Question

Use the \(95 \%\) rule and the fact that the summary statistics come from a distribution that is symmetric and bell-shaped to find an interval that is expected to contain about \(95 \%\) of the data values. A bell-shaped distribution with mean 1500 and standard deviation 300.

Short answer

The interval that contains about 95% of the data values is [900,2100].

Step-by-step solution

Identify the mean and standard deviation

From the exercise, the mean (\( \mu \)) is 1500 and the standard deviation (\( \sigma \)) is 300.

Apply the 95% rule

According to the 95% rule of the normal distribution, 95% of the data falls within two standard deviations of the mean. Therefore, we calculate the lower and upper limits of this interval as follows: \[ \mu - 2\sigma \] and \[ \mu + 2\sigma \]

Compute the interval

The lower limit is \( 1500 - 2 \cdot 300 = 900 \) and the upper limit is \( 1500 + 2 \cdot 300 = 2100 \). Therefore, the interval that contains about 95% of the data values is [900,2100].