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Answer :
To determine if Lian's sample meets the conditions for performing the hypothesis test about the proportion of people living in poverty in her city, we must consider three main conditions:
1. Random Sample: The sample should be a Simple Random Sample (SRS) from the population of interest. This is important to ensure that the sample accurately represents the population and that various potential biases are minimized. In Lian's case, it is stated that she obtained an SRS, so this condition is met.
2. Expected Counts of Successes and Failures: For the test to be valid, the expected number of successes (people classified as living in poverty) and failures (people not classified as living in poverty) should both be sufficiently large, typically at least 5. To check this:
- Calculate the expected number of successes:
[tex]\[
\text{Expected successes} = \text{sample size} \times p = 50 \times 0.09 = 4.5
\][/tex]
- Calculate the expected number of failures:
[tex]\[
\text{Expected failures} = \text{sample size} \times (1 - p) = 50 \times 0.91 = 45.5
\][/tex]
The expected number of successes here is 4.5, which is less than 5. Therefore, this condition is not met.
3. Independence of Observations: Each observation in the sample should be independent. A common rule of thumb to ensure this is that the sample size should be less than 10% of the total population size. If Lian's city is large enough such that 50 is less than 10% of the population, this condition is likely met.
Based on this information:
- Condition A (The data is a random sample from the population of interest) is met.
- Condition B (The expected counts of successes and failures are both sufficiently large) is not met since the expected number of successes is less than 5.
- Condition C (Individual observations can be considered independent) is met, assuming the sample size is less than 10% of the population.
Thus, the true answers are:
- A: True
- B: False
- C: True
1. Random Sample: The sample should be a Simple Random Sample (SRS) from the population of interest. This is important to ensure that the sample accurately represents the population and that various potential biases are minimized. In Lian's case, it is stated that she obtained an SRS, so this condition is met.
2. Expected Counts of Successes and Failures: For the test to be valid, the expected number of successes (people classified as living in poverty) and failures (people not classified as living in poverty) should both be sufficiently large, typically at least 5. To check this:
- Calculate the expected number of successes:
[tex]\[
\text{Expected successes} = \text{sample size} \times p = 50 \times 0.09 = 4.5
\][/tex]
- Calculate the expected number of failures:
[tex]\[
\text{Expected failures} = \text{sample size} \times (1 - p) = 50 \times 0.91 = 45.5
\][/tex]
The expected number of successes here is 4.5, which is less than 5. Therefore, this condition is not met.
3. Independence of Observations: Each observation in the sample should be independent. A common rule of thumb to ensure this is that the sample size should be less than 10% of the total population size. If Lian's city is large enough such that 50 is less than 10% of the population, this condition is likely met.
Based on this information:
- Condition A (The data is a random sample from the population of interest) is met.
- Condition B (The expected counts of successes and failures are both sufficiently large) is not met since the expected number of successes is less than 5.
- Condition C (Individual observations can be considered independent) is met, assuming the sample size is less than 10% of the population.
Thus, the true answers are:
- A: True
- B: False
- C: True
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