**Chapter 7. Distributions**

**1.You are watching: What is the probability that z is between 0.0 and 2.0?** Prepare a table reflecting the frequencies, percentages, and also cumulative percentages for the adhering to data. Make the bins 5 units large and make the first bin have the reduced limit of 41.

89, 69, 76, 88, 74, 41, 49, 68, 72, 83, 66, 69, 70, 77,

56, 61, 58, 56, 63, 61, 66, 74, 69, 76, 68, 72, 83, 56

Value | Frequency | Percent | Cum. Percent |

41-45 | 1 | 3.57% | 3.57% |

46-50 | 1 | 3.57% | 7.14% |

51-55 | 0 | 0.00% | 7.14% |

56-60 | 4 | 14.29% | 21.43% |

61-65 | 3 | 10.71% | 32.14% |

66-70 | 8 | 28.57% | 60.71% |

71-75 | 4 | 14.29% | 75.00% |

76-80 | 3 | 10.71% | 85.71% |

81-85 | 2 | 7.14% | 92.86% |

86-90 | 2 | 7.14% | 100.00% |

Total | 28 | 100.00% |

**2.** draw a histogram for the outcomes of #1. Do the bins 5 units broad and do the an initial bin have actually the lower limit the 41.

**3.** draw histograms because that the following species of distributions:

a. Symmetric single-peaked

b. Symmetric bimodal

c. Asymmetric single-peaked

d. Asymmetric bimodal distribution

e. Rectangular (flat) distribution

**4.** below are the last exam scores in percentages because that students in a statistics food that uses a different text.

male | |

81.61 | 56.66 |

83.68 | 75.30 |

59.51 | 84.02 |

70.79 | 60.25 |

63.45 | 89.66 |

64.99 | 77.15 |

62.88 | 86.19 |

84.59 | 66.80 |

82.94 | 88.48 |

38.82 | 73.54 |

75.39 | 64.06 |

69.35 | 51.72 |

66.76 | 93.92 |

65.19 | 87.65 |

73.22 | 90.34 |

69.50 | 89.62 |

89.24 | 44.35 |

74.92 | 72.00 |

83.21 | 54.10 |

83.23 | 90.46 |

82.79 | 77.64 |

89.03 | 62.54 |

72.34 | 82.96 |

**a. Construct separate frequency tables because that males and also females. Set them up so they have actually 12 bins.**

Females | |||

Value | Frequency | Percent | Cum. Percent |

35.00-39.99 | 0 | 0.00% | 0.00% |

40.00-44.99 | 1 | 4.55% | 4.55% |

45.00-49.99 | 0 | 0.00% | 4.55% |

50.00-54.99 | 1 | 4.55% | 9.09% |

55.00-59.99 | 0 | 0.00% | 9.09% |

60.00-64.99 | 1 | 4.55% | 13.64% |

65.00-69.99 | 3 | 13.64% | 27.27% |

70.00-74.99 | 4 | 18.18% | 45.45% |

75.00-79.99 | 1 | 4.55% | 50.00% |

80.00-84.99 | 4 | 18.18% | 68.18% |

85.00-89.99 | 4 | 18.18% | 86.36% |

90.00-94.99 | 3 | 13.64% | 100.00% |

Total | 22 | 100.00% |

Males | |||

Value | Frequency | Percent | Cum. Percent |

35.00-39.99 | 1 | 4.17% | 4.17% |

40.00-44.99 | 0 | 0.00% | 4.17% |

45.00-49.99 | 0 | 0.00% | 4.17% |

50.00-54.99 | 1 | 4.17% | 8.33% |

55.00-59.99 | 2 | 8.33% | 16.67% |

60.00-64.99 | 5 | 20.83% | 37.50% |

65.00-69.99 | 2 | 8.33% | 45.83% |

70.00-74.99 | 2 | 8.33% | 54.17% |

75.00-79.99 | 3 | 12.50% | 66.67% |

80.00-84.99 | 5 | 20.83% | 87.50% |

85.00-89.99 | 3 | 12.50% | 100.00% |

90.00-94.99 | 0 | 0.00% | 100.00% |

Total | 24 | 100.00% |

b. Attract a histogram that shows both distributions, prefer the one on the appropriate side of web page 58.

