Answer :
To calculate a confidence interval for a population proportion, several conditions must be met:
1. Randomness: The experiment should consist of independent and random trials. In this case, coin tosses are assumed to be independent, so this condition is satisfied.
2. 10% Condition: When sampling without replacement from a finite population, the sample should not exceed 10% of the population. For coin tosses, this condition is either assumed to be met or is not applicable. Hence, we consider it satisfied.
3. Large Counts (Success-Failure) Condition: This requires that both the number of successes and the number of failures be at least 10. With 10 coin tosses, there are [tex]$6$[/tex] heads (considered successes) and [tex]$4$[/tex] tails (failures). Since neither [tex]$6$[/tex] nor [tex]$4$[/tex] is at least [tex]$10$[/tex], this condition is not met.
Since the Large Counts Condition is not satisfied, the conditions for calculating a confidence interval for the population proportion are not fully met.
Thus, the correct answer is:
[tex]$$\text{No, the Large Counts Condition is not met.}$$[/tex]
1. Randomness: The experiment should consist of independent and random trials. In this case, coin tosses are assumed to be independent, so this condition is satisfied.
2. 10% Condition: When sampling without replacement from a finite population, the sample should not exceed 10% of the population. For coin tosses, this condition is either assumed to be met or is not applicable. Hence, we consider it satisfied.
3. Large Counts (Success-Failure) Condition: This requires that both the number of successes and the number of failures be at least 10. With 10 coin tosses, there are [tex]$6$[/tex] heads (considered successes) and [tex]$4$[/tex] tails (failures). Since neither [tex]$6$[/tex] nor [tex]$4$[/tex] is at least [tex]$10$[/tex], this condition is not met.
Since the Large Counts Condition is not satisfied, the conditions for calculating a confidence interval for the population proportion are not fully met.
Thus, the correct answer is:
[tex]$$\text{No, the Large Counts Condition is not met.}$$[/tex]
