Z-Scores Continued

Ch. 6 Day 2

AP Statistics

EQ: What is the normal distribution and why do we use it?

When is a z-score big?

The larger the z-score, the more unusual it is.

For symmetric data, the _______________ is usually a bit ___________ than the IQR.

Normal Models

• Are appropriate for distributions that are roughly symmetric or unimodal•We write a normal model as

_____________.• Z-scores use the Standard Normal

Model

Parameters vs. Statistics

Parameter

Statistics

A cereal manufacturer has a machine that fills the boxes. Boxes are labeled “16 ounces”, so the company wants to have that much cereal in each box, but since no packaging process is perfect, there will be minor variations. If the machine is set at exactly 16 ounces and the Normal model applies, then about ½ the boxes will be underweight, making consumers unhappy and exposing the company to bad publicity and possible lawsuits. To prevent underweight boxes, the manufacturer has to set the mean a little higher than 16 ounces.

Based on their experience with the packaging machine, the company believes that the amount of cereal in the boxes fit a Normal model with a standard deviation of 0.2 ounces. The manufacturer decides to set the machine to put an average of 16.3 ounces in each box. What fraction of the boxes will be underweight?

The company’s lawyers say that 6.7% is too high. They insist that no more than 4% of the boxes can be underweight. So the company needs to set the machine to put a little more cereal in each box. What mean setting do they need?

The company president vetoes that plan, saying the company should give away less free cereal, not more. Her goal is to set the machine no higher than 16.2 ounces and still have only 4% underweight boxes. The only way to accomplish this is to reduce the standard deviation. What standard deviation must the company achieve, and what does that mean about the machine?

Many professional schools require applicants to take a standardized test. Suppose that 1000 students take such a test. Several weeks after the test, Pete receives his score report: he got a 63, which placed him in the 73rd percentile. What does that mean?

If the heights of American men follow a normal distribution, and 99.7% have heights between 5 feet and 7 feet, what is your estimate of the standard deviation of the height of American men?

4 inches

The scores on a university examination are normally distributed with a mean of 62 and a standard deviation of 11. If the bottom 5% of students will fail the course, what is the lowest mark that a student can have and still be awarded a passing grade?

44

Let X denote the time taken for a computer link to be made between the terminal in an executive’s office and the computer at a remote factory site. It is known that X has a normal distribution with a mean of 15 seconds and a standard deviation of 3 seconds. On 90% of the occasions the computer link is made in less than???

18.84 seconds

Homework

P.129 # 18, 20, 24, 36, 40, 47