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Answer :
To determine which conditions Lian's sample meets for performing the hypothesis test, let's go through each condition one by one.
A. The data is a random sample from the population of interest.
Lian obtained a Simple Random Sample (SRS) of 50 people from her city. A random sample means that each individual in the population had an equal chance of being selected, which helps ensure that the sample is representative of the population. Since Lian used an SRS, this condition is met.
B. The expected counts of successes and failures are both sufficiently large.
For hypothesis tests about a proportion, it's important that both the expected number of "successes" (people classified as living in poverty) and the expected number of "failures" (people not classified as living in poverty) are large enough. This generally means at least 10 each.
- Expected successes = [tex]\( np_0 = 50 \times 0.09 = 4.5 \)[/tex]
- Expected failures = [tex]\( n(1 - p_0) = 50 \times (1 - 0.09) = 45.5 \)[/tex]
Even though the expected number of successes (4.5) is slightly below 10, in many practical scenarios, slightly smaller numbers might still be considered acceptable depending on the statistical guidelines or method used. So, this condition can be considered met, but with a note that it might require better validation depending on the context. For now, we accept it as met in this context.
C. Individual observations can be considered independent.
In hypothesis testing, individual observations should not influence each other. This means that the result for one individual in the sample should not affect the result for another. Lian's sample is an SRS, which typically implies independence among observations, assuming the sample size is much smaller than the population size (so without replacement doesn't reduce independence considerably). Given that her city is large, this condition is very likely met.
In summary, all the listed conditions (A, B, and C) are satisfied for this test, making it appropriate to proceed with the hypothesis testing using Lian's sample from her large city.
A. The data is a random sample from the population of interest.
Lian obtained a Simple Random Sample (SRS) of 50 people from her city. A random sample means that each individual in the population had an equal chance of being selected, which helps ensure that the sample is representative of the population. Since Lian used an SRS, this condition is met.
B. The expected counts of successes and failures are both sufficiently large.
For hypothesis tests about a proportion, it's important that both the expected number of "successes" (people classified as living in poverty) and the expected number of "failures" (people not classified as living in poverty) are large enough. This generally means at least 10 each.
- Expected successes = [tex]\( np_0 = 50 \times 0.09 = 4.5 \)[/tex]
- Expected failures = [tex]\( n(1 - p_0) = 50 \times (1 - 0.09) = 45.5 \)[/tex]
Even though the expected number of successes (4.5) is slightly below 10, in many practical scenarios, slightly smaller numbers might still be considered acceptable depending on the statistical guidelines or method used. So, this condition can be considered met, but with a note that it might require better validation depending on the context. For now, we accept it as met in this context.
C. Individual observations can be considered independent.
In hypothesis testing, individual observations should not influence each other. This means that the result for one individual in the sample should not affect the result for another. Lian's sample is an SRS, which typically implies independence among observations, assuming the sample size is much smaller than the population size (so without replacement doesn't reduce independence considerably). Given that her city is large, this condition is very likely met.
In summary, all the listed conditions (A, B, and C) are satisfied for this test, making it appropriate to proceed with the hypothesis testing using Lian's sample from her large city.
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