1. Section 7.3
Day 2

2. What is a standard normal distribution?

3. A standard normal distribution is a normal distribution with mean 0 and standard deviation 1.

4. Z-scores
A standard normal distribution is a normal distribution with mean 0 and standard deviation 1.
Variable along horizontal axis of standard normal distribution is a z-score.

5. Z-scores
For a normal distribution:
z =

6. Z-scores
For sampling distribution of sample mean (x):
z = =

7. Z-scores
For sampling distribution of the sample proportion (p):
z = =

8. Properties of the Sampling Distribution of the Number of Successes
If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of the number of successes X:
has mean μx = np
has standard error
will be approximately normal as long as n is large enough (both np and n(1 – p) are at least 10).

9. Properties of the Sampling Distribution of the Sample Proportion
If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of p has these properties:

10. Properties of the Sampling Distribution of the Sample Proportion
If a random sample of size n is selected from a population with proportion of successes p, then the sampling distribution of p has these properties:
Mean of the sampling distribution is equal to the mean of the population, or

11. Properties of the Sampling Distribution of the Sample Proportion
Standard error of the sampling distribution is:

12. Properties of the Sampling Distribution of the Sample Proportion
As the sample size gets larger, the shape of the sampling distribution becomes more normal and will be approximately normal if n is large enough (both np and n(1 – p) are at least 10).

13. Is it appropriate to use the normal
approximation for a sampling distribution of
the sample proportion if n = 30 and p = 0.7?
Explain.

14. Is it appropriate to use the normal
approximation for a sampling distribution of
the sample proportion if n = 30 and p = 0.7?
Explain.
np = 30(0.7) = 21
n(1 – p) = 30 (1 - .7) = 9
Not appropriate because n(1- p) is not at least 10.

15. A botanist anticipates that the proportion of red blossoms in her hybrid plants is In a random sample of 75 of her plants, 18% of the blossoms are red.
If the botanist is correct, the standard error of the sampling distribution of the sample proportion is approximately _________.

16. A botanist anticipates that the proportion of red blossoms in her hybrid plants is In a random sample of 75 of her plants, 18% of the blossoms are red.
Remember to use the population proportion, p, in this calculation.
Do not use the sample proportion!

17. A botanist anticipates that the proportion of red blossoms in her hybrid plants is In a random sample of 75 of her plants, 18% of the blossoms are red.

18. In fall 2004, 37% of the 38,859 first-year students attending the California State University system needed remedial work in mathematics.

19. In fall 2004, 37% of the 38,859 first-year students attending the California State University system needed remedial work in mathematics.
a) Suppose you select 136 students at random from this population of students. Make an accurate sketch, with a scale on the horizontal axis, of the sampling distribution of the number of students who need remedial work.

20. In fall 2004, 37% of the 38,859 first-year students attending the California State University system needed remedial work in mathematics. Select 136 students at random and make accurate sketch.
The sampling distribution for the sample
total can be considered approximately
normal because both np = 50.3 and
n(1- p) = 85.7 are 10 or greater.

21. In fall 2004, 37% of the 38,859 first-year students attending the California State University system needed remedial work in mathematics. Select 136 students at random and make accurate sketch.
Shape: approximately normal
Mean: µx = np = 136(0.37) = 50.32

23. In fall 2004, 37% of the 38,859 first-year students attending the California State University system needed remedial work in mathematics. Select 136 students at random and make accurate sketch.

24. In fall 2004, 37% of the 38,859 first-year students attending the California State University system needed remedial work in mathematics. Select 136 students at random.
What is the probability that 68 or fewer students in the sample of 136 need remedial work?

25. In fall 2004, 37% of the 38,859 first-year students attending the California State University system needed remedial work in mathematics. Select 136 students at random.
What is the probability that 68 or fewer students in the sample of 136 need remedial work?
normalcdf(-1E99, 68, 50.32, 5.63) ≈

26. In fall 2004, 37% of the 38,859 first-year students attending the California State University system needed remedial work in mathematics.
c) Suppose you select 2850 students at random. Make an accurate sketch, with a scale on the horizontal axis, of the sampling distribution of the proportion who need remedial work.

27. This sampling distribution can be
In fall 2004, 37% of the 38,859 first-year students attending the California State University system needed remedial work in mathematics.
c) Suppose you select 2850 students at random. Make an accurate sketch, with a scale on the horizontal axis, of the sampling distribution of the proportion who need remedial work.
This sampling distribution can be
considered approximately normal because
both np = and n(1 - p) = are 10 or greater.

28. In fall 2004, 37% of the 38,859 first-year students attending the California State University system needed remedial work in mathematics.
c) Suppose you select 2850 students at random. Make an accurate sketch, with a scale on the horizontal axis, of the sampling distribution of the proportion who need remedial work.

29. In fall 2004, 37% of the 38,859 first-year students attending the California State University system needed remedial work in mathematics.
c) Suppose you select 2850 students at random. Make an accurate sketch, with a scale on the horizontal axis, of the sampling distribution of the proportion who need remedial work.

30. In fall 2004, 37% of the 38,859 first-year students attending the California State University system needed remedial work in mathematics.
d) What is the probability of getting 54% or more who need remedial work in a random sample of 2850 students?

