A researcher wishes to construct a 95% confidence interval for a population mean. She selects a simple random sample of size 8 from the population. The population is normally distributed and σ is unknown. When constructing the confidence interval estimate, the researcher uses the distribution value from the normal distribution (in Minitab). The confidence level of her resulting confidence interval estimate will beGroup of answer choicesExactly 95%Greater than 95%Less than 90%Exactly 90%Less than 95% but greater than 90%.
Question
A researcher wishes to construct a 95% confidence interval for a population mean. She selects a simple random sample of size 8 from the population. The population is normally distributed and σ is unknown. When constructing the confidence interval estimate, the researcher uses the distribution value from the normal distribution (in Minitab). The confidence level of her resulting confidence interval estimate will beGroup of answer choicesExactly 95%Greater than 95%Less than 90%Exactly 90%Less than 95% but greater than 90%.
Solution
The correct answer is: "Less than 95% but greater than 90%."
Here's why:
When the population standard deviation (σ) is unknown and the sample size is small (n<30), we should use the t-distribution, not the normal distribution, to construct the confidence interval. The t-distribution is wider than the normal distribution, which means it gives a wider confidence interval for the same level of confidence.
If the researcher uses the normal distribution instead of the t-distribution, the confidence interval will be narrower than it should be. This means that the actual level of confidence will be lower than the intended level of 95%. However, because the sample size is not extremely small (n=8), the difference between the t-distribution and the normal distribution is not very large, so the actual level of confidence will still be fairly close to 95%, but it will be less than 95%.
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