Questions: A distribution of measurements is relatively mound-shaped with a mean of 70 and a standard deviation of 13. Use this information to find the proportion of measurements in the given interval. between 57 and 83. A distribution of measurements is relatively mound-shaped with a mean of 40 and a standard deviation of 11. Use this information to find the proportion of measurements in the given interval.

Solution Steps

1. For a distribution that is relatively mound-shaped (normal distribution), we can use the properties of the normal distribution to find the proportion of measurements within a given interval. We will standardize the interval limits using the z-score formula and then use a standard normal distribution table or a computational tool to find the corresponding probabilities.

2. Calculate the z-scores for the interval limits using the formula: \( z = \frac{(X - \mu)}{\sigma} \), where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

3. Use the z-scores to find the cumulative probabilities from a standard normal distribution, and then calculate the proportion of measurements within the interval by finding the difference between these probabilities.

We are given a normal distribution with a mean \(\mu = 70\) and a standard deviation \(\sigma = 13\). We need to find the proportion of measurements between the interval 57 and 83.

To find the proportion of measurements within the interval, we first calculate the z-scores for the lower and upper bounds of the interval using the formula: \[ z = \frac{X - \mu}{\sigma} \] For the lower bound \(X = 57\): \[ z_{\text{lower}} = \frac{57 - 70}{13} = -1.0 \] For the upper bound \(X = 83\): \[ z_{\text{upper}} = \frac{83 - 70}{13} = 1.0 \]

Using the standard normal distribution, the cumulative probability for \(z = -1.0\) is approximately 0.1587, and for \(z = 1.0\) it is approximately 0.8413. The proportion of measurements within the interval is the difference between these cumulative probabilities: \[ \text{Proportion} = 0.8413 - 0.1587 = 0.6826 \]

Final Answer