Finding values Given a Probability Chapter 5: The Normal Distribution 5.3: Normal Distribution: Finding Values Objectives: Find a z-score given the area under the normal curve Transform a z-score to an x-value Find a specific data value of a normal distribution given the probability Finding values Given a Probability In section 5.2 we were given a normally distributed random variable ____and we were asked to find a _____________________. In this section, we will be given a _____________ and we will be asked to find the value of the random variable _____. Example 1: Finding a z-Score Given an Area Find the z-score that corresponds to a cumulative area of 0.3632.
Example 2: Finding a z-Score Given an Area Find the z-score that has 10.75% of the distribution’s area to its right. Example 3: Finding a z-Score Given a Percentile Find the z-score that corresponds to P5.
Transforming a z-Score to an x-Score To transform a standard z-score to a data value x in a given population, use the formula and solve for x. Example 4: Finding an x-Value The speeds of vehicles along a stretch of highway are normally distributed, with a mean of 67 miles per hour and a standard deviation of 4 miles per hour. Find the speeds x corresponding to z-sores of 1.96, -2.33, and 0. Compare your results with the mean. Example 5: Finding an x-Value The monthly utility bills in a city are normally distributed, with 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. Compare your results with the mean. You can also use the normal distribution to find a ____________ _________ ______________ (___-________) for a _________ ___________________.
Example 6: Finding a Specific Data Value 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? Example 7: Finding a Specific Data Value The breaking distances of a sample of Honda Accords are normally distributed. On a dry surface, the mean breaking distance was 142 feet and the standard deviation was 6.51 feet. What is the longest breaking distance on a dry surface one of these Honda Accords could have and still be in the top 1%? Example 5: Finding a Specific Data Value In a randomly selected sample of 1169 men ages 35-44, the mean total cholesterol level was 210 milligrams per deciliter with a standard deviation of 38.6 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%. Example 6: Finding a Specific Data Value 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?