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Statistical Analysis on the TI-84 Finding the mean, median, standard deviation, range and mode of a set of data. Press the STAT button, and select EDIT Now, enter the following numbers into L1. 5,10,7,5,8,9,10,12,15,17,13,12 Statistical Analysis on the TI-84 Once the numbers have been entered, go to STAT again. Select STAT, then arrow to CALC, enter 1, then press Enter 3 times This calculates 1-variable stats You will see the following information: 𝑥 = 10.25 (this is the mean) ∑x = 123 (this is the sum of our list) ∑x2 = 1415 (we won’t use this) sx = 3.744693215 (stan. dev. of a sample) σx = 3.585270794 (stan. dev. of a population) Statistical Analysis on the TI-84 n = 12 (number of items – sample space) Arrow down for more info!! Min X = 5 (smallest value) Q1 (median of top half of data) Med = 10 (median of all data) Q3 (median of bottom half of data) Max X = 17 (largest value) To find the range, simply subtract Min X from Max X (in our case, 17 - 5 = 12) Statistical Analysis on the TI-84 The values that we will use for now are: 𝑥 = 10.25 σx = 3.58 (rounded) MinX = 5 MaxX = 17 Statistical Analysis on the TI-84 If you are going to need to find the mode, you would want to sort the data. STAT - 2 gets you to the SORT A function. Type in 2nd 1 (to sort L1) and press Enter It will say DONE. This will put the numbers into ascending order. STAT - 3 does the same thing, only in descending order. Apply to a Normal Distribution Applying this data to a normal distribution Also called a bell curve. 2.77 6.51 -3.74 10.25 -3.74 13.99 17.73 + 3.74 + 3.74 Standard Deviation Gap: Distance between each line from the mean Apply to a Normal Distribution Percentages that hold for ALL normal distributions. 34.1% 2.15% 13.6% 34.1% 13.6% 2.15% 68.2% of data occurs between -1 and 1 s.d. 95.4% of data occurs between -2 and 2 s.d. 99.7% of data occurs between -3 and 3 s.d. 𝑥−𝜇 𝑧= 𝜎 Apply to a Normal Distribution Z-Scores Used when we want to find the percentage of data above or below numbers that do not fall exactly on a standard deviation line. The number we are interested in minus the mean divided by the standard deviation. Normal Distribution The heights of adult American males are normally distributed with a mean of 69.5 inches and a standard deviation of 2.5 inches. What percent of adult American males are between 67 and 74.5 inches tall? What are the z-scores? (67-69.5)/2.5 and (74.5-69.5)/2.5 -1 and 2 What percentage of data points lie between -1 and 2 standard deviations? 34.1 + 34.1 + 13.6 = 81.8% μ = 69.5 σ = 2.5 64.5 .15% 67.0 69.5 72.0 74.5 .15% Normal Distribution In a group of 2000, about how many would you expect to be taller than 6 feet (72 inches)? What is the z-score? (72-69.5)/2.5 = ? z-score is 1. What percentage of data points will lie above 1 standard deviation from the mean? 13.6 + 2.15+.15 = 15.90% μ = 69.5 σ = 2.5 64.5 .15% 67.0 69.5 72.0 74.5 .15% Normal Distribution In a group of 2000, about how many would you expect to be taller than 6 feet (72 inches)? Notice that the question doesn’t ask for the percentage, it asks for the number of men out of a group of 2000. We would take the 2000 times the percentage to get our guess. 2000*.159 = 318 Normal Distribution Josh’s and Richard’s Algebra II grades for the third cycle have the same mean (70). But Josh’s standard deviation is 15.6, and Richard’s is 20.1. What does the data tell us about their grades? Richard σ = 20.1 29.8 38.8 49.9 70 54.4 70 Josh σ = 15.6 90.1 85.6 110.2 101.2 Josh was more consistent; Richard less so Using the Calculator: Example 1: The weights of newborn humans are normally distributed about the mean, 3250 grams. The standard deviation is 500 grams. Find the probability that a baby chosen at random weighs between 2250 and 4250 grams. 3250 2250 4250 95.4% of babies will be in the given range. Using the Calculator: Example 1: 2nd VARS 2 – Lower number = 2250 Upper number = 4250 Mean = 3250 S = 500 p = 0.954 Example 2: A company manufactures batteries having a lifespan that are normally distributed with a mean of 45 months and a standard deviation of 5 months. Find the probability that a battery chosen at random will have a lifespan of 50-55 months. Step 1: Find the lower and upper z-scores. 50−45 55−45 , 5 5 The z-scores are 1 and 2. 45 50 55 13.6% chance that the battery will have a lifespan between 50-55 months. Using the Calculator: Example 2: 2nd VARS 2 – Lower number = 50 Upper number = 55 Mean = 45 S= 5 p = 0.136 Stat – Edit – Enter numbers into L1 Stat – Calc – 1-Var stats Mean is 9.46 Standard Deviation (population) is 3.74 Mean is 9.46 Standard Deviation (population) is 3.74 Based on these numbers, how likely is it that the next snowfall of more than 6 inches is between 10 and 15 inches? 2nd VARS - 2 Lower - 10 Upper - 15 Mean - 9.46 s = 3.74 p = .373, or 37.3% chance Today's Assignment: Pages 737-738; #7-28 and 30-31 (Skip 29)