Question

consider the normal distribution defined by the given figure. find the mean \\( \mu \\) and the standard deviation \\( \sigma \\) of the distribution
\\( \mu = 4.5 \mathrm{kg} \\) (simplify your answer)
\\( \sigma=\square \mathrm{kg} \\) (simplify your answer)

Explanation:

Step1: Recall the empirical rule for normal distribution

The empirical rule states that for a normal distribution, approximately 68% of the data lies within \( \mu\pm\sigma\). That is, if the lower bound is \(L = \mu-\sigma\) and the upper bound is \(U=\mu + \sigma\), then \(U - L=2\sigma\).

Step2: Calculate the standard deviation \(\sigma\)

We know that \(L = 3.8\) kg and \(U = 5.2\) kg. Using the formula \(U - L=2\sigma\), substitute the values: \(5.2-3.8 = 2\sigma\). So, \(1.4=2\sigma\). Solving for \(\sigma\), we get \(\sigma=\frac{1.4}{2}\).

Answer:

\(\sigma = 0.7\) kg