A river runs through a certain city and divides the city into two parts, west and east. The population proportion of residents in the west who are in favor of building a new bridge across the river is known to be pW=0.30. The population proportion of residents in the east who are in favor of building a new bridge across the river is known to be pE=0.20. Two random samples of city residents of size 50, one sample from the west and one sample from the east, were taken to investigate opinions on the bridge, where pˆW=0.38 and pˆE=0.25 represent the sample proportions. For samples of size 50 from each population, which of the following is the best interpretation of the mean of the sampling distribution of the difference in the sample proportions (west minus east) of residents from the west and east who are in favor of building the bridge?
Answers
Answer:
For all random samples of size 50 residents from both populations, the difference between the sample proportion of residents from the west and the sample proportion of residents from the east will be 0.13.
A correct interpretation of the mean should not indicate that the difference between sample proportions will have the same value for all samples since the value of the difference in sample proportions will vary from one sample to the next.
Answer:
The difference between the sample proportion of inhabitants from the west and the sample proportion of residents from the east will be 0.13 for all random samples of size 50 from both groups.
Step 1: Identify the population proportions:
The population proportion of residents in the west who are in favor of building a new bridge across the river is known to be pW = 0.30.
The population proportion of residents in the east who are in favor of building a new bridge across the river is known to be pE = 0.20.
Step 2: Identify the sample proportions:
p^W = 0.38 represents the sample proportion of residents in the west who are in favor of building a new bridge across the river, based on a sample of size 50.
p^E = 0.25 represents the sample proportion of residents in the east who are in favor of building a new bridge across the river, based on a sample of size 50.
Step 3: Calculate the mean of the sampling distribution of the difference in the sample proportions:
The mean of the sampling distribution of the difference in the sample proportions (west minus east) is equal to the difference in population proportions (pW - pE), which is
0.38 - 0.25 = 0.13
Step 4: Interpret the mean of the sampling distribution of the difference in the sample proportions:
This represents the expected difference between the sample proportions if the samples were taken an infinite number of times. So, the best interpretation of the mean of the sampling distribution of the difference in the sample proportions is the expected difference between the sample proportions in favor of building the bridge, if the samples were taken an infinite number of times.
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