c. Do women do much better than men? What have the right to you say about this by evaluating the table and histogram friend made?

d. Attract the cumulative portion curves for the two groups on the histograms, using red for females and also blue for males.

e. Find the medians for the two groups by analyzing the cumulative portion curves. What perform the medians tell girlfriend in an answer to inquiry 4 c?

The average for females is 75.0 and also the median for males is around 67.0. The females it seems to be ~ to have actually done better than the males.

f. Room either of these distributions normal? How can you tell? If castle aren"t, what is it about them that makes them not normal?

Neither is normal. The circulation for males is bimodal and the one because that females is asymmetric and also skewed.

**5.** assume the average is 72 and also the standard deviation is 2.4.

a. What percent of cases lie in between the mean and also 2.0 traditional deviations above the mean?

This snapshot shows the area in between the typical (z = 0.0) and z = 2.0. You can see in the table the this area is .4772 of the full area under the regular curve. In a typical distribution, 47.72% that all cases have z-scores in between 0.0 and 2.0.

z | z to mean | smaller area | larger area |

0.00 | 0.0000 | 0.5000 | 0.5000 |

: | : | : | : |

1.90 | 0.4713 | 0.0287 | 0.9713 |

2.00 | 0.4772 | 0.0228 | 0.9772 |

b. What percent of instances are more than 1.8 traditional deviations far from the mean?

Do the right half of the distribution first. You desire the ones above z =1.8, therefore you usage the table to gain the "smaller area" for z = 1.8, which is .0359. You also want the area below 1.8, i beg your pardon is the exact same as the area above z = 1.8, so you main point .0359 by 2 and find that 7.18% are more than 1.8 typical deviations far from the mean.

z | z come mean | smaller area | larger area |

0.00 | 0.0000 | 0.5000 | 0.5000 |

: | : | : | : |

1.80 | 0.4641 | 0.0359 | 0.9641 |

1.90 | 0.4713 | 0.0287 | 0.9713 |

2.00 | 0.4772 | 0.0228 | 0.9772 |

c. What percent of cases are an ext than 1.2 and less than 1.8 traditional deviations below the mean?

First convert the ranges to z-scores. This is easy:

-1.2 and also -1.8.

Then look because that both of this on the table to gain the areas. Indigenous the average to z = 1.2 is .3849; from the median to z = 1.8 is .4641. Subtract the an initial from the second:

.4641 - .3849 = .0792

to see that 7.92% of cases are between 1.2 and 1.8 standard deviations below the mean.

z | z to mean | smaller area | larger area |

0.00 | 0.0000 | 0.5000 | 0.5000 |

: | : | : | : |

1.00 | 0.3413 | 0.1587 | 0.8413 |

1.10 | 0.3643 | 0.1357 | 0.8643 |

1.20 | 0.3849 | 0.1151 | 0.8849 |

: | : | : | : |

1.80 | 0.4641 | 0.0359 | 0.9641 |

1.90 | 0.4713 | 0.0287 | 0.9713 |

2.00 | 0.4772 | 0.0228 | 0.9772 |

d. If you desire the top fifteen percent of scores, how far above the average do you need to go?

This time you begin with the area (15%) and need the z-score that synchronizes to it. The z-score will divide the common curve into two sections; the smaller will be the height 15% and also the bigger will be the other 85%. Therefore look because that the number in the "smaller area" pillar closest come 0.15 and you watch 1.04 is the best. You need to be at least 1.04 conventional deviations over the mean to it is in in the optimal 15%.

z | z to mean | smaller area | larger area |

: | : | : | : |

1.00 | 0.3413 | 0.1587 | 0.8413 |

1.01 | 0.3438 | 0.1562 | 0.8438 |

1.02 | 0.3461 | 0.1539 | 0.8461 |

1.03 | 0.3485 | 0.1515 | 0.8485 |

1.04 | 0.3508 | 0.1492 | 0.8508 |

1.05 | 0.3531 | 0.1469 | 0.8531 |

**6.** If the scores ~ above an test are typically distributed, the median exam score is 86.4, the conventional deviation is 11.5, and also the sample dimension is 500, just how many people have scores over 95?

