The following data represent the age (in weeks) at which...

The following data represent the age (in weeks) at which babies first crawl based on a survey of 12 mothers conducted by Essential Baby. $$ \begin{array}{llllll} \hline 52 & 30 & 44 & 35 & 39 & 26 \\ \hline 47 & 37 & 56 & 26 & 39 & 28 \\ \hline \end{array} $$ (a) Draw a normal probability plot to determine if it is reasonable to conclude the data come from a population that is normally distributed. (b) Draw a boxplot to check for outliers. (c) Construct and interpret a $95 \%$ confidence interval for the mean age at which a baby first crawls.

The following data represent the age (in weeks) at which babies first crawl based on a survey of 12 mothers conducted by Essential Baby.
$$
\begin{array}{llllll}
\hline 52 & 30 & 44 & 35 & 39 & 26 \\
\hline 47 & 37 & 56 & 26 & 39 & 28 \\
\hline
\end{array}
$$
(a) Draw a normal probability plot to determine if it is reasonable to conclude the data come from a population that is normally distributed.
(b) Draw a boxplot to check for outliers.
(c) Construct and interpret a $95 \%$ confidence interval for the mean age at which a baby first crawls.

Key Concepts

Student’s t-Distribution and Small Sample Inference

When the sample size is small and the population standard deviation is unknown, the Student’s t-distribution is employed to construct confidence intervals for the mean. It adjusts for the increased variability and uncertainty inherent with smaller samples, making it a key tool in small sample statistical inference.

Boxplot and Outlier Detection

A boxplot is a graphical summary that shows the median, quartiles, and potential outliers in a dataset. It is used to quickly assess the central tendency, dispersion, and symmetry of the data. Outliers are typically indicated as points beyond the whiskers, which are determined by a multiple of the interquartile range (IQR). Identifying outliers is crucial as they can influence statistical analyses and insights.

Confidence Interval Construction for the Mean

Constructing a confidence interval for the mean involves estimating the range within which the true population mean is likely to lie based on a sample. This process takes into account the sample mean, variability, and sample size. For small samples drawn from a normally distributed population (or when normality is assumed), the t-distribution is typically used to account for extra uncertainty in the estimate.

Normality Assessment

This concept involves determining whether the data approximates a normal distribution. It is a critical assumption in many statistical procedures, especially inference techniques. Assessing normality can be done visually, for example by using a normal probability (or quantile-quantile) plot, where data points that closely follow a straight line suggest that the normality assumption is reasonable.

Quantile?Quantile (QQ) Plot

A QQ plot is a graphical tool that plots the observed quantiles of data against the expected quantiles from a normal distribution. This allows for visual inspection of any deviations from normality. If the data are normally distributed, the points will fall approximately along a straight line, whereas systematic departure from this line indicates potential skewness or outliers.

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