University:
Concordia University Wisconsin
Course:
MATH 2050 | Statistics I
Academic year:

2020
Views:

81

Pages:

7
Author:

rezancuk2j

5.5 Z-Scores Z - Scores [5.5] Unit 4: Lesson 5 In Summary Key Ideas The standard normal distribution is a normal distribution with mean, μ, of 0 and a standard deviation, σ, of 1. The area under the curve of a normal distribution is 1. Z-scores can be used to compare data from different normally distributed sets by converting their distributions to the standard normal distribution. [Diagram of normal distribution curve with percentages and standard deviations] Class Ex. #1 A nurse records the number of hours an infant sleeps during a day. He then records the data on a normal distribution curve shown below. The values shown on the horizontal axis differ by one standard deviation. [Normal Distribution Curve Diagram] Frequency labeled on y-axis "Mean" Labelled at center (12) Horizontal axis labeled "Number of Hours Slept" with A at 6 hours, B at 8 hours, 10 hours, 12 hours, 13 hours, C at 16 hours, D at 18 hours. A, B, 10, 12, 13, 14, C, D Standard deviations labeled -3, -2, -1, 0, 1, 2, 3 percentages marked under the curve a) What is the mean of the data? 12 hours b) What is the standard deviation? 2 hrs c) What are the values for A, B, C, and D? 6, 8, 16, 18 d) What percentage of a day, to the nearest hundredth, does the infant sleep: i) between 12 and 14 h? 34% ii) between 8 and 16 h? 95% (13.5 + 34 + 34 + 13.5) iii) less than 6 h? 0.15% (written as 0.158% on image) e) Why is it not possible at this time to determine the percentage of a day that the infant sleeps for less than 13 hours? This model is limited to whole standard deviations... for 1/2 deviations we need Z-scores z-scores A z-score for a data value describes the number of standard deviations above or below the mean. Displaying z-scores on the Normal Curve Consider Class Example #1 with a mean value of 12 and a standard deviation of 2. The data value 16 is 2 standard deviations above the mean and has a z-score of 2. • The data value 6 is 3 standard deviations below the mean and has a z-score of - 3 • The data value 12 is standard deviations away from the mean and has a z-score of 0. • The data value 13 is mathfrak o .5 standard deviations above the mean and has a z-score of 05 Unit 4: Lesson 5 z-score Formula z-scores can be calculated using the formula z = (x - μ) / σ where, z is the z-score x is the particular data value μ is the mean (average) σ is the standard deviation (From example) z (subscript 13 hours) = (13-12) / 2 = 0.5 (written on the side) z-score on graph Example 2: Determine at which location Hailey’s run time was better when compared to club results. Hailey and Serge belong to a running club in Vancouver. Part of their training involves a 200 m sprint. Below are normally distributed times for the 200 m sprint in Vancouver and on a recent trip to Lake Louise. At higher altitudes, run times improve. Location Altitude (m) Club Mean Time: μ for 200 m (s) Club Standard Deviation: σ (s) Hailey's Run Time (s) Serge's Run Time (s) Vancouver 4 25.75 0.62 24.95 25.45 Lake Louise 1661 25.57 0.60 24.77 26.24

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