Question

consider the normal distribution curve. which statements are true about the curve? check all that apply. the standard deviation of the data is 64. the variance of the data is 49. the median is 64. the data point 75 is less than one standard deviation from the mean. the data point 50 is two standard deviations away from the mean.

Explanation:

Step1: Recall the relationship between variance and standard deviation

The variance \(\sigma^{2}\) and standard deviation \(\sigma\). Given \(\sigma = 7\), then \(\sigma^{2}=7^{2}=49\).

Step2: Analyze the property of the normal distribution curve

In a normal distribution, the mean, median and mode are equal. From the graph, the center of the curve is at \(x = 64\), so the median is \(64\).

Step3: Calculate the range of one - standard deviation from the mean

The mean \(\mu=64\), one - standard deviation: \(\mu-\sigma=64 - 7=57\), \(\mu+\sigma=64 + 7=71\). The data point \(75>71\) (more than one - standard deviation from the mean).

Step4: Calculate the range of two - standard deviations from the mean

Two - standard deviation: \(\mu - 2\sigma=64-2\times7=64 - 14 = 50\), \(\mu+2\sigma=64 + 2\times7=64+14 = 78\).

Answer:

The variance of the data is 49. ; The median is 64. ; The data point 50 is two standard deviations away from the mean.