The letter Z by flickr user Admit One

Working the

scores

ShadeNorm(Lo z, Hi z)[shades area under std. normal curve]

ShadeNorm(Lo value, Hi value, mean, std. dev.)[shades area under modified normal curve]

µ+5σ

µ-5σ

σ

σ

The 68-95-99 Rule

• 68% of all the data in a normal distribution lie within the 1 standard deviation of the mean,

• 95% of all the data lie within 2 standard deviations of the mean, and

• 99.7% of all the data lie within 3 standard deviations of the mean.

Generally speaking, approximately:

Properties of a Normal Distribution

The mean mark for a large number of students is 69.3 percent with a standard deviation of 7 percent. What percent of the students have a 'B' mark (i.e., 70 percent to 79 percent)? Assume that the marks are normally distributed.

Case 1(a): Calculate the Percentage of Scores Between Two Given Scores HOMEWORK

Find the percent of z-scores in a standard normal distribution that are:

(d) between z = 0.55 and z = 0.15(c) between z = -1.11 and z = 0.92

(b) above z = -2.35(a) below z = 0.52

HOMEWORK

Find the z-score if the area under a standard normal curve:

(c)to the right of z is 0.785

(b) to the right of z is 0.305

(a) to the left of z is 0.812

HOMEWORK

Determine the values for z and x. HOMEWORK

The 3 meanings of a Shaded Normal Curve

The shaded area under a normal curve between two z-scores is interpreted, simultaneously, as:

zlow highz

• an area(the area under the normal curve)

area

The 3 meanings of a Shaded Normal Curve

The shaded area under a normal curve between two z-scores is interpreted, simultaneously, as:

zlow highz

• an area(the area under the normal curve)• a percentage(the percentage of all values in a data set that lie between two particular z-scores)

%

The 3 meanings of a Shaded Normal Curve

The shaded area under a normal curve between two z-scores is interpreted, simultaneously, as:

zlow highz

• an area(the area under the normal curve)• a percentage(the percentage of all values in a data set that lie between two particular z-scores)• a probability(the probability that a particular z-score falls between two given z-scores)

P(E)

The 3 meanings of a Shaded Normal Curve

The shaded area under a normal curve between two z-scores is interpreted, simultaneously, as:

zlow highz

• an area(the area under the normal curve)• a percentage(the percentage of all values in a data set that lie between two particular z-scores)• a probability(the probability that a particular z-score falls between two given z-scores)

P(E)%

area

1200 light bulbs were tested for the number of hours of life. The mean life was 640 hours with a standard deviation of 50 hours. Assume that "life in hours" of light bulbs is normally distributed.

Case 1(c): Calculate the Number of Scores

(b) How many light bulbs should be expected to last between 600 and 700 hours?

(a) What percent of light bulbs lasted between 600 and 700 hours?

Case 1(b): Calculate the Percentage of Passing ScoresThe mean mark for a large number of students is 69.3 percent with a standard deviation of 7 percent. What percent of the students have a passing mark if they must get 60 percent or better to pass? Assume that the marks are normally distributed.

HOMEWORK

(Case 2) If we know two z-scores of a standard normal distribution, we can find the percentage of scores that lie between them. The procedure is similar to that used in the previous examples.

Case 2(a): Calculate the Percentage of Scores Between Two Z-Scores

Sample question(s):What percent of scores lie between z = 0.87 and z = 2.57?

OR

What is the probability that a score will fall between z = 0.87 and z = 2.57?

OR

Find the area between z = 0.87 and z = 2.57 in a standard normal distribution.

HOMEWORK

Case 2(b): Calculate the Percentage of Scores Between Two Z-ScoresFind the probability of getting a z-score less than 0.75 in a standard normal distribution.

HOMEWORK

What is the z-score if the probability of getting less than this z-score is 0.750?

Case 3(b): Find the Z-Value that Corresponds to a Given ProbabilityHOMEWORK

Case 3(a): Find the Z-Score that Corresponds to a Given Probability

If we know the probability of an event, we can find the z-score that corresponds to this probability. This is the reverse of what we did in Case 2.

Sample question:

What is the z-score if the probability of getting more than this z-score is 0.350?

HOMEWORK