Assuming data come from a random sample, under which of the following conditions should we not calculate a confidence interval for a population mean? Population distribution is unknown and sample size is 50 individuals. Population distribution is unknown and sample size is 20 individuals. Population is normally distributed and sample size is 20 individuals. Population is normally distributed and sample size is 50 individuals.
Question
Assuming data come from a random sample, under which of the following conditions should we not calculate a confidence interval for a population mean? Population distribution is unknown and sample size is 50 individuals. Population distribution is unknown and sample size is 20 individuals. Population is normally distributed and sample size is 20 individuals. Population is normally distributed and sample size is 50 individuals.
Solution
We should not calculate a confidence interval for a population mean when the population distribution is unknown and the sample size is 20 individuals.
Here's why:
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When the population distribution is unknown and the sample size is 50 individuals: In this case, we can still calculate a confidence interval for the population mean. This is because of the Central Limit Theorem, which states that if the sample size is large enough (usually considered to be over 30), the sampling distribution of the mean will be approximately normally distributed, regardless of the shape of the population distribution.
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When the population distribution is unknown and the sample size is 20 individuals: In this case, we should not calculate a confidence interval for the population mean. The sample size is too small to invoke the Central Limit Theorem, and without knowing the shape of the population distribution, we cannot make assumptions about the sampling distribution of the mean.
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When the population is normally distributed and the sample size is 20 individuals: In this case, we can calculate a confidence interval for the population mean. Even though the sample size is small, the fact that the population is normally distributed allows us to make assumptions about the sampling distribution of the mean.
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When the population is normally distributed and the sample size is 50 individuals: In this case, we can also calculate a confidence interval for the population mean. The sample size is large enough to invoke the Central Limit Theorem, and the population is normally distributed.
Similar Questions
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