Probability basics. Let A be the event that you go to your favorite class tomorrow; let B be the event that it rains tomorrow; let C be the event that you skip all your classes tomorrow; and let D be the event that Joe Biden is president tomorrow. (a) Which two events are mutually exclusive? Explain. A and C, it's impossible for both of these events to happen at the same time. (b) Are A and C complementary events? Explain. No. They are not exact opposites (unless you only have one MWF class). (c) Which two of the above events are independent? Which two of the above events are dependent? In both cases, explain your reasoning. D and (A or B or C) are independent since the odds Biden is president tomorrow won't change if events A, B, or C happen. B and (A or C) are dependent since if it rains, you may be less likely to go to class. ### 3. Good luck! The probability of winning something when you buy a certain scratch-off lottery ticket is 0.2. You buy 80 of these tickets, scratch them off, and count how many won something. (a) Explain why this is a binomial experiment. 80 identical trials. Either win or lose each time. P(win) is the same each time. $x = \# \ of \ wins$ $n=80$ $p = .2$ $q = 1 - p = .8$ (b) Find the mean of the resulting binomial distribution. Interpret your answer. Be specific! $mean = np = 80(.2) = 16$. On average, in the long run, 16 tickets will win. (c) Why is it reasonable to assume that the distribution is approximately normal? Since $np = 16 \geq 15$ and $nq = 64 \geq 15$. (d) Find the standard deviation of this distribution, and use it, along with the mean, to sketch an approximate picture of the distribution. (Think Empirical Rule!) $s.d. = \sqrt{npq}$ $s.d. \approx 3.6$ *(Insert image of bell curve with mean = 16 and standard deviation = 3.6)* (e) Suppose that 20 of your tickets won something. Is this unusual? Use a z-score to answer this question. $z = \frac{20 - 16}{3.6} = 1.11 \mathbb{1} \ not \ unusual \ (since \ z < 2)$ ### 4. Shocking! In 1998, psychologists at FSU examined the effects of alcohol on the reactions of people to a threat. After the subjects reached a certain BAC, the psychologists placed them in a room and threatened them with electric shocks! The mean and standard deviation of the subjects' startle responses, *x*, were 37.9 and 12.4 milliseconds, respectively. Assume that these startle responses are normally distributed. (a) The probability that *x* is between 40 and 50 milliseconds is 0.2690. Represent this by a picture, drawn roughly to scale. *(Insert image of a bell curve centered at 37.9 with an area under the curve from 40 to 50 labeled .2690)* (b) What is the probability that a subject's startle response will be less than 50.3 milliseconds? (Use your above picture!) $34 + 34 + 13.5 + 2.5 = 84\%$ (c) The 90th percentile is represented by a z-score of 1.28. What is the corresponding startle response for a person in this study? Draw a picture representing this, drawn roughly to scale. $\frac{x - 37.9}{12.4} = 1.28$ $ x = 53.8$ *(Insert image of a bell curve centered at 37.9 with the area under the curve to the left of 53.8 labeled 90%)*