AP Statistics Monday, 21 September 2015 OBJECTIVE TSW examine density curves, z-scores, Chebyshev’s Rule, normal curves, and the empirical rule. ASSIGNMENT DUE (black tray) –WS Comparative Box Plots

Week of 09/21 – 25/2015 MONDAYInterpreting Center & Variability WS Mean and Variance TUESDAYWS Interpreting Center & Variability QUIZ: Means, Boxplots & Variability WEDReview WS AP Review FRIDAYTEST: Measuring and Interpreting Variability MON, 09/28Sampling Design TUES, 09/29Sampling Design WED, 09/30Experimental Design

Density Curves Can be created by smoothing histograms ALWAYS on or above the horizontal axis Has an area of exactly one underneath it Describes the proportion of observations that fall within a range of values Is often a description of the overall distribution Uses  &  to represent the mean & standard deviation

z score Standardized score Creates the standard normal density curve Has  = 0 &  = 1

What do these z scores mean?    below the mean 1.8  above the mean 6.1  above the mean 4.3  below the mean

Jonathan wants to work at Utopia Landfill. He must take a test to see if he is qualified for the job. The test has a normal distribution with  = 45 and  = 3.6. In order to qualify for the job, a person can not score lower than 2.5 standard deviations below the mean. Jonathan scores 35 on this test. Does he get the job? No, he scored 2.78 SD below the mean.

Chebyshev’s Rule The percentage of observations that are within k standard deviations of the mean is at least where k > 1 can be used with any distribution At least what percent of observations is within 2 standard deviations of the mean for any shape distribution? 75%

Chebyshev’s Rule - what to know Can be used with any shape distribution Gives an “At least...” estimate For 2 standard deviations – at least 75%

The scores on a statistics quiz are negatively skewed with a mean of 82 and standard deviation of 10. What can I say about the number of students who scored less than 60 on the quiz? z =  2.2, Chebyshev’s Rule: % Conclusions: At most 20% of the students scored less than 60. Students who scored less than 60 are at most in the 20 th percentile. At least 80% of the students scored higher than a 60.

Normal Curve Bell-shaped, symmetrical curve Transition points between cupping upward & downward occur at  +  and  –  As the standard deviation increases, the curve flattens & spreads As the standard deviation decreases, the curve gets taller & thinner

Empirical Rule Approximately 68% of the observations are within 1  of  Approximately 95% of the observations are within 2  of  Approximately 99.7% of the observations are within 3  of  See p. 181 Can ONLY be used with normal curves!

The height of male students at JVHS is approximately normally distributed with a mean of 71 inches and standard deviation of 2.5 inches. a) What percent of the male students are shorter than 66 inches? b) Taller than 73.5 inches? c) Between 66 & 73.5 inches? About 2.5% About 16% About 81.5%

Remember the bicycle problem? Assume that the phases are independent and are normal distributions. What percent of the total setup times will be more than minutes? PhaseMeanSD Unpacking Assembly Tuning About 2.5% First, find the mean & standard deviation for the total setup time.