Chapter 5: The Normal Curve
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The Standard Normal Curve

The normal distribution is the bell curve (or normal curve). The standard normal curve is the normal curve with a mean of zero and a standard deviation of one.

The probability that an observation falls into an interval I will be computed as the area of interval I under the normal curve.

The probability of an observation between 0 and z is 40%.

What is the probability of an observation greater than z?

If the probability of an observation less than z is 80% then what is the probability of an observation greater than z?

 

The probability of an observation less than z is 15%. 

What is the probability of an observation greater than z?

What is the probability of an observation between 0 and z?

 

 

 

Using the TI-83 to Compute Probabilities on the Normal Curve

 

 

What is the probability of an observation between 0 and 1.34 on the standard normal curve?

 

What is the probability of an observation less than z=1.34 on the standard normal curve?

 

 

 

 

Find the probability that an observation falls between 0 and .78 on the standard normal curve.

 

Find the probability that an observation falls between -.35 and 0 on the standard normal curve.

 

 

 

Find the probability that an observation falls between -1.62 and .59 on the standard normal curve.

 

 

 

 

Find the probability that an observation falls above 1.27 on the standard normal curve.

 

 

 

 

Find the probability that an observation falls below 1.83 on the standard normal curve.

 

 

 

 

Find the probability that an observation falls above -.58 on the standard normal curve.

 

 

Find the probability that an observation falls between .35 and 1.78 on the standard normal curve.

 

 

 

 

 

Find the 82nd percentile in the standard normal curve.

 

 

 

 

Find the 35th percentile in the standard normal curve.

 

 

 

 

What is the positive value z such that 33% of the standard normal curve falls between 0 and z?

 

Homework

Section 5-2:  9-28, 33-39, 43

The Non-Standard Normal Curve

Not every normal curve will be the standard normal curve. A normal curve is defined by two parameters, the mean and the standard deviation.

Consider a normal distribution where the mean is 56.8 and the standard deviation is 5.5. What is the probability of an observation greater than 60?

 

We need to convert the observation to standard units through the use of a z-score.

Now look up z=.58 and p=.7190. From our picture it is clear that the probability of an observation greater than 60 is 1-.7190=.281.

 

 

Consider a normal distribution where the mean is -36 and the standard deviation is 1.7. What is the probability of an observation between -35 and -36.5?

 

 

 

Consider a normal distribution where the mean is 10 and the standard deviation is 15. What is the probability of an observation larger than 20?

 

 

 

Consider a normal distribution where the mean is 55 and the standard deviation is 15. What is the 85th percentile? What is the 23rd percentile?

 

 

 

The average life span of a certain brand of tires is 30,000 miles with a standard deviation of 2,000 miles and follows a normal distribution.

 

Would it be unusual for a tire to last for 35,000 miles?

 

 

What is the probability that a tire will have a life span between 25,000 and 28,000 miles?

 

 

 

 

 

 

 

Suppose this company wishes to replace only 2 out of every 10,000 tires with its warrantee.  How many miles should it guarantee a tire will last?  Are there tires that have an unusually short lifespan yet not be covered under warrantee?  If yes, what percentage of the production falls into this category?

 

 

 

 

Homework

Section 5-3:  1-8, 10, 13, 14, 17-29

 

Syllabus Files ] Tips for Success ] Intro ] Chapter 2: Numerical Methods ] Chapter 3: Probability ] Chapter 4: Probability Distributions ] [ Chapter 5: The Normal Curve ] Chapter 5: Central Limit Theorem ] Chapter 4: The Binomial Probability Distribution ] Questionnaire ] Chapter 1 ] Chapter 6: Confidence Intervals ] Chapter 6: Required Sample Size ] Chapter 6: Estimation of Proportion ] Required Sample Size (Updated) ] Chapter 7: Hypothesis Testing ] Chapter 7: Applications of Hypothesis Testing ] Sample Tests ]