Role of Probability

• Concepts in probability allows us
  • to have a better understanding of statistical inference,
  • to quantify the strength or confidence in our conclusion.
• It is natural to study probability prior to studying statistical inference.
• Example 1.1. In a manufacturing process, 100 items are sampled and 10 are found to be defective.
• However, in the long run, the company can only tolerate 5% defective in the process.
• Suppose we learn that; if it does produce items 5% of which are defective, there is a probability of 0.0282 of obtaining 10 or more defective items in a random sample of 100 items from the process.
• The small probability suggests that the process indeed have a long-run defective exceeding 5%.
• Probability aids in translation of sample information into conclusions.

• Example 1.2. We want to determine if the use of nitrogen influences the growth of the roots?
• Experimental Design:
  • Two samples of 10 northern red oak seedlings are planted in a greenhouse, one containing seedlings treated with nitrogen and one containing no nitrogen.
  • Here, we have two samples from two populations.
  • All other environmental conditions are held constant.
• The stem weights in grams were recorded after the end of 140 days.
• Would the data set indicate that nitrogen is effective? We observed:
  • Four nitrogen observations are larger than any of the no-nitrogen observations (see underlined elements in Table 1).
  • Most of the no-nitrogen observations appear to be below the center of the data (see underlined element in Table 1).

Table 1: Observation of nitrogen influences.

No nitrogen	Nitrogen
0.32	0.26
0.53	0.43
0.28	0.47
0.37	0.49
0.47	0.52
0.43	0.75
0.36	0.79
0.42	0.86
0.38	0.62
0.43	0.46

• How this can be quantified or summarized in some sense?
• The conclusions may be summarized in a probability statement:
  The probability that data like these could be observed given that nitrogen has no effect is small, say 0.03.
• That would be strong evidence that the use of nitrogen does have influence.

Figure 1: Stem weight data. (o: the with nitrogen data. x: the without nitrogen data.

• For a statistical problem, the sample along with inferential statistics allow us to draw conclusions about the population, with inferential statistics making clear use of elements of probability. ( inductive in nature)
• For a probability problem, we can draw conclusions about characteristics of hypothetical data taken from the population based on known features of the population. ( deductive in nature)

Figure 2: Fundamental relationship between probability and inferential statistics.

• The procedure we recognize is that we may want to know about the parameter not from the entire population but from the sample.
• The procedure involves two main different jobs. Those are
  1. estimate a parameter of the population through sample,
  2. testing hypotheses (or conjectures/claims) about the parameter.
• Usually the above two procedures are called collectively statistical inference

Figure 3: The Cycle of Statistical Procedure.