Empirical rule, z-score, confidence interval Review

Empirical Rule CW

Z Score Formula

We use z-scores to compare data with different units. 

Z-scores standardize our values so we can compare unlike data.

Info about z-scores

- Z-scores tell you how far away you are from the mean in an approximately normal distribution

- z=-1: You are 1 SD below the mean

- z=2: You are 2 SD above the mean

- Large z-scores mean that the value is very unusual (usually 3 or greater)

Multiple Choice

You have a z-score of 2.5 for heights. This means that…

• You are 2.5 feet taller than others.

• You are 2.5 SD above the mean height.

• You are 2.5 SD below the mean height.

• You are 2.5 feet above the mean height.

• You are 2.5 feet below the mean height.

Multiple Choice

In the formula for z-score what does the x stand for?

• the data point in question

• the mean of the data set

• the standard deviation of the data set

• the mode

Multiple Choice

In the formula for z-score what does the symbol below stand for?

$$\mu$$  

• the data point in question

• the mean of the data set

• the standard deviation of the data set

• the mode

Multiple Choice

If the z - score is 1.5 the data point ________ the mean.

• above

• below

• the same as the mean

Multiple Choice

If the z - score is 0 the data point ________ the mean.

• above

• below

• the same as the mean

Definitions:

Confidence Interval- An interval estimate for an unknown population parameter. This depends on: (1) the desired confidence level, (2)information that is known about the distribution (for example, known standard deviation), (3) the sample and its size.

z-score - gives the number of standard deviations a value is away from the mean.

Multiple Choice

A 90% confidence interval for the average salary of all CEOs in the electronics industry was constructed using the results of a random survey of 45 CEOs. The interval was ($139,048, $154,144). Give a practical interpretation of the interval.

• 90% of the sampled CEOs have salaries that fell in the interval $139,048 to $154,144

• We are 90% confident that the mean salary of all CEOs in the electronics industry falls in the interval $139,048 to $154,144. 

• 90% of all CEOs in the electronics industry have salaries that fall between $139,048 to $154,144

• We are 90% confident that the mean salary of the sampled CEOs falls in the interval $139,048 to $154,144.

Multiple Choice

Use the following information and the Empirical Rule to estimate the answer.

The ages of golfers are normally distributed, with a mean of 38 and a standard deviation of 4.

Find the proportion of golfers that are between 30 and 46 years old.

• 68%

• 94%

• 95%

• 99.7%

Multiple Choice

At a local high school, GPA's are normally distributed with a mean of 2.9 and standard deviation of 0.6. What percentage of students at the high school have a GPA between 2.3 and 3.5?

• 68%
• 99.7%
• 95%
• 84%

Multiple Choice

The mean life of a tire is 30,000 km. The standard deviation is 2000 km.

Then, 68% of all tires will have a life between ___________ km and __________ km.

•  28,000 km and 32,000 km.
• 24,000 km and 34,000 km.
• 26,000 km and 34,000 km.
• 27,000 km and 31,000 km.

Multiple Choice

The shelf life of a particular dairy product is normally distributed with a mean of 12 days and a standard deviation of 3 days.

About what percent of the products last between 12 and 15 days?

• 68%
• 34%
• 16%
• 2.5%

Multiple Choice

Which interval shows 95% of the data given a mean of 15 and a standard deviation of 3? 

• 9 to 21
• 12 to 18
• 6 to 24
• 13 to 16

Multiple Choice

Which interval shows 99.7% of the data given a mean of 15 and a standard deviation of 3? 

• 9 to 21
• 12 to 18
• 6 to 24
• 13 to 16

Multiple Choice

What is the percentage of data that falls within three standard deviations of the mean?

• 99.7%

• 98.7%

• 95%

• 100%

Multiple Choice

What is the percentage of data that falls within two standard deviations of the mean?

• 68%

• 99.7%

• 95%

• 98%

Multiple Choice

What is the percentage of data that falls within one standard deviation of the mean?

• 67%

• 65%

• 72%

• 68%

Multiple Choice

The Empirical Rule's percents are:

• 68-95.7-99

• 68-95-99.7

• 68.7-95-99

Multiple Choice

The Empirical (68-95-99.7) Rule can be applied only if data is ....

• Normally Distributed

• Uniform

• Skewed

• Symmetric