1. Discrete Vs. Continuous
For the following, indicate if the variable of interest is discrete or continuous.
a. The time it takes for high school students to run one mile.
b. The number of dogs a family has as pets.
c. The ranking (on a scale of 1-3, with answers of 1, 2, or 3) of how excited you are to finish your degree.
d. The amount of money spent by women on clothing each month.
2. Discrete Random Variables
Suppose the grading for a course project can only have 7 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:
x
100
90
80
70
60
50
0
P(X = x)
0.100
0.400
0.340
0.100
0.050
0.005
0.005
a. What is the probability that X is AT MOST 60%?
b. What is the probability that X is AT LEAST 70%?
c. What is the probability that X is MORE than 70?
d. What is the probability that X is LESS than 60?
e. Compute the expected value for this random variable.
Hint: see page 269 of text.
c. It is known that 5% of a particular brand of batteries are defective. Suppose we receive a shipment of batteries. It is too expensive to test all of the batteries so we randomly select 8 from the crate.
i. Let X be the number of defective found from the 8 chosen.
a. Is X a binomial random variable? Yes
b. Why or why not? X represents either a success or failure
ii. Calculate by hand the probability that out of the 8 batteries, exactly one will be defective. Round your answer to four decimal places.
Hint: Review pages 290 - 291 of the textbook for a demonstration of the calculation.
iii. Use software to find the probability in part (ii) above. Copy and paste your output.
Hint: Your answer should match the probability found above. If not, try again! Let me know if you need help! Remember, we are find an exact probability...exactly 1 battery is defective.
iv. Calculate by hand the probability that AT MOST 1 battery is found to be defective.
Hint: This means we can find 0 or 1 defective batteries.
v. Use software to find the probability in part (iv).
Hint: You can use information from (iii) for part of this question. Your answer should match the probability found above. If not, try again! Let me know if you need help!
vi. Calculate by hand the probability that AT LEAST one battery out of the 8 is defective.
Hint: What is the complement of AT LEAST one? You will be able to use some of the data found in (v).
4. Continuous Random Variables - Empirical Rule
Suppose the Math SAT scores (SATM) for Penn State student follow a normal distribution with an average score of 550 with a standard deviation of 30.
Using this information find intervals that contain approximately 68%, 95%, and 99.7% of the scores. Make sure to show your work for each interval.
a. 68% interval:
b. 95% interval:
c. 99.7% interval:
5. Continuous Random Variables -Finding Probabilities Using the Normal Distribution
Suppose the Math SAT scores (SATM) for Penn State student follow a normal distribution with an average score of 550 with a standard deviation of 30.
a. Find the cumulative probability that a SATM score is less than 525. Use software. Copy and paste your output.
Hint: You are finding P(X ≤ 525).
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 without a background in statistics.
c. Find the probability that a SATM score for a randomly selected student is more than 595.
Hint: You are finding P(X >595). First use software to find P(X ≤ 595) as you did in part (a). Then, use that probability to complete the question.
d. Write a sentence interpreting the value you found in (c). You should write the sentence so that it makes sense to someone without a background in statistics.
e. Using the information you found in parts (a) and (c), find the probability that a SATM score is between 525 and 595.
f. Write a sentence interpreting the value you found in the (e). You should write the sentence so that it makes sense to someone without a background in statistics.
g. Find the z-score for a SATM of 595 by hand. Round your answer to two decimal places.
Hint: See page 282 of the textbook for the formula!
h. Using the Standard Normal Table (A-1 and A-2 in the textbook) and the z-score found in (g), find the probability that a student has a score of 595 or less.
Hint: We need to find P(X ≤ 595) = P(Z ≤ answer from (g)).
i. Using software, find the probability in (h), P(X ≤ 595). Copy and paste your software output.
Hint: Your answer should match part (h). If not, try again! Contact me if you need help!
j. What is the SATM value that 10% of PSU students scored less than. In other words, find the 10th percentile for SATM scores.
Hint: First find the z-score that corresponds to a probability of 10% or 0.10 using the inside of the Standard Normal Table. Then, use the z-score found, the mean, and the standard deviation to solve for the observed value, or x, in the z-score formula on page 282.
k. Write a sentence that interprets this value.