We know that 65% of all Americans prefer chocolate over vanilla ice cream. Suppose that 1000 people were randomly selected. The Sampling Distribution of the sample proportion is
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0.01508
the distribution of all possible sample proportions that could be obtained from repeated sampling. In this case, since 65% of all Americans prefer chocolate over vanilla ice cream, the expected value of the sample proportion would also be 65%.
The Sampling Distribution of the sample proportion follows a normal distribution when certain conditions are met. One important condition is that the sample size should be sufficiently large, typically at least 30. In this case, since 1000 people were randomly selected, we can assume that the sample size is large enough for the Sampling Distribution to be approximately normal.
The standard deviation of the Sampling Distribution can be calculated using the formula:
σ_p = √(p * (1-p) / n)
where p is the population proportion (65% or 0.65 in this case) and n is the sample size (1000).
So, the Sampling Distribution of the sample proportion in this scenario would be approximately normal with an expected value of 0.65 and a standard deviation of √(0.65 * (1-0.65) / 1000).
To understand the sampling distribution of the sample proportion in this scenario, we need to consider the concept of sampling.
Sampling is the process of selecting a subset of individuals from a larger population in order to make inferences about the population as a whole. In this case, we have a population of all Americans, and we want to make inferences about their preferences for chocolate vs. vanilla ice cream.
The sampling distribution refers to the distribution of sample statistics, like the sample proportion, that we would get if we were to take multiple samples of the same size from the population.
In this scenario, we are told that 65% of all Americans prefer chocolate over vanilla ice cream. So, if we were to take a single random sample of 1000 Americans, we would expect that about 65% of them would prefer chocolate.
However, if we were to take multiple random samples of 1000 Americans and calculate the proportion of each sample that prefers chocolate, we would likely get slightly different values each time. The sampling distribution of the sample proportion is the distribution of these different sample proportions.
The sampling distribution of the sample proportion can be approximated using the Central Limit Theorem. This theorem states that, under certain conditions, the sampling distribution of the sample proportion will be approximately normal, regardless of the shape of the population distribution. These conditions include having a random sample and a sufficiently large sample size.
In this case, we are assuming that the sample of 1000 people was randomly selected from the population of all Americans. If we also assume that the sample size of 1000 is sufficiently large, then we can expect the sampling distribution of the sample proportion to be approximately normal.
The mean of the sampling distribution of the sample proportion will be equal to the population proportion, which in this case is 65%. The standard deviation of the sampling distribution, also known as the standard error, can be calculated using the formula:
Standard Error = sqrt((p * (1 - p)) / n)
where p is the population proportion (65% or 0.65), and n is the sample size (1000).
So, to summarize, the sampling distribution of the sample proportion in this scenario is approximately normal with a mean of 0.65 and a standard deviation of sqrt((0.65 * (1 - 0.65)) / 1000).