Let's put our heads together by flickr user Normski's

All Together Nowa statistics workshop

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

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

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)

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

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.

An orange producer who calls himself Doctor Juice grows an exclusive variety of oranges which are sorted into three categories and sold at different prices.

The diameters of the oranges are distributed normally with a mean of 84 mm and a standard deviation of 12 mm.

(a) What percent of the oranges are sorted into the small category?

(b) What is the minimum diameter (rounded to the nearest millmeter) of a Jumbo Orange?

(c) What is the expected income from 2000 unsorted oranges,

Description Size Price per orangeSmall less than 75mm 12 centsJumbo largest 12% 45 centsRegular all others 35 cents

HOMEWORK

Forty students measured the width of the gym, and wrote their measurements in centimetres, rounded to the nearest cm. The measurements are recorded on the table below.

Draw a histogram of the data. Using the properties of a Normal Distribution, determine if the data is approximately normal.

2251 2249 2250 2247 2253 2248 2249 22532254 2247 2250 2253 2248 2255 2249 22492250 2251 2252 2250 2249 2250 2247 22502250 2252 2253 2255 2254 2248 2248 22422249 2245 2251 2246 2250 2246 2251 2246

HOMEWORK

A college aptitude test is scaled so that its scores approximate a normal distribution with a mean of 500 and standard deviation of 100.

(b) Find the score x, such that 76 percent of the students have a score less than x.

(a) Find the probability that a student selected at random will score 800 or more points. HOMEWORK

The weights of babies born in a certain hospital average 8 lb 1 oz, with a standard deviation of 12 oz. Assume that the weights are normally distributed.

(c) Find the weight, W, such that the percentage of babies with a birth weight less than W is 25 percent.

(b) Find the weight, W, such that the percentage of babies with a birth weight greater than W is 60 percent.

(a) Find the percentage of babies with a birth weight between 7 and 9 pounds.

HOMEWORK