Finding Z – scores& Normal Distribution

Using the Standard Normal Distribution

Week 9

Chapter’s 5.1, 5.2, 5.3

Normal Distribution• Normal Distribution - is a very important

statistical data distribution pattern occurring in many natural phenomena, such as height, blood pressure, grades, IQ, baby birth weights, etc.

• Normal Curve - when graphing the normal distribution as a histogram, it will create a bell-shaped curve known as a normal curve.

• It is based on Probability! You’ll see!

Normal Distribution Curve:

                                                                                                   

  

What is this curve all about?• The shape of the curve is bell-shaped • The graph falls off evenly on either side of the

mean.  (symmetrical)• 50% of the distribution lies on the left of the mean• 50% lies to the right of the mean. (above)• The spread of the normal distribution is controlled

by the standard deviation.• The mean and the median are the same in a normal

distribution. (and even the mode)

Features of Standard Normal Curve• Mean is the center

• 68% of the area is within one S.D.

• 95% of area is within two S.D.’s

• 99% of area is within 3 S.D.’s

• As each tail increases/decreases, the graph approaches zero (y axis), but never equals zero on each end.

• For each of these problems we will need pull-out table IV in the back of text

What is a Z – Score?• Z-score’s allow us a method of converting,

proportionally, a study sample to the whole population.

• Z-Score’s are the exact number of standard deviations that the given value is away from the mean of a NORMAL CURVE.

• Table IV always solves for the area to the left of the Z-Score!

Finding the area to the left of a Z

(Ex. 1) – Find the area under the standard normal curve that lies to the left of Z=1.34.

Finding the area to the right of a Z

(Ex. 2) - Find the area under the standard normal curve that lies to the right of Z = -1.07.

Finding the area in-between two Z’s

(Ex. 3) - Find the area under the standard normal curve that lies between Z=-2.04 and Z=1.25.

Formula:

x = data value

u = population mean

ixZ

Practice examples:• For each of the following examples, Look

for the words "normally distributed" in a question before using Table IV to solve them.

• Don’t forget - Table IV always solves for the area to the left of the Z-Score!

Finding ProbabilitiesThe shaded area under the curve is equal to the

probability of the specific event occurring.• Ex (4) - A shoe manufacturer collected data

regarding men's shoe sizes and found that the distribution of sizes exactly fits a normal curve.  If the mean shoe size is 11 and the standard deviation is 1.5.

(a)What is the probability of randomly selecting a man with a shoe size smaller than 9.5?

(b)If I surveyed 40 men, how many would be expected to wear smaller than 9.5?

How did we get that answer:

ixZ

• -1.00 is a Z-score (# of S.D.’s from the mean) that refers to the area to the left of that position. Find it in Table IV.

• -1.00 = .1587

•We want the area to the left of that curve, so, this is the answer. Table IV gives us the answer for area to the left of the curve.

(b) .1587(40) = 6.3 = 6

15.1

115.9

Z

This is how many SD’s

from the mean

Ex (5) – Gas mileage of vehicles follows a normal curve. A Ford Escape claims to get 25 mpg highway, with a standard deviation of 1.6 mpg. A Ford Escape is selected at random.

(a) What is the probability that it will get more than 28 mpg?

(b) If I sampled 250 Ford Escapes, how many would I expect to get more than 28 mpg?

How did we get that answer:

ixZ 875.1

6.1

2528

Z

•1.875 is a Z-score (# of S.D.’s from the mean) that refers to the area to the left of that position.

1.875 = .9696

•We want the area to the right of that curve, thus

1- .9696 = .0304

Ex (6) – This past week gas prices followed a normal distribution curve and averaged $3.73 per gallon, with a standard deviation of 3 cents. What percentage of gas stations charge between $ 3.68 and $ 3.77?

Ex (7) – This week gas prices followed a normal distribution curve and averaged $3.71 per gallon, with a standard deviation of 3 cents.

(a) What percent of stations charge at least $3.77?

(b) What percent will charge less than $3.71?

(c) What percent will charge less than $3.69?

(d) What percent will charge in-between $3.67and $3.75 per gallon?

(e) If I sampled 30 gas stations, how many would charge between $3.67and $3.75 per gallon?

NOW – Going back from probabilities to Z-Scores:

Chapter 5.3

– Finding Z-scores from probabilities

– Transforming a Z-score to an X-value

• Look up .9406 on Table 4

• What Z-score corresponds to this area?

Finding Z-score’s of area to the left(Ex. 8)

(a) Find the Z-score so that the area to the left is 10.75%

(b) Find the Z-score that represents the 75th percentile?

(c) Find the Z-score so that the area to the left is .88

(d) Find the Z-score so that the area to the left is .9880

Transforming a Z-score to an x-value

zx

45 5.2Try: 90th percentile

Look for the three ingredients to solve for x: Population mean, standard deviation, and you will need the Z-score that corresponds to the

given percent (or probability)

(Ex. 9) – The national average on the math portion of the SAT is a 510 with a standard deviation of 130. SAT scores follow a normal curve.

(a) What score represents the 90th percentile?

(b) What score will place you at the 35th percentile?

Finding Z-score’s of area to the left

Finding Z-score’s of area to the Right

(Ex. 10) – Find the Z score so that the area under the standard normal curve to the right is .7881

Find the Z score of area to the Right

(Ex. 11) – A batch of Northern Pike at a local fish hatchery has a mean length of a 8 inches just as they are released to the wild. Their lengths are normally distributed with a standard deviation of 1.25 inches.

What is the shortest length that could still be considered part of the top 15% of lengths?

Find the Z score of area in-between two Z’s

(Ex. 12) – Find the Z score that divides the middle 90% of the area under the standard normal curve.

Critical Two-tailed Z value:

- Used to find the remaining percent on the outside of the area under the curve.

- 90% is equal to 1-.90=.10

- .10/2 = .05 = 5%

2

Z

%1

M&M PackagingEx (13) – A bag of M&M’s contains 40 candies

with a Standard deviation of 3 candies. The packaging machine is considered un-calibrated if it packages bags outside of 80%, centered about the mean. What interval must the candies be between for sale?

The Central Limit Theorem:

• As the sample sizes increases, the sampling distribution becomes more accurate in representation of the entire population.

• Thus, As additional observations are added to the sample, the difference of the Sample mean and the population mean approaches ZERO. ( No difference)