Homework Help: Questions and Answers: A researcher is wondering whether the smoking habits of young adults (18 – 25 years of age) in a certain city in the US. are the same as the proportion of the general population of young adults in the U.S. A recent study stated that the proportion of young adults who reported smoking at least twice a week or more in the last month was 0.16. The researcher collected data from a random sample of 75 adults in the city of interest. Check that the conditions hold so that the sampling distribution of the z-statistic will approximately follow the standard normal distribution. Are the conditions satisfied? If not, choose the condition that is not satisfied.
a) Yes, all the conditions are satisfied
b) No, the population of interest is not large enough to assume independence
c) No, the researcher did not collect a random sample
d) No, the researcher did not collect a large enough sample
Answer:
To determine whether the conditions are satisfied for the sampling distribution of the z-statistic to approximately follow the standard normal distribution, we need to check the following conditions:
- Random Sampling: The sample should be a random sample from the population.
- Independence: The sample observations should be independent. This is generally satisfied if the sample size is less than 10% of the population.
- Sample Size (Normality): The sample size should be large enough so that both np and n(1−p) are at least 10, where n is the sample size and p is the population proportion.
Let’s go through each condition step-by-step:
1. Random Sampling
- The problem states that the researcher collected data from a random sample of 75 adults in the city. Thus, this condition is satisfied.
2. Independence
- The independence condition requires that the sample size should be less than 10% of the population. Given that we are sampling from a city and assuming the population of young adults in the city is large, this condition is likely satisfied. However, since we are not given the actual population size, we assume this condition is met unless explicitly stated otherwise.
3. Sample Size (Normality)
- Calculate np and n(1−p) using n = 75 and p = 0.16:
np = 75 × 0.16 = 12
n(1−p) = 75 × (1−0.16) = 75 × 0.84 = 63 - Both np = 12 and n(1−p) = 63 are greater than 10, so this condition is satisfied.
Given Options Analysis
a) Yes, all the conditions are satisfied
- This appears to be correct based on our analysis.
b) No, the population of interest is not large enough to assume independence
- This is incorrect; we can reasonably assume the population is large enough.
c) No, the researcher did not collect a random sample
- This is incorrect; the question states that a random sample was collected.
d) No, the researcher did not collect a large enough sample
- This is incorrect; the sample size of 75 is sufficient.
Final Answer
All the conditions for the sampling distribution of the z-statistic to approximately follow the standard normal distribution are satisfied. Therefore, the answer is:
a) Yes, all the conditions are satisfied
The sample is random and large enough, the population can be assumed to be much larger than the sample, and we have a known population parameter.
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