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Name: _______________________________________________________________ Date: _____________________ Period: _______ Chapter 14: Probability & Statistics Topic #5: Normal Distribution Normal Distribution is a very important statistical data distribution pattern occurring in many natural phenomena, such as height, weight, blood pressure, IQ scores, SAT scores, and so on. Certain data when graphed as a histogram creates a bell-shaped curve known as a normal curve, or a normal distribution. Normal distributions are symmetrical with a single central peak at the mean (average) of the data. The shape of the curve is described as bell-shaped with the graph falling off evenly on either side of the mean. Fifty percent of the distribution lies to the left of the mean and fifty percent lies to the right of the mean. The spread of a normal distribution is controlled by the standard deviation, 𝜎. The smaller the standard deviation the more concentrated the data. The mean and the median are the same in a normal distribution. Testing Tips: - If you see the words “normally distributed” in a Regents problem, you should consider using the normal curve graph to solve the problem. Data is more “consistent” when the standard deviation is smaller. Utilize the curve on the reference sheet. Mark it up as much as you need to (in pencil, in case you need it for another question!) Name: _______________________________________________________________ Examples: Date: _____________________ Period: _______ 1. If the scores of the 40 students on the math test are normally distributed with a mean of 85 and a standard deviation of 2: ̅ = 𝟖𝟓 𝑿 𝝈=𝟐 79 81 83 85 87 89 91 a) What is the probability that a student will receive between an 84 and an 87 on the test? 19.1% + 19.1% + 15.0% = 53.2% b) How many students got above an 88 on the test? 4.4% + 1.7% + 0.5% + 0.1% = 6.7% 6.7% of 40 students = 2.68 Two students got above an 88 on the test 2. The NuBolt Company manufactures nuts and bolts. The measures of the diameters of the bolts manufactured produce a normal distribution. The mean size of a certain bolt is 3 centimeters, with a standard deviation of 0.1 centimeters. Bolts that vary from the mean by more than 0.3 centimeters cannot be sold. If the company manufactures 150,000 of the 3-centimeter bolts, approximately how many of them cannot be sold? 3. In a study of 200 students at St. Francis Prep, the researchers found that the hours that these students studied a week were normally distributed, with a mean of 4.5 hours and a standard deviation of 1 hour. Determine if there were 6 students who studied less than 2.5 hours. Name: _______________________________________________________________ Date: _____________________ Period: _______ 4. In a national sit-up test with a normal distribution, the mean was 42 sit-ups per minute and the standard deviation was 2.6. Which score could be expected to occur less than 5% of the time? (1) 50 (2) 45 (3) 39 (4) 37 5. Each year, the College Board publishes the mean SAT score and the standard deviation for the students taking the test. SAT scores are normally distributed. Assume for a group of students that the mean SAT score is 500 with a standard deviation of approximately 100 points. In 2007, approximately 1,500,000 students took the SAT. Approximately how many of them would be expected to score: a. between 400 and 450 b. between 500 and 550 c. between 650 and 750 d. between 300 and 400 6. The mean score in a national jump rope competition is 82.75 jumps per minute and the standard deviation is 2.25. If the scores were normally distributed, which of the following scores would be most likely to occur? (1) 90 (2) 87.25 (3) 80.5 (4) 77 7. The number of hours students watch TV a week has a normal distribution with a mean of 20.5 hours and a standard deviation of 3 hours. If Dylan watches 25 hours of TV a week, what percentile will he lie in?