Stats - Dearborn Public Schools



Statistics 5-3: 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.

C) Find the z-score for which 87.64% 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) [pic] B) [pic] C) [pic]

This means that there is

a z-score that has an area Closest area _________ Closest area _________

of _______ to its left.

Closest area _________ z = __________ z = __________

z = __________

TIY 2: Find the z-score that corresponds to each percentile.

A) [pic] B) [pic] C) [pic]

Closest area _________ Closest area _________ Closest area _________

z = __________ z = __________ z = __________

Objective 2: I can transform a 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.

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?

Step 1: Find the z-score that corresponds to the given percent.

Step 2: Use the formula and plug in ____, _____, and _____, to find x.

Step 3: Remember that x is a data value in your data set and should be labeled as such in your answer.

Ex 5: 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%?

Ex 6: 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%.

Ex 7: 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?

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