10A The Normal Distribution, 10B Probabilities Using a Calculator Unit 3: Statistical Applications 10A, 10B 4/9/2019 2:27 PM

Scenario Weights of oranges may vary due to several factors, including: genetics different times when the flowers were fertilized different amounts of sunlight reaching the leaves and fruit different weather conditions such as the prevailing winds The result is that most of the fruit will have weights close to the mean, while fewer oranges will be much heavier or much lighter. If you collect enough weight measurements, then the data will likely represent a normal distribution. 10A, 10B 4/9/2019 2:27 PM

Normal Distribution the most important distribution for quantitative continuous variables continuous variable X uppercase represents all data (population) x lowercase represents a piece of data many naturally occurring phenomena have a distribution that is normal, or approximately normal physical attributes of a population (height, weight, and arm length) crop yields scores for tests taken by a large population weights, lengths, or quantities for items manufactured or assembled by machines yields a bell-shaped distribution which is symmetric about the mean 10A, 10B 4/9/2019 2:27 PM

Histograms 10A, 10B 4/9/2019 2:27 PM

Empirical Rule 10A, 10B 4/9/2019 2:27 PM

Notation Copy If a continuous variable X is normally distributed (bell-shaped) and the parameters are mean: “mu” standard deviation: “sigma” then we say: and its probability density function is: 10A, 10B 4/9/2019 2:27 PM

mean: “mu” standard deviation: “sigma” then we say: and its probability density function is: 4/9/2019 2:27 PM

Probabilities using a Calculator Copy Suppose How would we find since 8 and 11 are not multiples of one standard deviation either below or above the mean? On Calculator: 2nd > Distr > normalcdf( TI-83+ If lower bound is then use -1E99 If upper bound is then use 1E99 YOU MUST SHOW WORK BY DRAWING, LABELING, AND SHADING A DIAGRAM “CALCULATOR SPEAK” NOT ALLOWED IN WRITTEN WORK 10A, 10B 4/9/2019 2:27 PM

Example The chest measurements of 18 year old footballers are normally distributed with a mean of 95 cm and a standard deviation of 8 cm. (a) Find the percentage of footballers with chest measurements between: 87 cm and 103 cm 103 cm and 111 cm (b) Find the probability that the chest measurement of a randomly chosen footballer is between 87 cm and 111 cm. (c) A football league consists of 39 players who are 18 years old. Estimate the number of players that have a chest measurement between 87 cm and 111 cm. 10A, 10B 4/9/2019 2:27 PM

Common Percentages Copy to obtain with GDC, use the standard normal distribution 10A, 10B 4/9/2019 2:27 PM

Practice p. 303: 2,3,5,6,7 p. 307: 3,6,8 Read and follow all instructions. List the page and problem numbers alongside your work and answers in your notes. Use the back of the book to check your answers. Copy 10A, 10B 4/9/2019 2:27 PM