A simple random sample of size 100 is taken from a...

A simple random sample of size 100 is taken from a population with unknown standard deviation. A normal probability plot of the data displays significant curvature but no outliers. Can you reasonably apply the $t$ -interval procedure? Explain your answer.

Key Concepts

t-Interval Procedure

The t-interval procedure is used to construct a confidence interval for the population mean when the population standard deviation is unknown. It relies on the t-distribution, which accounts for additional uncertainty introduced by estimating the standard deviation from the sample rather than knowing it beforehand.

Normality Assumption

A core assumption for the validity of the t-interval procedure is that the underlying population distribution is normal. This assumption ensures that the sample mean follows a t-distribution, particularly when sample sizes are small. Departures from normality can potentially invalidate the inference provided by the t-procedure.

Assessment of Normality with Graphical Methods

Graphical diagnostics such as normal probability plots are used to assess whether the data appear to come from a normal distribution. Significant curvature in these plots suggests systematic departures from normality, such as skewness or kurtosis, which may violate the assumption necessary for the t-interval procedure.

Robustness and Sample Size Considerations

While the t-interval procedure is somewhat robust to minor deviations from normality, especially with moderately large sample sizes, serious departures indicated by clear patterns in diagnostic plots may undermine the procedure’s reliability. When the sample size is large, the central limit theorem provides some justification for using the procedure; however, substantial non-normality can still lead to poor approximations.

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