First, calculation the z-score. It would be:

95 - 86.4 = 8.6

and then:

8.6 /11.5 = .7478

The "smaller area" for z = .75 is 0.2266 or 22.66%. Finally, 22.66% of 500 is 113.3 which rounds to 113 people.

z | z to mean | smaller area | larger area |

: | : | : | : |

0.73 | 0.2673 | 0.2327 | 0.7673 |

0.74 | 0.2704 | 0.2296 | 0.7704 |

0.75 | 0.2734 | 0.2266 | 0.7734 |

0.76 | 0.2764 | 0.2236 | 0.7764 |

0.77 | 0.2794 | 0.2206 | 0.7794 |

7. If the scores on an exam are generally distributed, the typical is 83.1, conventional deviation is 13, and sample dimension is 600, how many civilization have scores between 95 and also 105?

First, calculate the z-scores. For 95, it is:

(95 - 83.1) / 13 = .9154 = z

For 105, that is:

(105 - 83.1) / 13 = 1.685 = z

For raw scores the 95 and also 105, the z-scores will be .9154 and 1.685. The area for z = .92 is .3212; the area because that z = 1.69 is .4545, for this reason to discover the area between 95 and also 105, subtract 32.12% native 45.45% to get 13.33%. To uncover out how countless people, main point 13.33% by 600 to gain 79.98, which rounds to 80 people.

z | z to mean | smaller area | larger area |

: | : | : | : |

0.90 | 0.3159 | 0.1841 | 0.8159 |

0.91 | 0.3186 | 0.1814 | 0.8186 |

0.92 | 0.3212 | 0.1788 | 0.8212 |

0.93 | 0.3238 | 0.1762 | 0.8238 |

: | : | : | : |

1.67 | 0.4525 | 0.0475 | 0.9525 |

1.68 | 0.4535 | 0.0465 | 0.9535 |

1.69 | 0.4545 | 0.0455 | 0.9545 |

1.70 | 0.4554 | 0.0446 | 0.9554 |

**8. What is the z-score that the person in the 80th percentile?**

**This time you begin with the area (80%) and need the z-score that corresponds to it. The z-score will certainly divide the common curve into two sections; the larger will it is in the bottom 80% and also the smaller will it is in the other 20%. So look for the number in the "larger area" pillar closest come 0.80 and also you watch 0.84 is the best. You must be at the very least .84 standard deviations above the typical to it is in in the 80th percentile or the optimal 20%.See more: 42 Is 28 Percent Of What Percent Of 42 Is 28 ? :: Homework Help And Answers**

z | z to mean | smaller area | larger area |

: | : | : | : |

0.83 | 0.2967 | 0.2033 | 0.7967 |

0.84 | 0.2995 | 0.2005 | 0.7995 |

0.85 | 0.3023 | 0.1977 | 0.8023 |

0.86 | 0.3051 | 0.1949 | 0.8051 |

0.87 | 0.3078 | 0.1922 | 0.8078 |

**9. Walk the answer to concern 8 depend on the mean and standard deviation? Why or why not?**

**10. If your z-score is 1.85, what percentile room you in?**

**Since the bigger area for 1.85 is .9678, 96.78% of anyone else is below you. This method you space in the 97th percentile.**

z | z to mean | smaller area | larger area |

: | : | : | : |

1.84 | 0.4671 | 0.0329 | 0.9671 |

1.85 | 0.4678 | 0.0322 | 0.9678 |

1.86 | 0.4686 | 0.0314 | 0.9686 |