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Warm Up 1.What is the distance between the points (2, -5) and (-4, 7)? 2. Determine the center and radius for the circle with (-5, 2) and (3, -2) as endpoints.

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Presentation on theme: "Warm Up 1.What is the distance between the points (2, -5) and (-4, 7)? 2. Determine the center and radius for the circle with (-5, 2) and (3, -2) as endpoints."— Presentation transcript:

1 Warm Up 1.What is the distance between the points (2, -5) and (-4, 7)? 2. Determine the center and radius for the circle with (-5, 2) and (3, -2) as endpoints of the diameter.

2 6.2 Normal Distributions

3 Total area under curve equals 1. Why?

4

5 EXAMPLE 1 Find a normal probability P ( – 2σ x )xx 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). = = 0.475

6 GUIDED PRACTICE A normal distribution has mean and standard deviation σ. Find the indicated probability for a randomly selected x -value from the distribution. x 1.1. P ( )xx 0.5 ANSWER

7 GUIDED PRACTICE for Examples 1 and ANSWER %

8 GUIDED PRACTICE for Examples 1 and P( < < + 2σ )xxx ANSWER

9 GUIDED PRACTICE for Examples 1 and P( – σ < x < )xx 0.34 ANSWER

10 GUIDED PRACTICE for Examples 1 and P (x – 3σ) x ANSWER

11 GUIDED PRACTICE for Examples 1 and P (x > + σ) x 0.16 ANSWER

12 EXAMPLE 2 Interpret normally distribute data 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 ? Readings higher than 200 are considered undesirable. About what percent of the readings are undesirable? b.

13 EXAMPLE 2 a. About what percent of the women have readings between 158 and 186 ?

14 EXAMPLE 2 Readings higher than 200 are considered undesirable. About what percent of the readings are undesirable? b.

15 GUIDED PRACTICE for Examples 1 and WHAT IF? In Example 2, what percent of the women have readings between 172 and 200 ?

16

17 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

18 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. –1.6 z = x – x 50 – = STEP 2 Use: the table to find P(x < 50) P(z < – 1.6).

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

20 GUIDED PRACTICE for Example 3 8. WHAT IF? In Example 3, find the probability that at most 90 seals were observed during a survey ANSWER

21 GUIDED PRACTICE for Example 3 9. REASONING: Explain why it makes sense that P(z < 0) = 0.5.

22 Homework Page 221 #1,4,6,12

23 For use after Lesson ANSWER 2. The average donation during a fund drive was $75. The donations were normally distributed with a standard deviation of $15. Use a standard normal table to find the probability that a donation is at most $115. ANSWER A normal distribution has mean x and standard deviation. For a randomly selected x -value from the distribution, find P(x x – 2 ).


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