Question

the graph shows a distribution of data. which statement about the data is true? the data has a standard deviation of 0.1. the mean of the data is greater than 0.5. a value of 0.7 is within 1 standard deviation of the mean. a value of 0.9 is more than 2 standard deviations from the mean.

Explanation:

Step1: Analyze the mean

From the graph, the mean \(\mu = 0.5\). So the statement "The mean of the data is greater than \(0.5\)" is false.

Step2: Analyze the standard - deviation

Assume the standard deviation \(\sigma\). If \(\mu = 0.5\), for a normal - like distribution (symmetric bell - shaped curve shown in the graph), if \(\sigma=0.1\), the interval \((\mu - \sigma,\mu+\sigma)=(0.4,0.6)\), \((\mu - 2\sigma,\mu + 2\sigma)=(0.3,0.7)\), \((\mu-3\sigma,\mu + 3\sigma)=(0.2,0.8)\).
The statement "A value of \(0.7\) is within \(1\) standard deviation of the mean" is false since \(0.7
otin(0.4,0.6)\) (assuming \(\sigma = 0.1\)).
For the statement "The data has a standard deviation of \(0.1\)", we cannot confirm this just from the graph's visual shape (the standard deviation is a measure of spread, and without numerical data points related to variance calculation, we can't say for sure).
For the statement "A value of \(0.9\) is more than \(2\) standard deviations from the mean":
If \(\mu = 0.5\) and assume \(\sigma=0.1\), \(\mu+2\sigma=0.5 + 2\times0.1=0.7\), \(\mu - 2\sigma=0.5-2\times0.1 = 0.3\). Since \(0.9>0.7\) (if \(\sigma = 0.1\)), \(|0.9 - 0.5|=0.4\) and if \(\sigma = 0.1\), \(2\sigma=0.2\), \(0.4>0.2\)

Answer:

A value of \(0.9\) is more than \(2\) standard deviations from the mean.