Suppose that we have a data set with a mean (µ) of 5 and a standard deviation (σ) of 4.Using the empirical rule, estimate how likely it is that a sample statistic picked at random will fall between -3 and 1.Group of answer choices68%16%13.5%50%

To answer this question, we need to use the empirical rule, which states that for a normal distribution:

• About 68% of the data falls within one standard deviation (σ) of the mean (µ).
• About 95% of the data falls within two standard deviations (σ) of the mean (µ).
• About 99.7% of the data falls within three standard deviations (σ) of the mean (µ).

Given:

• Mean (µ) = 5
• Standard deviation (σ) = 4

We need to find the probability that a sample statistic falls between -3 and 1.

First, we calculate how many standard deviations away from the mean these values are:

1. For -3: $z = \frac{-3 - 5}{4} = \frac{-8}{4} = -2$

2. For 1: $z = \frac{1 - 5}{4} = \frac{-4}{4} = -1$

So, -3 is 2 standard deviations below the mean, and 1 is 1 standard deviation below the mean.

According to the empirical rule:

• About 68% of the data falls within ±1σ of the mean.
• About 95% of the data falls within ±2σ of the mean.

The range from -3 to 1 is from -2σ to -1σ. This is the lower half of the range from -2σ to +2σ, which is 95%. Since the empirical rule is symmetric, the range from -2σ to -1σ is half of the range from -2σ to +2σ.

Therefore, the probability that a sample statistic falls between -3 and 1 is: $\frac{95\%}{2} = 47.5\%$

However, this is not one of the given choices. The closest match is 16%, which represents the probability of falling between -2σ and -1σ in one tail of the distribution.

Thus, the correct answer is: 16%