Statistics5-3: Normal Distributions—Finding Values
Objective 1: I can find a z-score given the area under the standard normal curve.
In section 5.2, we found the ______that a given ______would fall into a given ______by finding the area under the standard normal curve. But what if we are given a ______and need to find a corresponding value?
Example 1:
A) Find the z-score that corresponds to a cumulative area of 0.3632.
B) Find the z-score that has 10.75% of the distribution’s area to its right.
TIY 1:
A) Find the z-score that has 96.16% of the distribution’s area to the right.
B) Find the z-score for which 95% of the distribution’s area lies between –z and z.
We can also find z-scores that correspond to any percentile. Recall from chapter 2 that the nth percentile contains n% of the area under the standard normal curve to its ______.
For example, if a child scores in the 83rd percentile, then
Example 2: Find the z-score that corresponds to each percentile.
A) B) C)
TIY 2: Find the z-score that corresponds to each percentile.
A) B) C)
Objective 2: I can transforma z-score into an x-value.
Recall that to transform an x-value from a data set that is normally distributed, we use the formula:
If we know the z-score and need to work backwards to find the x-value, we can transform that formula and solve it for x.
Example 3: The speeds of vehicles along a stretch of highway are normally distributed, with a mean speed of 67 mph and a standard deviation of 4 mph. Find the speeds x corresponding to z-scores of 1.96, -2.33, and 0. Interpret your results.
TIY 3: 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?
Objective 3: I can find a specific data value for a given probability.
You can also use the standard normal distribution to find a specific data value, or the ______, for a given probability. When a college says they take only the top 5% of applicants based on ACT scores, what does that mean? How do you know what score you need? This is what we are going to look at in this objective.
Example 4:
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?
TIY 4: The braking distances of a sample of Honda Accords are normally distributed. On a dry surface, the mean braking distance was 142 feet and the standard deviation was 6.51 feet. What is the longest braking distance on a dry surface one of these Accords could have and still be in the top 5%?
Example 5: In a randomly selected sample of 1169 men, the mean cholesterol level was 210 mg/dl with a standard deviation of 38.6 mg/dl. Assume that cholesterol levels are normally distributed. Find the highest cholesterol level a man could have and be in the lowest 1%.
TIY 5: 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?