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It is common knowledge that a fair penny will land heads up [tex]50\%[/tex] of the time and tails up [tex]50\%[/tex] of the time. It is very unlikely for a penny to land on its edge when flipped, so a probability of 0 is assigned to this outcome. A curious student suspects that 5 pennies glued together will land on their edge [tex]50\%[/tex] of the time. To investigate this claim, the student securely glues together 5 pennies and flips the penny stack 100 times. Of the 100 flips, the penny stack lands on its edge 46 times. The student would like to know if the data provide convincing evidence that the true proportion of flips for which the penny stack will land on its edge differs from [tex]0.5[/tex].

What are the appropriate hypotheses for this test?

A. [tex]H_0: p=0.5[/tex] versus [tex]H_{a}: p\ \textless \ 0.5[/tex], where [tex]p=[/tex] the true proportion of flips for which the penny stack will land on its edge.

B. [tex]H_0: p=0.5[/tex] versus [tex]H_{a}: p\ \textgreater \ 0.5[/tex], where [tex]p=[/tex] the true proportion of flips for which the penny stack will land on its edge.

C. [tex]H_0: p=0.5[/tex] versus [tex]H_a: p \neq 0.5[/tex], where [tex]p=[/tex] the true proportion of flips for which the penny stack will land on its edge.

D. [tex]H_0: p \neq 0.5[/tex] versus [tex]H_{a}: p=0.5[/tex], where [tex]p=[/tex] the true proportion of flips for which the penny stack will land on its edge.

Sagot :

GinnyAnswer
To determine the appropriate hypotheses for the given scenario, we need to look at the problem and verify which hypothesis setup matches the investigation.

### Scenario Analysis:

1. Statement of Problem:
- A student suspects that 5 pennies glued together will land on their edge \(50\%\) of the time.
- The student conducts an experiment by flipping the stack 100 times, and it lands on its edge 46 times.
- The student wishes to know if this result provides convincing evidence that the true proportion, \(p\), of flips resulting in the stack landing on its edge differs from \(0.5\).

2. Defining Hypotheses:
- The null hypothesis (\(H_0\)) typically states that there is no effect or no difference. It represents the status quo or the situation where the student's suspicion is not true.
- The alternative hypothesis (\(H_a\)) represents what the student is trying to find evidence for; in this case, that the true proportion differs from \(0.5\).

3. Possible Hypotheses:
- \(H_0: p = 0.5\) versus \(H_a: p < 0.5\)
- \(H_0: p = 0.5\) versus \(H_a: p > 0.5\)
- \(H_0: p = 0.5\) versus \(H_a: p \neq 0.5\)
- \(H_0: p \neq 0.5\) versus \(H_a: p = 0.5\)

### Appropriate Hypotheses:
Given that the student is investigating whether the true proportion of flips for which the stack lands on its edge differs from \(0.5\), the alternative hypothesis should be a two-tailed hypothesis, suggesting that \(p\) could be either greater than or less than \(0.5\).

Thus, the correct hypotheses for this test should be:

- Null Hypothesis (\(H_0\)): \(p = 0.5\)
- Alternative Hypothesis (\(H_a\)): \(p \neq 0.5\)

Where \(p\) is the true proportion of flips for which the penny stack will land on its edge.

### Conclusion:
The appropriate hypotheses to test whether the observed data provide convincing evidence that the true proportion of flips for which the penny stack will land on its edge differs from \(0.5\) are:

[tex]\[ H_0: p = 0.5 \][/tex]
[tex]\[ H_a: p \neq 0.5 \][/tex]

Hence, the selected option is:
[tex]\[ \textbf{H_0: p=0.5 versus H_a: p \neq 0.5, where } p \text{ is the true proportion of flips for which the penny stack will land on its edge.} \][/tex]
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