**Z-Score Workshop**

If an individual value, x, is less than the mean, its z-score is a negative number. If an individual value, x, is greater than the mean, its z-score is a positive number.

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**What is a z-score?**

**Three types of**

z-score problems

z-score problems

**Less than example**

**Greater than example**

Less than example

Calculate z-score

Greater than example

Calculate z-score

Acknowledgments

For more information, contact Germanna Tutoring Services:

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Data for the number of movies people watch in a month is collected from a 100 person sample group. The sample data indicates that the mean of the sample is 15 with a sample standard deviation of 3. Find the probability that a person watches

less than

22 movies per month.

Data for the number of pieces of candy people eat in a month is collected from a 120 person sample group. If the mean of the sample is 20 and the sample standard deviation is 5, what is the probability of a person eating

less than

23 pieces of candy?

Formula:

z-score for 23 pieces of candy: z = 23-20 = 0.60

5

All of the employees of a company are asked how many books they read per month. The data collected from this population indicates that the population mean is 4.25 with a population standard deviation of 2. Find the probability for a person who reads

more than

2.5 books per month.

All of the employees at a company are asked how many hours per week they watch television. The data collected from this population results in a mean of 6 with standard deviation of 2.2. Find the probability that a person watches

more than

2.5 hours of television per week.

Formula:

z-score for 2.5 hours: z = 2.5-6 = -1.59

2.2

Z-scores correspond to a specific area under the bell curve. This area can be found in a z-score table.

**Between example**

Data for the number of movies people watch in a month is collected from a 100 person sample group. The sample data indicates that the mean of the sample is 15 with a sample standard deviation of 6.5. Find the probability that person watches

between

6 and 22 movies per month.

**Between example**

All of the students at a college are asked how many electronic devices that they own. The data collected for this population indicated that the population mean is 10 with a population standard deviation of 4.5. Find the probability that a person owns

between

2 and 12 devices.

Calculate z-score

Area = 0.7257

Probability = 0.7257

Area = 0.7761

Probability = 0.7761

Area = 0.6325

Probability = 0.6325

Use z-score table

Find area/probability

Use TI-83/84 Calculator

Press 2nd and then the Vars key.

Choose option 2: normalcdf(

Type normalcdf(-99999, 2.33)

Press Enter

The probability that a person watches less than 22 movies per month is .9901.

Calculate z-score

Find area/probability

Use TI-83/84 Calculator

Press 2nd and then the Vars key.

Choose option 2: normalcdf(

Type normalcdf(-99999, 0.60)

Press Enter

Use z-score table

Use z-score table

Finding area/probability

Use TI-83/84 Calculator

Press 2nd and then the Vars key.

Choose option 2: normalcdf(

Type normalcdf(-0.88,99999)

Press Enter

The probability that a person reads more than 2.5 books a month is 0.8106.

Find area/probability

Use z-score table

Use TI-83/84 Calculator

Press 2nd and then the Vars key.

Choose option 2: normalcdf(

Type normalcdf(-1.38, 1.08)

Press Enter

The probability that someone watches between 6 and 22 movies per month is 0.7761.

Calculate z-score

Formula:

z-score for 6 movies: z = 6-15=-1.38

6.5

z-scores for 22 movies: z = 22-15 = 1.08

6.5

1-0.1894 to find the area to the right of a -0.88 z-score.

Find area/probability

Use TI-83/84 Calculator

Press 2nd and then the Vars key.

Choose option 2: normalcdf(

Type normalcdf(-1.59,99999)

Press Enter

Use z-score table

1-0.0559 to find the area to the right of a -1.59 z-score.

In order to find the area between two z-scores, take the area of the right most z-score (1.08) and subtract the area of the left most z-score (-1.38) from it. Thus we would do 0.8599-0.0838 to find the area between our two z-scores.

Formula:

z-score for 2 devices: z = 2-10 = -1.78

4.5

z-score for 12 devices: z = 12-10 = 0.44

4.5

Find area/probability

Use z-score table

In order to find the area between two z-scores, take the area of the right most z-score (0.44) and subtract the area of the left most z-score (-1.78) from it. Thus we would do 0.6700-0.0375 to find the area between our two z-scores.

Use TI-83/84 Calculator

Press 2nd and then the Vars key.

Choose option 2: normalcdf(

Type normalcdf(-1.78, 0.44)

Press Enter

Information about z-score properties and images of z-score tables obtained from:

Triola, Mario F.

Elementary Statisics, 12th

edition

. Boston, MA: Pearson Education,

Inc. E-book.

Less than

Greater than

Between

Formula:

z-score for 22 movies: z = 22-15 = 2.33

3

Z-score = 2.33

z-score=0.60

The probability that a person eats less than 23 pieces of candy is 0.7257.

z-score = -0.88

z-score = -1.59

0.9441 probability that a person watches more than 2.5 hours of television a week.

Calculate z-score

Formula:

z-score for 2.5 books: z = 2.5-4.25 = -0.88

2

The probability that a person owns between 2 and 12 devices is 0.6235

z-score = -1.38

z-score = 1.08

Z-score formula

z-score = -1.78

z-score = 0.44

z-score for population:

z-score for sample: