# STAT Lab Activity 4 – Probability Distributions (Solution) done june 2015

| October 3, 2018

Lab Activity 4 – Probability DistributionsNotes:· Lab Activity 4 does not require a dataset· Lab Activity 4 does require Minitab/SPSSI. Discrete Vs. Continuous1. (4 points)For the following, indicate if the variable of interest is discrete or continuous and why? If it is neither, please indicate as such.a) The amount of coffee (ounces) someone drinks in one day.b) The length of time it takes for a package to arrive from a particular website.c) The number of classes a student missed during the Spring semester.d) The number of questions one got correct on a 20 question multiple choice quiz.II. Discrete Random Variables2. (5 points)General Discrete Random VariableSuppose the grading for a course project can only have 6 possible outcomes; 100, 90, 80, 70, 60, 50, and 0. Let the random variable X be the scores on the project. Suppose X has the following distribution:x10090807060500P(X=x)0.050.350.30.10.10.080.02a. What is the probability that X is AT MOST 70?b. What is the probability that X is AT LEAST 80?c. What is the probability that X is MORE than 80?d. What is the probability that X is LESS than 70?e. Compute the expected value for this random variable (see page 269 of text).3. (10 points)Binomial Random Variablea) Write down the criteria for a random variable to be a Binomial random variable. There are 3 criteria shown in the text and the fourth criteria is that each trial has 2 possible outcomesb) Indicate below if the variable of interest is a Binomial random variable.i. A box contains 10 red balls and 5 green balls. Seven balls are chosen at random, without replacement, and their color noted. We are interested in the number of green balls chosen out of the 7.ii. A fair die is rolled 15 times and the number of 5s out of the 15 rolls is recorded. Assume the rolls are independent.iii. The Heights of college-aged women are recorded and the interest is to compare heights of women by their GPA.c) It is known that 5% of a particular brand of fuses is defective. Suppose we receive a shipment of fuses. It is too expensive to test all of the fuses so we randomly select 10 fuses from the crate.i. Let X be the number of defective fuses found from the 10 chosen. Is X a Binomial random variable?ii. Calculate by hand the probability that out of the 10 fuses, exactly one fuse will be defective. Show all work (calculations) to receive creditiii. Use software to find the probability in part ii. Copy and paste your computer outputiv. Calculate by hand the probability that AT MOST 1 fuse is found to be defective. Hint: This means we can find 0 or 1 defective fuses. Show all work (calculations) to receive credit.v. Use software to find the probability in part iv. Copy and paste your computer output for credit.vi. Calculate by hand the probability that AT LEAST one fuse out of the 10 is defective. Hint: What is the complement of AT LEAST one? Show all work (calculations) for credit.III. Continuous Random Variables4. (3 points)Empirical RuleSuppose the Math SAT scores (SATM) for Penn State students follow a normal distribution with an average score of 600 with a standard deviation of 90. Assume that SATM scores can go higher than 800 and compute your intervals, etcUsing this information find intervals that contain approximately 68%, 95%, and 99.7% of the scores. Make sure to show your work (calculations) for each interval.5. (11 points)Suppose the Math SAT scores (SATM) for Penn State students follow a normal distribution with an average score of 600 with a standard deviation of 90. Find probabilities and percentiles using the Normal Distributiona) Find the cumulative probability that a SATM score is less than 500. You should use Minitab/SPSS to find this. Hint: You are finding P(X < 500) which is labeled in the software as P(X ? 500).b) Write a sentence interpreting the value you found in the previous part. You should write the sentence so that it makes sense to someone not taking a statistics course. Note that for continuous random variables, the cumulative probability is the probability up to but not including the value.c) Find the probability that a SATM score for a randomly selected student is more than 630. To find this probability you should proceed as you did for (a) and use that to find the requested probability. Hint: You are finding P(X > 630).d) Write a sentence interpreting the value you found in the (c). You should write the sentence so that it makes sense to someone not taking a statistics course.e) Using the information you found in parts (a) and (c), find the probability that a SATM score is between 500 and 630.f) Write a sentence interpreting the value you found in part (e). You should write the sentence so that it makes sense to someone not taking a statistics course.g) Find the z-score for a SATM of 700.h) Using the Normal table and the z-score, find the probability that a student has a score less than 700.i) Find this probability using software. Copy and paste your computer output.j) Suppose we want to find the SATM score that 20% of the PSU students scores less than. In other words, find the 20th percentile for SATM scores. You may use software to find your answer. Copy and paste your computer output.k) Write a sentence that interprets this value.

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