Homework 7

Please complete all of these assignments, to be submitted all together on Canvas by Monday, March 24. Number each assignment clearly (#1, #2, #3, etc). Write clearly or type your answers and where applicable, show your work. The daily breakdown is merely a guideline, but in my experience you will successfully retain more knowledge and increase your skills significantly if you try to work about 10-20 minutes per day rather than cramming a week’s worth of assignments into one or two hours the day before it’s due.

1. (5 points) 

• A dataset has a mean of 50 and a standard deviation of 8. Find the z-score for a value of 58.
• A student scores 77 on a test where the mean is 65 and the standard deviation is 7. What is the z-score?
• The average height of a basketball team is 6’2” with a standard deviation of 4 inches. A player is 6’5”. What is their z-score?

2. (5 points) 

• A factory produces screws with a mean length of 2cm and a standard deviation of 0.4cm. Find the standard error for a sample of 25 screws.
• A college student studies an average of 15 hours per week with a standard deviation of 4 hours. What is the standard error for a sample of 40 students?
• The average monthly rent in a city is $1,200 with a standard deviation of $200. If a random sample of 50 apartments is taken, what is the probability that the sample mean rent is above $1,250?

3. (5 points)

• A class’s test scores have a mean of 78 and a standard deviation of 12. Find the z-score for a student who scored 90.
• A sample of 64 students is taken from a population where the mean SAT score is 1,100 with a standard deviation of 250. Find the standard error.
• A company’s customer service response time is normally distributed with a mean of 8 minutes and a standard deviation of 2 minutes. What is the probability that a randomly selected customer waits more than 10 minutes?
• The lifespan of a certain brand of lightbulbs follows a normal distribution with a mean of 1,000 hours and a standard deviation of 100 hours. What proportion of lightbulbs last between 900 and 1,200 hours?

4. (5 points)

• A standardized test has a mean score of 500 and a standard deviation of 100. A student scores 650 on the test. Find the z-score for this student’s score.
• The average height of adult women in a city is 64 inches with a standard deviation of 3 inches.
  • What is the z-score for a woman who is 70 inches tall?
  • What is the z-score for a woman who is 60 inches tall?
• The average IQ score is 100 with a standard deviation of 15. Assume IQ scores follow a normal distribution.
  • Using the empirical rule (68-95-99.7 rule), what percentage of people have an IQ between 85 and 115?
  • What percentage of people have an IQ above 130?

5. (5 points) 

• Decide whether each scenario represents theoretical or experimental probability:
  • A coin is flipped 100 times, and it lands on heads 43 times.
  • The probability of rolling a 5 on a fair 6-sided die is 1/6.
  • A spinner with 4 equal sections (red, blue, green, and yellow) is spun 50 times, and it lands on red 18 times.

• Two fair 6-sided dice are rolled. Find the probability of:

  • Rolling a sum of 7.
  • Rolling a sum less than 5.
  • Rolling doubles (both dice show the same number).

6. (5 points)

• A bag contains 5 red, 3 blue, and 2 green marbles. If you randomly pick one marble, what is the probability of selecting:
  • A red marble?
  • A blue marble?
  • Not a green marble?
• A fair 6-sided die is rolled. What is the probability of:
  • Rolling a 4?
  • Rolling an even number?
  • Rolling a number greater than 2?