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The authors of the paper "Development of Nutritionally At-Risk Young Children Is Predicted by Malaria, Anemia, and Stunting in Pemba, Zanzibar" (The Journal of Nutrition [2009]: 763-772) studied factors that might be related to dietary deficiencies in children. Children were observed for a fixed period of time, and the amount of time spent in various activities was recorded. One variable of interest was the amount of time (in minutes) a child spent fussing. Data for 15 children, consistent with summary quantities in the paper, are given in the accompanying table. Normal scores for a sample size of 15 are also given. $$ \begin{array}{|cc|} \hline \text { Fussing Time }(x) & \text { Normal Score } \\ \hline 0.05 & -1.739 \\ 0.10 & -1.245 \\ 0.15 & -0.946 \\ 0.40 & -0.714 \\ 0.70 & -0.515 \\ 1.05 & -0.335 \\ 1.95 & -0.165 \\ 2.15 & 0.000 \\ 3.70 & 0.165 \\ 3.90 & 0.335 \\ 4.50 & 0.515 \\ 6.00 & 0.714 \\ 8.00 & 0.946 \\ 11.00 & 1.245 \\ 14.00 & 1.739 \\ \hline \end{array} $$ a. Construct a normal probability plot for the fussing time data. Does the plot look linear? Do you agree with the authors of the paper that the fussing time distribution is not normal? b. Calculate the correlation coefficient for the (normal score, \(x\) ) pairs. Compare this value to the appropriate critical \(r\) value from Table 6.2 to determine if it is reasonable to think that the fussing time distribution is approximately normal.

Step by step solution

01

Create a scatter plot of the "Fussing Time" and "Normal Score" pairs.

In order to create the normal probability plot, we need to plot the data points using the "Fussing Time" values as the x-axis and the "Normal Score" values as the y-axis.

02

Interpret the scatter plot to check if it looks linear.

After creating the scatter plot, we will determine if the data points follow a linear pattern. If the data points have a linear pattern, this means the data is normally distributed and vice versa. #b. Calculating correlation coefficient and comparing with critical r value to determine if the data is normal#

03

Calculate the correlation coefficient for the given (normal score, fussing time) pairs.

In order to calculate the correlation coefficient, we will use the following formula: \( r = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sqrt{\sum (x - \bar{x})^2 \sum (y - \bar{y})^2}} \) Where r is the correlation coefficient, x is the fussing time, y is the normal score for the given data, and \( \bar{x} \) and \( \bar{y} \) are the mean values of x and y respectively. This will give us a value between -1 and 1, indicating the strength and direction of the relationship between the two variables.

04

Compare the correlation coefficient (r) to the appropriate critical r value to determine normality.

To determine if the relationship between the fussing time and corresponding normal scores is approximately normal, compare the computed correlation coefficient to the critical r value in the table. If the correlation coefficient is greater than the critical r value, it means the data is likely to be normally distributed. If not, it means the distribution may not be normal, and other statistical methods may need to be applied.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Probability Plot

A normal probability plot is a graphical tool used to determine whether a sample of data is approximately normally distributed. When creating a normal probability plot, you plot your data values on the x-axis against the expected normal scores on the y-axis. In the context of our exercise, the fussing time is plotted against the provided normal scores.

In an ideal normal probability plot, the points should fall approximately along a straight line if the data is normally distributed. It's crucial to examine the plot carefully: deviations from a straight line suggest deviations from normality. However, minor deviations can occur in real-world data and still indicate an approximately normal distribution. The key is in the pattern and correlation of the plotted points. Consistently curving away from linearity might suggest a non-normal distribution, as observed in the exercise with fussing times.

Correlation Coefficient

The correlation coefficient, often denoted as \( r \), is a measure that indicates the extent to which two variables are linearly related. Calculating \( r \) helps determine if the data points on the normal probability plot align closely with a straight line.

Using the formula \( r = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sqrt{\sum (x - \bar{x})^2 \sum (y - \bar{y})^2}} \), where \( x \) is the fussing time and \( y \) is the normal score, we evaluate the linear relationship between these variables by computing a correlation coefficient. The resultant \( r \) value ranges from -1 to 1, where values close to 1 or -1 indicate a strong positive or negative linear relationship, respectively. Meanwhile, values near 0 suggest little or no linear relationship. For normally distributed data, \( r \) should be close to 1.

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric around the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. In the context of fussing times, to assume normality implies that the majority of fussing times center around a central value, with fewer instances occurring as you move further away from this center.

The shape of a normal distribution is a bell curve, commonly referred to as the Gaussian distribution. If the fussing times were normally distributed, their normal probability plot would likely show a straight line, as mentioned previously.

• The mean, median, and mode of a perfectly normal distribution are all equal.
• Approximately 68% of the data falls within one standard deviation of the mean.
• About 95% is within two standard deviations, and 99.7% falls within three standard deviations.

Normal distribution is fundamental in statistical analysis, making it crucial to recognize when data approximates this distribution.

Critical Value

A critical value helps us decide whether a computed statistic is within a range that we would expect from a specified distribution, under a null hypothesis. In this exercise, the critical value refers to a threshold correlation coefficient value from a statistical table.

By comparing our calculated correlation coefficient \( r \) with the critical value, we can judge if the data distribution is approximately normal. If the correlation coefficient surpasses this critical value, we can assume the sample data is likely drawn from a normal distribution. This comparison is essential because even if a plot seems linear, statistical confirmation helps with precise decision-making.

Understanding the critical value's role helps in distinguishing between mere visual inspection and quantitatively backed evaluation in statistical analyses.