Review: Unit 1 Normal Distribution



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Name:_______________________ Review: Unit 1 - Normal Distribution U1 D4 c/w

  • What is the total area under the normal curve?




  • Draw two normal curves that have the same mean but different standard deviations. Describe the similarities and differences



  • Draw two normal curves that have different means but the same standard deviations. How are these curves similar?



  • What is the mean of the standard normal distribution? What is the standard deviation of the standard normal distribution?


Use the Empirical Rule for normal distributions to answer the following questions:

  • Consumer Reports Magazine wrote an article stating that monthly charges for cell phone plans in the U.S. are normally distributed with a mean of $62 and a standard deviation of $18.

a) Draw a picture of the normal curve with the cell phone charges for 1, 2 and 3 standard deviations above and below the mean.

b) What percent of people in the U.S. have a cell phone bill between $62 and $80 per month?

c) Between what two cell phone charges are the middle 95% of people?

d) What percent of people in the U.S. have a monthly cell phone bill between $26 and $44?

e) Find the cell phone bill that 99.85% of people’s bills fall below.


  • Human pregnancies are normally distributed and last a mean average of 266 days and a standard deviation of 16 days.

a) Draw a picture of the normal curve with the pregnancy lengths for 1, 2 and 3 standard deviations above and below the mean.

b) What percent of pregnancies last between 218 days and 266 days?

c) Find the range of pregnancy lengths that the middle 68% of people fall between.

d) A travelling salesman came home from a long business trip and was amazed to hear that his wife was pregnant and expecting a baby. This was especially amazing since it had been 314 days since he had last seen his wife. His wife claims that the baby is just very late in coming. What is the probability that a pregnancy would last 314 days or more? What does this tell you about the wife’s claim?



Questions requiring students to use methods other than the Empirical Rule:

  • A survey was conducted to measure the height of U.S. men. In the survey, respondents were grouped by age. In the 20–29 age group, the heights were normally distributed, with a mean of 69.2 inches and a standard deviation of 2.9 inches. A study participant is randomly selected. (Source: U.S. National Center for Health Statistics)

(a) Find the probability that his height is less than 66 inches.

(b) Find the probability that his height is between 66 and 72 inches.

(c) Find the probability that his height is more than 72 inches.

(d) Find the height of a man who is taller than 20% of adult men.




  • The lengths of Atlantic croaker fish are normally distributed, with a mean of 10 inches and a standard deviation of 2 inches. An Atlantic croaker fish is randomly selected. (Adapted from National Marine Fisheries Service, Fisheries Statistics and Economics Division)

(a) Find the probability that the length of the fish is less than 7 inches.

(b) Find the probability that the length of the fish is between 7 and 15 inches.

(c) Find the probability that the length of the fish is more than 15 inches.

(d) If 200 Atlantic croakers are randomly selected, about how many would you expect to be shorter than 8 inches?




  • In a recent year, the ACT scores for high school students with a 3.50 to 4.00 grade point average were normally distributed, with a mean of 24.1 and a standard deviation of 4.3. A student with a 3.50 to 4.00 grade point average who took the ACT during this time is randomly selected. (Source: ACT, Inc.)

(a) Find the probability that the student’s ACT score is less than 20.

(b) Find the probability that the student’s ACT score is between 22 and 27.

(c) If 500 students took the ACT, about how many would you expect to find that scored more than 29?


  • The weights of adult male rhesus monkeys are normally distributed, with a mean of 15 pounds and a standard deviation of 3 pounds. A rhesus monkey is randomly selected.

(a) Find the probability that the monkey’s weight is less than 13 pounds.

(b) Find the probability that the weight is between 15 and 17 pounds.

(c) If 50 rhesus monkeys are randomly selected, about how many would you expect to weigh less than 12 pounds?

(d) A particularly large monkey weighs in the top 1% of all monkeys. How much does he weigh?




  • A survey was conducted to measure the number of hours per week adults in the United States spend on home computers. In the survey, the number of hours was normally distributed, with a mean of 7 hours and a standard deviation of 1 hour. A survey participant is randomly selected.

(a) Find the probability that the hours spent on the home computer by the participant are less than 4.5 hours per week.

(b) Find the probability that the hours spent on the home computer by the participant are between 4.5 and 9.5 hours per week.

(c) Find the probability that the hours spent on the home computer by the participant are more than 12 hours per week.

(d) How much time would a random adult have to spend on his home computer to be in the 85% percentile of all adults?




  • The monthly utility bills in a city are normally distributed, with a mean of $100 and a standard deviation of $12. A utility bill is randomly selected.

(a) Find the probability that the utility bill is less than $80.

(b) Find the probability that the utility bill is between $80 and $115.

(c) Find the probability that the utility bill is more than $115.

(d) What percent of the utility bills are more than $125?

(e) If 300 utility bills are randomly selected, about how many would you expect to be less than $90?


  • SAT math scores are normally distributed; the mean is 519 and the standard deviation is 115.

(a) What percent of the SAT math scores are less than 500?

(b) If 1500 SAT math scores are randomly selected, about how many would you expect to be greater than 600?




  • The life span of a battery is normally distributed, with a mean of 2000 hours and a standard deviation of 30 hours. What percent of batteries have a life span that is more than 2065 hours? Would it be unusual for a battery to have a life span that is more than 2065 hours? Explain your reasoning.



  • Assume the mean annual consumption of peanuts is normally distributed, with a mean of 5.9 pounds per person and a standard deviation of 1.8 pounds per person. What percent of people annually consume less than 3.1 pounds of peanuts per person? Would it be unusual for a person to consume less than 3.1 pounds of peanuts in a year? Explain your reasoning.


Questions involving z-scores:


  • For a distribution of raw scores with a mean of 45, the Z-score for a raw score of 55 is calculated to be -2.00. Regardless of the value of the standard deviation, why must this Z-score be incorrect?




  • A distribution of scores has a standard deviation of 10. Find the z-scores corresponding to the following values:

A score that is 10 points below the mean

A score that is 30 points below the mean

 


  • On a statistics exam, you have a score of 73. If the mean of the exam is 65 would you prefer the standard deviation of the scores to be 8 or 16? Why?

 

  • A normal distribution has a mean of 120 and a standard deviation of 20. For this distribution

What score separates the top 40% of the scores from the rest?

What score corresponds to the 90th percentile?



What range of scores would form the middle 60% of this distribution?



  • A potato chip company sells a family size bag of chips in which the claimed weight of the product is 24 oz. Because it is difficult to get an exact weight on a bag of chips, we expect some variation in the weights of the bags. Machines package the chips so that the weights of the bags should be normally distributed with a mean of 24 oz. and a standard deviation of 0.27oz. You buy fifty bags of chips and find that all but five of them weigh 23.4oz or less. What would you tell this company with regards to the filling machines? Justify your response by using specific references to a normal distribution and its values.


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