Review 1 2003.11.5
Chapter 1:
1. Elements, Variable, and Observations:
2. Type of Data: Qualitative Data and Quantitative Data
(a) Qualitative data may be nonnumeric or numeric.
(b) Quantitative data are always numeric.
(c) Arithmetic operations are only meaningful with quantitative data.
Chapter 2: Figure 2.22, p. 66.
1. Summarizing qualitative data:
l Frequency distribution, relative frequency distribution, and percent frequency distribution.
l Bar plot and Pie plot.
2. Summarizing quantitative data:
l Frequency distribution, relative frequency distribution, percent frequency distribution, cumulative frequency distribution, cumulative relative frequency distribution, cumulative percent frequency distribution
l Histogram, Ogive, and stem-and leaf display.
Chapters 3
Measures of Location, Dispersion, Exploratory Data Analysis, Measure of Relative Location, Weighted and Grouped Mean and Variance
Chapter 4:
l Tabular and Graphical Methods: Crosstabulation (qualitative and quantitative data) and Scatter Diagram (only quantitative data).
l Numerical Method: Covariance and Correlation Coefficient.
Chapter 5:
1. Multiple Step Experiments, Permutations, and Combinations.
2. Event, Addition Law, Mutually Exclusive Events and Independent Event.
3. Bayes’ Theorem
Example:
Table 1.1 (p. 5) in the textbook!!
25 elements (25 companies): Advanced Comm. Systems, Ag-Chem Equipment Co.,…,Webco Industries Inc..
5 variables : Exchange, Ticker Symbol, Annual Sales, Share Price,
Price/Earnings Ratio.
25 observations: (OTC, ACSC, 75.10, 0.32, 39.10), (OTC, AGCH, 321.10, 0.48, 23.40),…, (AMEX, WEB, 153.50, 0.88, 7.50).
Qualitative variables: Exchange, Ticker Symbol.
Quantitative variables: Annual Sales, Share Price, Price/Earnings Ratio.
Example:
The amount of time (in minutes) that a sample of students spends watching television per day is given below.
40 / 25 / 35 / 30 / 20 / 40 / 30 / 2040 / 10 / 30 / 20 / 10 / 5 / 20
(a) Compute the mean
(b) The standard deviation.
(c) The coefficient of variation.
(d) The 40th percentile.
(e) The mode.
(f) The interquartile range.
(g) Construct a frequency distribution, a cumulative frequency distribution and a relative frequency distribution. Let the first class be 1-10. (10%)
(h) Based on the frequency table obtained in (g), compute the mean and variance for the grouped data. Compare with the results in (a) and (b). (10%)
[solution:]
(a)
(b)
(c)
.
(d)
1. The data are
5 / 10 / 10 / 20 / 20 / 20 / 20 / 2530 / 30 / 30 / 35 / 40 / 40 / 40
2.
Thus,
is the 40th percentile.
(e)The mode is 20.
(f) Since
,
.
(g)
Class / Frequency / Cumulative Frequency / Relative Frequency1-10 / 3 / 3 / 0.2
11-20 / 4 / 7 / 4/15
21-30 / 4 / 11 / 4/15
31-40 / 4 / 15 / 4/15
(h)
and
Thus, .
The group mean and the group standard deviation are close to the original mean and standard deviation.
Example:
The flashlight batteries produced by one of the manufacturers are known to have an average life of 60 hours with a standard deviation of 4 hours.
(a) At least what percentage of batteries will have a life of 54 to 66 hours?
(b) At least what percentage of the batteries will have a life of 52 to 68 hours?
(c) Determine an interval for the batteries’ lives that will be true for at least 80% of the batteries.
[solution:]
Denote
(a)
Thus, by Chebyshev’s theorem, within 1.5 standard deviation, there is at least
of batteries.
(b)
Thus, by Chebyshev’s theorem, within 2 standard deviation, there is at least
of batteries.
(c)
Thus, within standard deviation, there is at least 80% of batteries. Therefore,
.
Example:
Assume you are taking two courses this semester (S and C). The probability that you will pass course S is 0.835, the probability that you will pass both courses is 0.276. The probability that you will pass at least one of the courses is 0.981.
(a) What is the probability that you will pass course C?
(b) Is the passing of the two courses independent event?
(c) Are the events of passing the courses mutually exclusive? Explain.
[solution:]
(a)
Let A be the event of passing course S and B be the event of passing course C. Thus,
.
.
(b)
Thus, events A and B are not independent. That is, passing of two courses are not independent events.
(c)
Since , events A and B are not mutually exclusive.
Example:
In a random sample of Tung Hai University students 50% indicated they are business majors, 40% engineering majors, and 10% other majors. Of the business majors, 60% were female; whereas, 30% of engineering majors were females. Finally, 20% of the other majors were female. Given that a person is female, what is the probability that she is an engineering major?
[solution:]
Let
A1: the students are engineering majors
A2: the students are business majors
A3: the students are other majors.
B: the students are female.
Originally, we know
.
Then, by Bayes’ theorem,
2