<p>Statistics 5.3 Normal Distributions: Finding Values</p><p>LEQ: How do you find a specific data value for a given probability?</p><p>Procedure:</p><p>1. Finding z-Scores:</p><p> a. Examples 1 – 4: Finding a z-Score given an area.</p><p>1. Find the z-score that corresponds to a cumulative area of 0.3632.</p><p>2. Find the z-score that has 10.75% of the distribution’s area to its right.</p><p>3. Find the z-score that has 96.16% of the distribution’s area to the right.</p><p>4. Find the z-score for which 95% of the distribution’s area lies between z and –z.</p><p> b. Examples 5 – 7: Finding a z-score given a percentile.</p><p>Find the z-score that corresponds to each percentile.</p><p>5. P5</p><p>6. P50</p><p>7. P90 2. Transforming a z-Score to an x-Value:</p><p> c. Definition 1: Transforming a z-Score to an x-Value: </p><p>To transform a standard z-score to a data value x in a given population, use the formula</p><p> d. Example 8 & 9: Finding an x-value corresponding to a z-score.</p><p>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.</p><p>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? 3. Finding a Specific Data Value for a Given Probability:</p><p> e. Example 10 & 11: Finding a specific data value.</p><p>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?</p><p>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%? f. Examples 12 & 13: Finding a specific data value.</p><p>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%.</p><p>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?</p><p>2. HW: p. 242 (3 – 45 mo3)</p>