Z-scores and Empirical Rule Notes - blogs



Z-scores and Empirical Rule Notes Name: _______________________

Empirical Rule says that …

68% of the data in a normally distributed data set is within 1 standard deviation.

95% of the data in a normally distributed data set is within 2 standard deviations.

99.7% of the data in a normally distributed data set is within 3 standard deviations.

Types of questions involving the Empirical Rule:

1. The scores on a university examination are normally distributed with a mean of 70 and a standard deviation of 10. If the middle 68% of students will get a “C”, what is the lowest mark that a student can have and still be awarded a C?

To solve: The middle 68% of students are within 1 standard deviation of the mean

according to the Empirical Rule. The question wants to know the LOWEST mark

that a student can get o receive a C, so you must subtract 1 standard deviation from the mean.

70 – 10= 60

2. The lifetime of lightbulbs of a particular type are normally distributed with a mean of 100

mmHg and a standard deviation of 6 mmHg. What percentage of 18-year-old women have a systolic blood pressure between 88 mmHg and 112 mmHg?

To solve: You must decide how many standard deviations away each of the given blood

pressures are from the mean. Start by looking at the left of the mean. 100 – 6 = 94.

That’s not far enough, so you subtract another standard deviation. 94 – 6 = 88. Because 88 is two standard deviations away from the mean, 95% of 18-year-old women have a systolic blood pressure between 88 mmHg and 112 mmHg.

Z-scores: [pic]

Z-scores are used to normalize data, or to convert all data to a common unit. A z-score tells you how many standard deviations away your data is from the mean.

Types of z-score questions:

1. Lewis earned 80 on his biology midterm and a 71 on his history midterm. In the biology class the mean score was 75 with a standard deviation of 4. In the history class the mean score was 73 with a standard deviation of 3.

a. Convert each score to a standard z-score.

To solve: Biology: [pic] History: [pic]

b. On which test did he do better compared to the rest of the class?

Solution: In both classes he did worse than the average because both z-scores were negative.

However, he did better compared to the rest of the class in History because his z

score is smaller.

Z-table:

The z-table gives you the area, probably, or percent of data that is below the said value. You use the z-table when you see one of the above bold words.

How to use the z-table: You need to have one number before the decimal, and two numbers after the decimal.

• If there is no number before the decimal, put a 0 before the decimal.

• If there is only one number after the decimal, add a 0 on the end.

• If there is no decimal, add one then add two 0s after it.

• If there are more than two numbers after the decimal you have to ROUND.

o Look at the 4th number after the decimal. If it is a 4 or below, keep the 3rd number the same.

▪ Example: 1.45345698 becomes 1.45 because 3 is the 4th number.

o Look at the 4th number after the decimal. If it is a 5 or above, raise the 3rd number 1 higher than it was before.

▪ Example: -0.86795643 becomes -0.87 because 7 is the 4th number.

Types of z-table questions:

A class of 217 students participated in a softball throw for the distance test. The mean performance of the group was 173 and the standard deviation was 31. Based on this data, answer the following questions:

a. What percentage of students was able to throw the softball between 151 and 180?

To solve: Because the questions asks for the percentage, you must use your z-table. In

order to use your z-table, you must convert your throw values to z-scores.

[pic] when rounded [pic] when rounded

Next find both of these z-scores on the z-table: 0.2389 and 0.5910

To find the percentage between two numbers you subtract the lower from the higher:

0.5910 – 0.2389 = 0.3521 = 35.21%

b. What percentage of the students could throw farther than 200 feet?

To solve: Find the z-score first: [pic]. Then find the z-table value: 0.8078

To find the percentage of students who throw farther than 200, you must subtract

your z-table value from 1. 1 – 0.8078 = 0.1922 = 19.22%

c. What percentage of the students could only throw less than 114 feet?

To solve: The z-table values give you the percent that throws less than the given amount.

Therefore, once you find the z-score, simply look it up on your z-table.

Z-score: [pic] Then find the z-table value: 0.0287 = 2.87%

Normal Distribution:

A normal distribution looks like a bell curve. In order to use the Empirical Rule or a Z-table

your data must be normally distributed.

Use what you know about a normal distribution to answer the following questions:

1. Which graph above has a larger mean?

Solution: Graph B has a larger mean because the mean is located in the middle of each

graph and the mean of graph B is located further to the right.

2. Which graph has a larger standard deviation?

Solution: Graph B has a larger standard deviation because it is more spread apart.

Exercises:

1. What percent of data is within 1 std deviation of the mean?

2. What percent of data is within 2 std deviations of mean?

3. What percent of data is within 3 std deviations of mean?

4. The scores on a university examination are normally distributed with a mean of 62 and a standard deviation of 11. If the middle 68% of students will get a “C”, what is the lowest mark that a student can have and still be awarded a C?

5. The lifetimes of lightbulbs of a particular type are normally distributed with a mean of 360 hours and a standard deviation of 5 hours. What percentage of the bulbs have lifetimes that lie within 2 standard deviation of the mean?

A) 31% B) 84% C) 68% D) 95%

6. The systolic blood pressure of 18-year-old women is normally distributed with a mean of 120 mmHg and a standard deviation of 12 mmHg. What percentage of 18-year-old women have a systolic blood pressure between 96 mmHg and 144 mmHg?

A) 99% B) 68% C) 95% D) 99.99%

7. Lewis earned 85 on his biology midterm and 81 on his history midterm. In the biology

class the mean score was 79 with a standard deviation of 5. In the history class the mean

score was 76 with a standard deviation of 3.

(a) Convert each score to a standard z score.

(b) On which test did he do better compared to the rest of the class?

4. On one measure of attractiveness, scores are normally distributed with a mean of 3.93

and a standard deviation of .75. Find the probability of randomly selecting a subject with

a measure of attractiveness that is greater than 2.75.

5. The serum cholesterol levels in men aged 18 to 24 are normally distributed with a mean of 178.1 and a standard deviation of 40.7. If a man aged 18 to 24 is randomly selected, find the probability that his serum cholesterol level is between 100 and 200.

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