A simple random sample of size 17 is taken from a population...

A simple random sample of size 17 is taken from a population with unknown standard deviation. A normal probability plot of the data reveals an outlier but is otherwise roughly linear. Can you reasonably apply the $t$ -interval procedure? Explain your answer.

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Key Concepts

Small Sample Size Considerations

With a small sample size, such as 17 observations, the analysis is more sensitive to departures from assumptions, including the presence of outliers. In such cases, even a single outlier can have a disproportionate effect on the calculations, making it important to assess whether the outlier is a valid observation or a data error before proceeding.

Impact of Outliers

Outliers can greatly influence the sample mean and variance, potentially violating the normality assumption. Even if a normal probability plot appears roughly linear overall, the presence of an outlier suggests that there may be departures from normality or unusual variability that could affect the validity of the t-interval procedure.

Assumption of Normality

A key assumption of the t-interval procedure is that the sample comes from a normally distributed population. This assumption is crucial, especially with small sample sizes, because deviations from normality can significantly impact the reliability of the interval estimates.

t-interval Procedure

The t-interval procedure is used to estimate the mean of a normally distributed population when the population standard deviation is unknown. It relies on the t-distribution, which adjusts for uncertainty due to a small sample size by having heavier tails than the normal distribution.

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