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Explore standardized scores and z-scores to compare SAT Math scores with ACT Math scores. Understand the concept of z-scores, mean, and standard deviation. Determine qualification for a job based on z-scores. Analyze math achievement test scores and identify the test where a student performed the best. Learn about density curves and normal distribution. Use the empirical rule to estimate proportions of observations within a certain range. Apply these concepts to real-world scenarios.
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Interpreting Center & Variability
Suppose you take the SAT test and the ACT test. Not using the chart they provide, can you directly compare your SAT Math score to your ACT math score? Why or why not? We need to standardized these scores so that we can compare them.
z score Standardized score Has m = 0 & s = 1
Let’s explore . . . So what does the z-score tell you? Suppose the mean and standard deviation of a distribution are m = 50 & s = 5. If the x-value is 55, what is the z-score? If the x-value is 45, what is the z-score? If the x-value is 60, what is the z-score? 1 -1 2
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. 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
Sally is taking two different math achievement tests with different means and standard deviations. The mean score on test A was 56 with a standard deviation of 3.5, while the mean score on test B was 65 with a standard deviation of 2.8. Sally scored a 62 on test A and a 69 on test B. On which test did Sally score the best? She did better on test A.
Density Curves • Can be created by smoothing histograms • ALWAYS on or above the horizontal axis • Has an area of exactly one underneath it • Uses m & s to represent the mean & standard deviation • Describes the proportion of observations that fall within a range of values • Is often a description of the overall distribution
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 Put the following into your calculator: (Window: x: [0,20] & y: [0,0.3]) Y1: normalpdf(X,10,2) Y2: normalpdf(X,10,1.5) Y3: normalpdf(X,10,3) What happens? • Press 2nd VARS to find normalpdf(X,m,s) Let’s use our calculator to graph some 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 Can ONLY be used with normal curves!
The height of male students at SHS 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 44.96 minutes? First, find the mean & standard deviation for the total setup time. 2.5%
