Green House by the Laundry by Dylan James

Working with z-scores and the Normal Curve ...it's everywhere!

The normal distributions used up to this point have all been non-standard normal distributions. This means that the mean and standard deviation of the distribution are the mean and standard deviation of the data being studied.

In a standard normal distribution, the scale on the x-axis is the z-score (a standard score) where 'z = 0' is the mean, and the standard deviation is '1'. The distribution is a probability distribution where the area under the curve = 1. This means there is a 100 percent chance of every score being included in this distribution. Other probabilities are illustrated below.

The Standard Normal Distribution

Example: The graph at right represents the marks of a large number of students where the mean mark is 69.3 percent and the standard deviation is 7 percent, and the distribution of marks is approximately normal.

62.3% 69.3% 76.3%

A Modified Normal Distribution

North American women have a mean height of 161.5 cm and a standard deviation of 6.3 cm.

(b) The manufacturer designs the seats to fit women with a maximum z-score of 2.8. How tall is a woman with a z-score of 2.8?

(a) A car designer designs car seats to fit women taller than 159.0 cm. What is the z-score of a woman who is 159.0 cm tall?

RECALL

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 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

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.

The 3 "meanings" of a Shaded Normal Curve

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

z low 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)

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?

QUIZThe following are the number of steak dinners served on 50 consecutive Sundays at a restaurant.

(a) Draw a suitable histogram that has five bars.

QUIZ

(b) Determine how closely the data approximates the normal distribution using the properties of the Normal Distribution.

The following are the number of steak dinners served on 50 consecutive Sundays at a restaurant.

(2) The diagram shows a normal distribution with a mean of 36 and a standard deviation of 6. The values represent the number of standard deviations above and below the mean. Replace the numbers with raw scores.

(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.

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

Find the probability of getting a z-score less than 0.75 in a standard normal distribution.

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?

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

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

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

A club called "The Beanstalk Club" has a minimum height requirement of 5'10''. Women in North America have a mean height of 5' 5.5'', and a standard deviation of 2.5''. What percentage of women are eligible? Assume that the heights of women normally distributed. Include a sketch with your answer.

HOMEWORK