Question

The number of contaminating particles on a silicon wafer prior to a certain rinsing process was determined for each wafer in a sample of size 100, resulting in the following frequencies:

Number of particles	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Frequency	1	3	3	12	11	15	16	10	12	4	5	3	2	2	1

(a) What proportion of the sampled wafers had at least one particle? At least five particles? (Round your answers to two decimal places.)

at least one particle
at least five particles

(b) What proportion of the sampled wafers had between five and ten particles, inclusive? Strictly between five and ten particles? (Round your answers to two decimal places.)

between five and ten particles, inclusive

strictly between five and ten particles   (c) Draw a histogram using relative frequency on the vertical axis. How would you describe the shape of the histogram? The histogram is fairly symmetric and bimodal. The histogram is negatively skewed and bimodal.     The histogram is negatively skewed and unimodal. The histogram is fairly symmetric and unimodal. The histogram is positively skewed and unimodal. The histogram is positively skewed and bimodal.

Answer 1

Concepts and reason

A frequency distribution is a table that displays the frequency of various outcomes in a sample. Each entry in the table contains the frequency or count of the occurrences of values within a particular group or interval that summarizes the distribution of values in the sample.

Histogram is used to represent the univariate dataset graphically. Frequency distribution is a representation, either in a graphical or tabular format that displays the number of observations within a given interval.

Fundamentals

Frequency distribution is a representation, either in a graphical or tabular format that displays the number of observations within a given interval. The intervals must be mutually exclusive and exhaustive, and the interval size depends on the data being analysed and the goals of the analyst.

The Histogram consist rectangles, whose area is proportional to frequency of a value and whose width is equal to the class interval.

The probabilities for contaminating particles on a silicon wafer prior to a certain rinsing process were determined as follows:

Number of particles	Frequency (X)	Probability p(x)
0	1	0.01
1	3	0.03
2	3	0.03
3	12	0.12
4	11	0.11
5	15	0.15
6	16	0.16
7	10	0.10
8	12	0.12
9	4	0.04
10	5	0.05
11	3	0.03
12	2	0.02
13	2	0.02
14	1	0.01
Total	100	1.00

(a.1)

The proportion of sampled wafers that had at least one particle is,

(a.2)

The proportion of sampled wafers that had at least five particles is,

(b.1)

The proportion of the sampled wafers had between five and ten particles, inclusive is,

(b.2)

The proportion of the sampled wafers had strictly between five and ten particles is,

(c)

The histogram using relative frequency on the vertical axis is,

The shape of the histogram is fairly symmetric and unimodal.

Ans: Part a.1

The proportion of sampled wafers that had at least one particle is 0.99.

Part a.2

The proportion of sampled wafers that had at least five particles is 0.70.

Part b.1

The proportion of the sampled wafers had between five and ten particles, inclusive is 0.62.

Part b.2

The proportion of the sampled wafers had strictly between five and ten particles is 0.42.

Part c

The histogram using relative frequency is,

The histogram is fairly symmetric and unimodal.