Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Unit 7: Task 4 Name ________________________________ Let’s Be Normal Learning Task Until this task, we have focused on distributions of discrete data. We will now direct our attention to continuous data. Where a discrete variable has a finite number of possible values, a continuous variable can assume all values in a given interval of values. Therefore, a continuous random variable can assume an infinite number of values. We will focus our attention specifically on continuous random variables with distributions that are approximately normal. Remember from last semester, that normal distributions are symmetric, bell-shaped curves that follow the Empirical Rule. The Empirical Rule for Normal Distributions states that 68% of the data values will fall within one standard deviation of the mean, 95% of the data values will fall within two standard deviations of the mean, and 99.7% of the data values will fall within three standard deviations of the mean. Dotplots were used to explore this type of distribution and you spent time determining whether or not a given distribution was approximately normal. In this task, we will use probability histograms and approximate these histograms to a smooth curve that displays the shape of the distribution without the lumpiness of the histogram. We will also assume that all of the data we use is approximately normally distributed. 1. This is an example of a probability histogram of a continuous random variable with an approximate mean of 5 and standard deviation of 1. A normal curve with the same mean and standard deviation has been overlaid on the histogram. a. What do you notice about these two graphs? 2. First, you must realize that normally distributed data may have any value for its mean and standard deviation. Below are two graphs with three sets of normal curves. Unit 7: Task 4 Name ________________________________ a. In the first set, all three curves have the same mean, 10, but different standard deviations. Approximate the standard deviation of each of the three curves. Tallest Curve: __________ Middle Curve:___________ Smallest Curve:_________ b. In this set, each curve has the same standard deviation. Determine the standard deviation and then each mean. Left Curve: ______ ________ Middle Curve: Right Curve: ______ ________ ______ ________ c. Based on these two examples, make a conjecture regarding the effect changing the mean has on a normal distribution. Make a conjecture regarding the effect changing the standard deviation has on a normal distribution. Unit 7: Task 4 Name ________________________________ The Standard Normal Distribution is a normal distribution with a mean of 0 and a standard deviation of 1. To more easily compute the probability of a particular observation given a normally distributed variable, we can transform any normal distribution to this standard normal distribution using the following formula: z X When you find this value for a given value, it is referred to as the z-score. The z-score is a standard score for a data value that indicates the number of standard deviations that the data value is away from its respective mean. 3. Use the formula and definition for z-score to calculate the z-score for the mean of a variable. 4. Looking at the standard normal curve below, you see that half of the area of the curve lies below the mean of the distribution, therefore P( X ) 0.5 . Another way to express this probability is P( z 0) 0.5 . a. Now let’s say that you want to know the probability of an observation falling below 15 for a variable that is normally distributed with a mean of 15 and standard deviation of 3. Unit 7: Task 4 Name ________________________________ 5. We can use the z-score to find the probability of many other events. Let’s explore those now. a. Suppose that the mean time a typical American teenager spends doing homework each week is 4.2 hours. Assume the standard deviation is 0.9 hour. Assuming the variable is normally distributed, find the percentage of American teenagers who spend less than 3.5 hours doing homework each week. First, find the z-score for X = 3.5. Now, use the method you teacher has shown you to determine the probability of P(X < 3.5). Sketch a normal curve for this situation and shade the probability you have just found on the curve. b. The average height of adult American males is 69.2 inches. If the standard deviation is 3.1 inches, determine the probability that a randomly selected adult American male will be at most 71 inches tall. Assume a normal distribution. 6. We can also use z-scores to find the percentage or probability of events above a given observation. Due to the symmetry of the curve, We can use the Empirical Rule to we can quickly deduce that the determine that the probability of probability of a value occurring above a value happening one or more the mean is 0.50. standard deviations above the mean is 0.16, therefore . P( z 1) 0.16 Unit 7: Task 4 Name ________________________________ We can also compute P ( z a ) using the Complement Rule that you first learned in Accelerated Mathematics I. P( z a) 1 P( z a ) Use this rule to compute the following probabilities. a. The average on the most recent test Mrs. Cox gave her Physics students was 73 with a standard deviation of 8.2. Assume the test scores were normally distributed. Determine the probability that a student in Mrs. Cox’s class scored an A on the test. height of at least 64 inches. 7. We can also determine the probability between two values of a random variable. P(2 z 1) P( z 1) P( z 2) P(1.5 z 0.75) ___________ Unit 7: Task 4 Name ________________________________ a. According to the College Board, Georgia seniors graduating in 2008 had a mean Math SAT score of 493 with a standard deviation of 108. Assuming the distribution of these scores is normal, find the probability of a member of the 2008 graduating class in Georgia scoring between 500 and 800 on the Math portion of the SAT Reasoning Test. b. According the same College Board report, the population of American 2008 high probability that of a randomly selected senior from this population scoring between 500 and 800 on the Math portion of the SAT Reasoning Test. c. Compare the probabilities from parts a and b. Explain the differences in the two probabilities.