Density Curves Questions Ch1

Math 2311 Sections 4.1, 4.2 and 4.3 4.1 - Density Curves What do we know about density curves? area under curve = 1 Example: Suppose we have a density curve defined for x≥0 defined by the line y = x. Sketch: coordinate plane with a line y = x drawn from the origin. The area under the line from x = 0 to x = sqrt(2) is shaded. What percent of observations fall below 1? .5 What is the probability that x≥5? 0 Where is the median of this curve? coordinate plane with a line y = x drawn from the origin. The area under the line from x = 0 to x = 1 is shaded. A = .5 1/2 x^2 = .5 x^2 = 1 x = 1 4.2 - The Normal Distribution A density curve that is symmetric, single peaked and bell shaped is called a normal distribution. The normal distribution with mean μ and standard deviation σ is represented by N(μ, σ).   The Empirical Rule: The Empirical Rule states if a distribution has a normal distribution, Approximately 68% of all observations fall within one standard deviation of the mean. Approximately 95% of all observations fall within two standard deviations of the mean.   Approximately 99.7% of all observations fall within three standard deviations of the mean.   bell curve with mean μ and standard deviations marked at μ ± σ, μ ± 2σ, and μ ± 3σ. Percentages of 68%, 95%, and 99.7% are labeled within the corresponding intervals. Example: The length of time needed to complete a certain test is normally distributed with mean 60 minutes and standard deviation 10 minutes. What is the probability that someone will take between 40 and 80 minutes to complete the test? Sketch the distribution and shade in the area in question.   bell curve with mean 60 and standard deviations marked at 40, 50, 70, and 80. The area between 40 and 80 is shaded. The label “.95” is written above the shaded area. Find the interval that contains the middle 68% of completion times for all people taking the test. [50, 70] What percent of people take more than 80 minutes to complete the test?   .025 What if our values are not exactly within one, two or three standard deviations from the mean? Probabilities for these can still be found a number of ways, one of which we will explore in the next section. Using R, the probability can be found with the command pnorm(x, μ, σ). Note that the command in R only gives the probability that X is less than a given value. If we need to find the probability that X is greater than the given value, we will need to subtract the answer from 1.   With the TI-83 and TI-84 calculator, the command is normalcdf(lower_limit, upper_limit, μ, σ).   Continuing the example above: What is the probability that someone will take less than 45 minutes to complete the test? P(X<45) = normalcdf(-10000000,45,60,10) = 0.0668 = pnorm(45,60,10)   What is the probability that someone will take more than 30 minutes to complete the test? P(X>30) = normalcdf(30,100000000,60,10) = 1 - P(X≤30) = 1 - pnorm(30,60,10) = 0.999 How long would it take someone to finish the test if they are in the top 10% of the times? 72.8 = qnorm(0.9,60,10) Popper 07 If a group of students have test scores that are normally distributed with a mean of 82 and a standard deviation of 4, half of the students made below a grade of: a. 78 b. 74 c. 82 d. 70 e. none of these   If a student made a grade of 86, what is their percentile rank? a. 68th percentile b. 84th percentile c. 90th percentile d. 52nd percentile e. none of these