Download Chapter 7 Reading Notes

AP Statistics
Chapter 7
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Assignment
Section
Page
Name
Problems
I
7.1
351–354
7.1, 7.4, 7.6, 7.7
II
7.2
355–359
7.9, 7.11, 7.14, (7.18a-ec)
III
7.3
361–365
7.25, 7.26
IV
7.4
366–377
7.29, (7.33calc), 7.35, 7.40, 7.41
V
7.5
380–389
7.47, 7.51, 7.55, 7.57, 7.61
VI
7.6
392–407
7.67, 7.69, 7.71, 7.73
VII
7.7
409–416
7.81, 7.82
Review
Due Date
7.103 – 7.107
7.1 Random Variables p. 351
1.
Write a single sentence that compares a statistical random variable with an algebra variable.
2.
Give an example of a discrete random variable and an example of a continuous random variable.
3.
What is the mathematical advantage of continuous variables?
7.2 Probability Distributions for Discrete Random Variables p. 355
4.
The probability distribution for a random variable is …
5.
Name the three ways of displaying a discrete probability distribution.
6.
Use the information given in example 7.5 to calculate the P(GEEE)
7.
Use example 7.5 and the answer to the previous question to calculate P(3). (Provide full support for your answer by
starting with your answer to the previous question.)
8.
What is the probability at most 3 of the four customers choose electric models?
9.
Verify a discrete probability distribution property by calculating
4
 p(x) for the hot tub example.
x= 0
7.3 Probability Distributions for Continuous Random Variables p. 361
The diagram below is a more detailed version of the one on page 362. Use it to answers questions 10 to 12.
Figure 17.5 Probability distribution for birth weight:
(a) Weight measured to the nearest pound;
(b) Weight measured to the nearest tenth of a pound;
(c) Limiting curve as measurement accuracy increases; shaded area = P(6 ≤ weight ≤ 8)
10. In Figure 17.5 (a), calculate the probability that a randomly selected baby weighs between 6 and 8 pound (inclusive). Use
appropriate notation.
11. Make a prediction for the predicted probability if Figure 17.5 (b) were used instead. Do you think probabilities calculated
using histograms with measurements accurate to the nearest tenth of a pound, as in (b), are more or less accurate or about
the same as calculations done with measurements accurate to the nearest inch? Explain.
12. Will calculations based on a continuous curve, as in (c), be more precise than previous calculations?
13. What is the distribution for a continuous random variable function f (x) called? (The footnote contains an example of f(x).
You can see why the book never shows this function!)

To calculate continuous curve probabilities calculus is used to find the area under the curve instead of summing histogram bar heights. The computation is:
8
P(6 £ x £ 8) =
Ú1
6
1
-
e
(x - 7)2
2g12
dx = .6826
2p
You will NEVER need to perform any math even close to this. You will learn a major shortcut to avoid calculus.
14. How is probability defined on the f (x) curve?
15. Use Example 7.7 to calculate P(x < 4.5) .
16. Compare P(a £ x £ b) and P(a < x < b) for continuous and discrete distributions.
17. How does a cumulative area differ from an area between two values?
7.4 Mean and Standard Deviation of a Random Variable  p. 366
18. What is the difference between the mean x and the mean mx ? What is the difference between s and s x ?
19. The expression for x is________________based on_______________; the mean value of a
random variable x is _______________ based on_______________.
20. What are expected value and E (x) ?
21. What shape is the distribution of Apgar scores?
22. Record the mean and variance formulas for discrete random variables.
23. Use Example 7.12 to calculate the probability that that a randomly selected child’s score is within one standard deviation of
the mean.
24. Why will we not be calculating mx and s x for continuous distributions?
[The final two pages of 4.7 address one of the most confusing areas of statistics: The Mean and Variance of Linear
Functions and Linear Combinations. Be sure to read every word and understand every example. ]
25. Is there any aspect of the Mean, Variance and Standard Deviation of a Linear Function summary box that seems strange
(aside from the typo at the top that labels variance as s x instead of s x2 )?
26. How does a random variable that results form a linear function differ from a random variable that results from a linear
combination?
27. When creating summary statistics for Linear Combinations, do standard deviations add? Do variances? Always, sometimes,
never?
7.5 Binomial and Geometric Distributions  p. 380
28. What does dichotomous mean?
29. Define a trial.
30. What is the difference between the two different types of situations?
31. State the four binomial properties (requirements) in as few words as possible.
32. The binomial variable records…
33. Using the five customers example, what is the difference between P(SFFFF) and P(x=1)? (In other words, are they both
answering the same question?)
34. How many ways can two girls (x) be arranged in a seven-child family?
35. In example 7.17 will P(4 ≤ x ≤ 7) be the same as P(4 < x < 7)? If no, use the information in the example to find P(4 < x <
7). (No new calculations are required.)
36. [[SKIP appendix table 10.]]
37. •Sampling Without Replacement: Under what conditions may sampling without replacement be approximated with the
binomial methods?
38. Like any distribution, the mean is used to represent the center of the distribution and the standard deviation represents the
typical amount of variability around that center. Calculate the mean μx and standard deviation σx for the computer monitor
situation (Example 7.17).
39. Which value of π produces the most variability?
40. Geometric: Not interested in
but rather…
41. Of the four binomial requirements, which is not needed for geometric situations?
42. If 75% of statistics students complete their reading notes, what is the probability that the first successful selection of a
student with completed reading notes occurs on the 3 rd student selected? On or before the 3rd student selected? (Assume
independence. What must be true for this assumption?)
7.6 Normal Distributions  p. 392
43. Normal distributions are…1.
2.
3.
44. How are different normal distributions distinguished?
45. As with all continuous probability distributions…
46. Why is the reference standard normal distribution needed?
47.
Use the z* table on page 395 to find:
a. P(z > 1.5)
b. P(z ≥ 1.03)
(Answers: .0668, .1515)
48. The table on page 395 only contains positive z* values. Why is the smallest probability in this table 0.5?
49. Use Appendix Table 2 and Examples 7.21 and 7.22 to find:
a. P(z < -2.18)
b. P(-2 < z < 2)
c. P(z > 2.5)
(Answers: .0146, .9544, .0062)
50.
Use Example 7.23 to find the z* value that locates the…
a. smallest 3%
b. largest 2%
c. most extreme 1%
(Answers: -1.88, 2.05, ±2.575)
51. Simply calculating z-scores for the endpoints of an interval is called…
52. Use example 7.25 to calculate the proportion of children from μ = 100 and σ = 6 between 90 and 115 cm.
53.
What proportion of the population has Stanford-Binet IQ scores below 60?
(Answer: .9463)
54.
What registration time represents the bottom 1%?
(Answer: .0038)
7.7 Checking for Normality  p. 409
55. A normal probability plot is just a scatterplot of…
56. To test for normally distributed data, the normal probability plot should have what feature?
57. What minimum correlation coefficient satisfies the normality test on a normal probability plot with 22 values?
58. Explain (one sentence) transforming data.
59. What is the benefit of transforming to achieve normality?
60. What transformation is especially useful for right (positively) skewed data?