Suppose the weights of a group of students are normally distributed with a mean of $\mu =70 \mathrm{kg}$ and a standard deviation of $\sigma =5 \mathrm{kg}$. Which of the following weights are within $2$ standard deviations of the mean? Select three options.

• A. $85$ kg
• B. $75$ kg
• C. $80$ kg
• D. $60$ kg

Answer: (D)

Verified step by step guidance

1. Identify the mean (\mu) and standard deviation (\sigma) of the normal distribution. Here, \mu = 70 \text{ kg} and \sigma = 5 \text{ kg}.
2. Calculate the range that lies within 2 standard deviations from the mean. This range is given by: \(\mu - 2\sigma\) to \(\mu + 2\sigma\).
3. Substitute the values into the formula: lower bound = \$70 - 2 \times 5\(, upper bound = \)70 + 2 \times 5$.
4. Simplify the bounds to find the interval of weights within 2 standard deviations of the mean.
5. Check each given weight to see if it falls within this interval. Those weights that lie between the lower and upper bounds are within 2 standard deviations of the mean.