Forensic evidence, such as DNA analysis and fingerprints, is often introduced as part of the evidence
presented in criminal trials. A paper describes a study to learn about how potential jurors view forensic
evidence. Each person in a sample of 153 adult Americans was asked if they agreed with the following
statement: "If forensic evidence suggests a defendant is guilty, this should be enough to convict even if
other evidence (e.g., eyewitness testimony, alibi) suggest otherwise." Fifty-two responded that they agreed
(either strongly agreed or somewhat agreed) with this statement. Suppose that it is reasonable to regard
this sample as a random sample from the population of potential jurors and that you are interested in
learning about the proportion of all potential jurors who agree with the given statement.
The following table is similar to the table that appears in Examples 8.4 and 8.5 and is meant to summarize
what you know about the sampling distribution of p̂ in the situation just described. The "What You Know"
information has been provided. Complete the table by filling in the "How You Know It" column.
Guide On Rating System
Vote
Table:
---------------------------
| | What You Know | How You Know It |
|-------------------------|
| n | 153 | |
| x | 52 | |
| p̂ | ? | |
| σ(p̂) | ? | |
---------------------------
To fill in the "How You Know It" column, we need to use the information given in the problem and apply the formulas for proportions.
1. Calculation of p̂:
The proportion of people who agree with the statement in the sample is given as 52 out of 153 adults. Therefore, we can calculate p̂ as x/n.
p̂ = 52/153 ≈ 0.340
2. Calculation of σ(p̂):
The standard deviation of the sample proportion (σ(p̂)) can be calculated using the formula:
σ(p̂) = sqrt((p̂(1-p̂))/n)
Substituting the known values into the formula, we get:
σ(p̂) = sqrt((0.340(1-0.340))/153) ≈ 0.036
Completing the table:
---------------------------
| | What You Know | How You Know It |
|-------------------------|
| n | 153 | |
| x | 52 | |
| p̂ | 0.340 | x/n |
| σ(p̂) | 0.036 | sqrt((p̂(1-p̂))/n) |
---------------------------
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