Normal Distribution Examine the distribution shown in the histogram from Exercise 6. Does it appear that the sample data are from a population with a normal distribution? Why or why not?

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Step 1: Observe the histogram provided. A normal distribution typically has a bell-shaped curve, where the frequencies are highest at the center and taper off symmetrically towards the tails. In this histogram, the frequencies do not follow this pattern.

Step 2: Note the peaks and troughs in the histogram. For a normal distribution, the frequencies should gradually decrease as you move away from the center. However, the histogram shows irregular peaks, such as the high frequency at digit 5 and lower frequencies at digits 3 and 4.

Step 3: Assess the symmetry of the distribution. A normal distribution is symmetric around its mean. In this histogram, the frequencies are not symmetric; for example, the frequencies for digits 0, 1, and 2 differ significantly from those for digits 7, 8, and 9.

Step 4: Consider the spread of the data. A normal distribution has a smooth, continuous spread. The histogram here shows a jagged pattern with abrupt changes in frequency, which is inconsistent with a normal distribution.

Step 5: Conclude based on the observations. The irregular shape, lack of symmetry, and uneven spread of frequencies suggest that the sample data are not from a population with a normal distribution.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, where the mean, median, and mode are all equal. In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, which is crucial for understanding the spread and central tendency of the data.

Histogram

A histogram is a graphical representation of the distribution of numerical data, where the data is divided into bins or intervals. The height of each bar represents the frequency of data points within that interval. Histograms are useful for visualizing the shape of the data distribution, identifying patterns, and detecting outliers, which can help in assessing whether the data follows a normal distribution.

Skewness

Skewness refers to the asymmetry of the distribution of values in a dataset. A distribution is said to be positively skewed if it has a longer tail on the right side, while a negatively skewed distribution has a longer tail on the left. Assessing skewness is important when determining if a dataset approximates a normal distribution, as significant skewness indicates deviations from normality, which can affect statistical analyses.

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