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Section 5.3 Normal Distributions: Finding Values Examples 1 & 2 1. Find the z-score that corresponds to a cumulative area of 0.3632. 2. Find the z-score that has 10.75% of the distribution’s area to its right. Example 3 & 4 3. Find the z-score that has 96.16% of the distribution’s area to the right. 4. Find the z-score for which 95% of the distribution’s area lies between z and –z. Examples 5 – 7 5. P5 6. P50 7. P90 Transforming a z-Score to an xValue Definition 1: Transforming a z-Score to an x-Value: To transform a standard z-score to a data value x in a given population, use the formula 𝑥 = 𝜇 + 𝑧𝜎 Example 8 The speeds of vehicles along a stretch of highway are normally distributed, with a mean of 56 miles per hour and a standard deviation of 4 miles per hour. Find the speeds x corresponding to z-scores of 1.96, 2.33, and 0. Interpret your results. 63.84 is above the mean, 46.68 is below the mean, 56 is the mean. TOTD Use the Standard Normal Table to find the z-score that corresponds to the given cumulative area or percentile. 𝑃35 Find the indicated z-score. Find the z-score that has 78.5% of the distribution’s area to its right. Example 9 The monthly utility bills in a city are normally distributed, with a mean of $70 and a standard deviation of $8. Find the x-values that correspond to z-scores of -0.75, 4.29, and -1.82. What can you conclude? Negative z-scores represent bills that are lower than the mean. Example 10 Scores for a civil service exam are normally distributed, with a mean of 75 and a standard deviation of 6.5. To be eligible for civil service employment, you must score in the top 5%. What is the lowest score you can earn and still be eligible for employment? The lowest score you can earn and still be eligible for employment is 86. Example 11 The braking distances of a sample of Ford F-150s are normally distributed. On a dry surface, the mean braking distance was 158 feet and the standard deviation was 6.51 feet. What is the longest braking distance on a dry surface one of these Ford F-150s could have and still be in the top 1%? The longest breaking distance on a dry surface for an F-150 in the top 1% is 143 ft. Example 12 In a randomly selected sample of 1169 men ages 35-44, the mean total cholesterol level was 205 milligrams per deciliter with a standard deviation of 39.2 milligrams per deciliter. Assume the total cholesterol levels are normally distributed. Find the highest total cholesterol level a man in this 35-44 age group can have and be in the lowest 1%. The value that separates the lowest 1% of total cholesterol levels for men in the 35 – 44 age group from the highest 99% is about 114. Example 13 The length of time employees have worked at a corporation is normally distributed, with a mean of 11.2 years and a standard deviation of 2.1 years. In a company cutback, the lowest 10% in seniority are laid off. What is the maximum length of time an employee could have worked and still be laid off? The maximum length of time an employee could have worked and still be laid off is 8.5 years. TOTD Find the indicated area under the standard normal curve. To the right of z = 1.645 Between z = -1.53 and z = 0