Interpreting Center & Variability.

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Presentation transcript:

Interpreting Center & Variability

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 m & s to represent the mean & standard deviation

z score Standardized score Creates the standard normal density curve Has m = 0 & s = 1

What do these z scores mean? -2.3 1.8 6.1 -4.3 2.3 s below the mean 1.8 s above the mean 6.1 s above the mean 4.3 s below the mean

Jonathan wants to work at Utopia Landfill 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 m = 45 and s = 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

At least what percent of observations is within 2 standard deviations of the mean for any shape distribution? 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 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%

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

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

The height of male students at PWSH 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%

First, find the mean & standard deviation for the total setup time. 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 44.96 minutes? First, find the mean & standard deviation for the total setup time. Phase Mean SD Unpacking 3.5 0.7 Assembly 21.8 2.4 Tuning 12.3 2.7 2.5%