31. In fall 2004, 37% of the 38,859 first-year students attending the California State University system needed remedial work in mathematics.
What is the probability of getting 54% or more who need remedial work in a random sample of 2850 students?
normalcdf(0.54, 1E99, 0.37, 0.009) ≈
7.55E-80 ≈ 7.55 x or nearly 0.

32. Page 457, E43
a. Do the means of the sampling distributions depend on p? On n?

33. Page 457, E43 a. The means of the sampling distributions
definitely depend on p, as the first three center close to p = 0.2 and the second three center close to p = 0.4.
On n?

34. Page 457, E43 a. The means of the sampling distributions
definitely depend on p, as the first three center close to p = 0.2 and the second three center close to p = 0.4.
The means do not depend on sample size as the sampling distributions have means equal to p regardless of the sample size.

35. Page 457, E43
b. How do the spreads of the sampling distributions depend on p and n?

36. Page 457, E43 b. The spreads of the sampling distributions
decrease as n increases for both values of p.
The spreads also depend upon the value of p, however. For each sample size, the spread for p = 0.4 has a larger standard error than the one for p = 0.2.

37. Page 457, E43
c. How do the shapes of the sampling distributions depend on p and n?

38. Page 457, E43
c. For p = 0.2, the shape is quite skewed for n = 5 and some slight skewness remains at n = 25. For n = 100, the shape is basically symmetric.
For p = 0.4, the shape shows a slight skewness at n = 5 but is fairly symmetric at n = 25 and beyond.
The farther p is from 0.5, the more skewness in the sampling distribution of the sample proportion.

39. Page 457, E43
c. For p = 0.2, the shape is quite skewed for n = 5 and some slight skewness remains at n = 25. For n = 100, the shape is basically symmetric.
For p = 0.4, the shape shows a slight skewness at n = 5 but is fairly symmetric at n = 25 and beyond.
The farther p is from 0.5, the more skewness in the sampling distribution of the sample proportion.

40. Page 457, E43
d. For which combination(s) of p and n would you be willing to use the rule that roughly 95% of the values lie within two standard errors of the mean?

41. Page 457, E43
For which combination(s) of p and n would you be willing to use the rule that roughly 95% of the values lie within two standard errors of the mean?
The rule does not work well for samples of size 5 for either value of p but works well for samples of size 25 or more for both values of p.

42. To be a member of the U. S. Senate, you must be at least 30 years old
To be a member of the U.S. Senate, you must be at least 30 years old. In 2000, 9 of the 100 members of the U.S. Senate were women. Is this a reasonably likely event if gender plays no role in whether a person becomes a U.S. Senator?
Mean: 0.53
Standard error: 0.05

43. To be a member of the U. S. Senate, you must be at least 30 years old
To be a member of the U.S. Senate, you must be at least 30 years old. In 2000, 9 of the 100 members of the U.S. Senate were women. Is this a reasonably likely event if gender plays no role in whether a person becomes a U.S. Senator?
normalcdf(lower bound, upper bound, mean, standard error)

44. To be a member of the U. S. Senate, you must be at least 30 years old
To be a member of the U.S. Senate, you must be at least 30 years old. In 2000, 9 of the 100 members of the U.S. Senate were women. Is this a reasonably likely event if gender plays no role in whether a person becomes a U.S. Senator?
normalcdf(-1E99, 0.09, 0.53, 0.05) = ?

45. To be a member of the U. S. Senate, you must be at least 30 years old
To be a member of the U.S. Senate, you must be at least 30 years old. In 2000, 9 of the 100 members of the U.S. Senate were women. Is this a reasonably likely event if gender plays no role in whether a person becomes a U.S. Senator?
normalcdf(-1E99, 0.09, 0.53, 0.05) = 0
Reasonably likely event or not?

46. To be a member of the U. S. Senate, you must be at least 30 years old
To be a member of the U.S. Senate, you must be at least 30 years old. In 2000, 9 of the 100 members of the U.S. Senate were women. Is this a reasonably likely event if gender plays no role in whether a person becomes a U.S. Senator?
normalcdf(-1E99, 0.09, 0.53, 0.05) = 0
The probability of getting 9 or fewer women just by chance is 0 so this is not a reasonably likely event.

47. About 60% of Mississippians wear seat belts
About 60% of Mississippians wear seat belts. What proportion of seat belt users would be reasonably likely to occur in a random sample
of 40 drivers?
of 100 drivers?
of 400 drivers?

48. About 60% of Mississippians wear seat belts
About 60% of Mississippians wear seat belts. What proportion of seat belt users would be reasonably likely to occur in a random sample
of 40 drivers?
of 100 drivers?
of 400 drivers?
Because the sampling distributions are approximately normal, in each case 95% of the potential values of the sample proportion will lie within 1.96 standard errors of the mean

49. About 60% of Mississippians wear seat belts
About 60% of Mississippians wear seat belts. What proportion of seat belt users would be reasonably likely to occur in a random sample
a. of 40 drivers? b. of 100 drivers? c. of 400 drivers?

50. Questions?