Question

using the graph as a reference, select all statements that are true about a normal distribution of data. normal distributions are symmetrical about the mean, \\( \mu \\). the total area under the curve is 100. ninety - five percent of the data lies within 1 standard deviation of the mean. the probability that an event is within 1 standard deviation above the mean is 34%. all data sets are normally distributed. (graph: normal distribution curve with labels \\( \mu - 3\sigma \\), \\( \mu - 2\sigma \\), \\( \mu - \sigma \\), \\( \mu \\), \\( \mu + \sigma \\), \\( \mu + 2\sigma \\), \\( \mu + 3\sigma \\) and areas 68%, 95%, 99.7% )

Explanation:

Step1: Analyze each statement

• Statement 1: Normal distributions are symmetric about the mean (\(\mu\)). From the normal distribution graph, the curve is symmetric around \(\mu\) (the peak is at \(\mu\) and the left and right sides mirror), so this is true.
• Statement 2: The total area under the curve of a normal distribution is 1 (representing probability 1 or 100% in terms of proportion, but the area value is 1, not 100). So this is false.
• Statement 3: From the empirical rule (68 - 95 - 99.7 rule), 95% of data lies within \(2\) standard deviations of the mean, and 68% within \(1\) standard deviation. So "Ninety - five percent of the data lies within 1 standard deviation of the mean" is false.
• Statement 4: The total area within \(1\) standard deviation of the mean is 68% (from \(\mu-\sigma\) to \(\mu + \sigma\)). Since the distribution is symmetric, the area from \(\mu\) to \(\mu+\sigma\) is half of 68%, which is 34%. So the probability that an event is within 1 standard deviation above the mean is 34%, this is true.
• Statement 5: Not all data sets are normally distributed. There are many non - normal distributions (e.g., skewed distributions). So this is false.

Answer:

• Normal distributions are symmetrical about the mean, \(\mu\).
• The probability that an event is within 1 standard deviation above the mean is 34%.