7.4 Use Normal 7.4 Use Normal DistributionsDistributions

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1.) 1.) #9#9

2.) 2.) #11#11

Normal DistributionNormal Distribution

A bell-shaped curve is called a normal curve. It is symmetric about the mean. The percentage of the area in each standard deviation is shown above.

Empirical RuleEmpirical Rule(68 – 95 (68 – 95 99.7) 99.7)

Within 1 from the mean lies 68% of the dataWithin 2 from the mean lies 95% of the data

Within 3 from the mean lies 99.7% of the data

Standard Normal DistributionStandard Normal Distribution

Mean is 0 and standard deviation is 1. This formula is used to find the z-scores which “Normalize” a data point to it’s exact number of standard deviations away from the mean it is.

(Z-score is the # of standard deviations away from the mean)

Analyzing the dataAnalyzing the data

Suppose SAT scores among college Suppose SAT scores among college students are normally distributed with a students are normally distributed with a mean of 500 and a standard deviation of mean of 500 and a standard deviation of 100. If a student scores a 700, what 100. If a student scores a 700, what would be her would be her zz-score?-score?

Analyzing the dataAnalyzing the data

Suppose SAT scores among college students Suppose SAT scores among college students are normally distributed with a mean of 500 and are normally distributed with a mean of 500 and a standard deviation of 100. If a student scores a standard deviation of 100. If a student scores a 700, what would be her a 700, what would be her zz-score?-score?

700 5002

100z

Her z-score would be 2 which means her score is two standard deviations above the mean.

Analyzing the dataAnalyzing the data

• A set of math test scores has a mean of 70 and a standard deviation of 8.

• A set of English test scores has a mean of 74 and a standard deviation of 16.

For which test would a score of 78 have a higher standing?Answer Now

Analyzing the dataAnalyzing the data

78-70math -score = 1

8z

To solve: Find the z-score for each test.

78-74English -score= .25

16z

A set of math test scores has a mean of 70 and a standard deviation of 8. A set of English test scores has a mean of 74 and a standard deviation of 16. For which test would a score of 78 have a higher standing?

The math score would have the highest standing since it is 1 standard deviation above the mean while the English score is only .25 standard deviation above the mean.

Analyzing the dataAnalyzing the data

What will be the miles per gallon for a Toyota Camry when the average mpg is 23, it has a z value of 1.5 and a standard deviation of 2?

Answer Now

Analyzing the dataAnalyzing the data

What will be the miles per gallon for a Toyota Camry when the average mpg is 23, it has a z value of 1.5 and a standard deviation of 2?

231.5

2

x

xz

Using the formula for z-scores:

3 2 63 2x x

The Toyota Camry would be expected to use 26 mpg of gasoline.

Analyzing the dataAnalyzing the data

A group of data with normal distribution has a mean of 45. If one element of the data is 60, will the z-score be positive or negative?

Answer Now

Analyzing the dataAnalyzing the data

A group of data with normal distribution has a mean of 45. If one element of the data is 60, will the z-score be positive or negative?

The z-score must be positive since the element of the data set is above the mean.

EXAMPLE 1 Find a normal probability

SOLUTION

A normal distribution has mean x and standard deviation σ. For a randomly selected x-value from the distribution, find P(x – 2σ ≤ x ≤ x).

= 0.135 + 0.34 = 0.475

EXAMPLE 2 Interpret normally distribute data

The blood cholesterol readings for a group of women are normally distributed with a mean of 172 mg/dl and a standard deviation of 14 mg/dl.

A. About what percent of the women have readings between 158 and 186? (Hint: Find out how far away 158 and 186 are from the mean.)

B. The readings of 158 and 186 represent one standard deviation on either side of the mean, as shown below. So, 68% of the women have readings between 158 and 186.

EXAMPLE 2 Interpret normally distribute data

b. A reading of 200 is two standard deviations to the right of the mean, as shown. So, the percent of

readings that are undesirable is 2.35% + 0.15%, or 2.5%.

Readings higher than 200 are considered undesirable. About what percent of the readings are undesirable?

b.

GUIDED PRACTICE for Examples 1 and 2

A normal distribution has mean and standard deviation σ. Find the indicated probability for a randomly selected x-value from the distribution.

x

1. P( ≤ )x x

0.5

2. P( > )x x

0.5

0.475ANSWER

4. P( – σ < x < ) x x

0.34ANSWER

GUIDED PRACTICE for Examples 1 and 2

5. P(x ≤ – 3σ)x

0.0015ANSWER

6. P(x > + σ)x

0.16ANSWER

EXAMPLE 3 Use a z-score and the standard normal table

Scientists conducted aerial surveys of a seal sanctuary and recorded the number x of seals they observed during each survey. The numbers of seals observed were normally distributed with a mean of 73 seals and a standard deviation of 14.1 seals. Find the probability that at most 50 seals were observed during a survey.

Biology

EXAMPLE 3 Use a z-score and the standard normal table

SOLUTION

STEP 1 Find: the z-score corresponding to an x-value of 50.

–1.6z = x – x 50 – 7314.1=

STEP 2 Use: the table to find P(x < 50) P(z < – 1.6).

The table shows that P(z < – 1.6) = 0.0548. So, the probability that at most 50 seals were observed during a survey is about 0.0548.

GUIDED PRACTICE for Example 3

9. REASONING: Explain why it makes sense that P(z < 0) = 0.5.

A z-score of 0 indicates that the z-score and the mean are the same. Therefore, the area under the normal curve is divided into two equal parts with the mean and the z-score being equal to 0.5.

ANSWER

Wrap it up!Wrap it up!Percentages:Percentages:

Use theses tablesUse theses tablescalculate the z-scorescalculate the z-scoresFind the percentages Find the percentages

Using either table.Using either table.Finding z-score:Finding z-score:

Tells you how many standard deviations away Tells you how many standard deviations away from the mean a data point is.from the mean a data point is.

Is how we “Normalize” data.Is how we “Normalize” data.

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P266 2-18 EvenP266 2-18 Even

P266 1-19 OddP266 1-19 Odd

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