6.3 Use Normal Distributions. Page 399 What is a normal distribution? What is standard normal distribution? What does the z-score represent?

Transcription

1 6.3 Use Normal Distributions Page 399 What is a normal distribution? What is standard normal distribution? What does the z-score represent?

2 Normal Distribution and Normal Curve Normal distribution is one type of probability distribution. It is modeled by a bell shaped curve called a normal curve. A normal curve is symmetric about the mean.

3 Areas Under a Normal Curve A normal distribution with mean x and standard deviation σ has the following properties: The total area under the related normal curve is 1. About 68% of the area lies within 1 standard deviaion of the mean About 95% of the area lies within 2 standard deviations of the mean About 99.7% of the area lies within 3 standard deviations of the mean. See page 399 Key Concept.

4 The total area under the related normal curve is 1. About 68% of the area lies within 1 standard deviaion of the mean About 95% of the area lies within 2 standard deviations of the mean About 99.7% of the area lies within 3 standard deviations of the mean.

5 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). SOLUTION The probability that a randomly selected x-value lies between x 2σ and x is the shaded area under the normal curve shown. x x = = 0.475

6 Health 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? SOLUTION a. 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 (34% + 34% = 68%). Mg/dl = milligrams per deciliter

7 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. b. Readings less than 158 are considered desirable. About what percent of the readings are undesirable? b. A reading of 158 is one standard deviation to the left of the mean, as shown. So, the percent of readings that are desirable is 0.15% + 2.3% %, or 16%. Thus 84%of the readings are undesirable.

8 A normal distribution has mean x and standard deviation σ. Find the indicated probability for a randomly selected x-value from the distribution. 1. P( x x ) P( x x) = P( x 3σ) + P( x 2σ) + P( x σ) = ANSWER 0.5

9 A normal distribution has mean x and standard deviation σ. Find the indicated probability for a randomly selected x-value from the distribution. 2. P( x > x ) P( x > x) = P( x + σ) + P( x + 2σ) + P( x + 3σ) = ANSWER 0.5

10 A normal distribution has mean x and standard deviation σ. Find the indicated probability for a randomly selected x-value from the distribution. 3. P( x < x < x + 2σ ) P( x < x< x + 2σ )= P( x+ σ) + P( x + 2σ) = ANSWER 0.475

11 Standard Normal Distribution

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13 Standard Normal Table

14 Biology

15 SOLUTION STEP 1 Find: the z-score corresponding to an x-value of 50. z= x x = STEP 2 Use: the table to find P(x < 50) P(z < 1.6). The table shows that P(z < 1.6) = So, the probability that at most 50 seals were observed during a survey is about

16 Using the previous problem, find the probability that at most 90 seals were observed during a survey. z= x x = Use: the table to find P(x < 90) P(z < 1.2). The table shows that P(z < 1.2) = So, the probability that at most 90 seals were observed during a survey is about ANSWER

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

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19 6.3 Assignment Page 402, 3-14, 19-24