# (Answered) MATH399N Week 1 Assignment: Introduction to the Empirical Rule and Chebyshev’s Theorem

Question

At a carnival, contestants are asked to keep rolling a pair of dice until they roll snake eyes.  The number of rolls needed has a mean of 36 rolls, with a standard deviation of 5.4 rolls. The distribution of the number of rolls needed is not assumed to be symmetric.

Between what two numbers of rolls does Chebyshev’s Theorem guarantee that we will find at least 75% of the contestants?

Round your answers to the nearest tenth. Enter the bounds in ascending order.

# Question

According to Chebyshev’s Theorem, for any distribution, at least what proportion of data are within k=2.5 standard deviations of the mean?

# Question

The members of Maggie’s choir have a mean height of 58 inches, with a standard deviation of 4 inches.  The choir includes both children and adults, and the distribution of their heights is not symmetric.

Between what two heights does Chebyshev’s Theorem guarantee that we will find at least approximately 89% of the choir members?
Round your answers to the nearest whole number. Enter the bounds in ascending order.

# Question

Toyotas manufactured in the 1990s have a mean lifetime of 22.6 years, with a standard deviation of 3.1 years. The distribution of their lifetimes is not assumed to be symmetric.

Between what two lifetimes does Chebyshev’s Theorem guarantee that we will find at least 95% of the Toyotas?

Round your answers to the nearest hundredth. Enter the bounds in ascending order.

# Question

Suppose that the distribution of snake lengths in a certain park is not assumed to be symmetric.

According to Chebyshev’s Theorem, at least what percentage of snake lengths are within k=2.9 standard deviations of the mean?

# Question

A random sample of adult female reaction times has a sample mean of x¯=394.3 milliseconds and sample standard deviation of s=84.6 milliseconds.  Use the Empirical Rule to determine the approximate percentage of adult female reaction times that lie between 140.5 and 648.1 milliseconds.

# Question

A random sample of SAT scores has a sample mean of x¯=1060 and sample standard deviation of s=195.  Use the Empirical Rule to estimate the approximate percentage of SAT scores that are less than 865.

# Question

A random sample of vehicle mileage expectancies has a sample mean of x¯=169,200 miles and sample standard deviation of s=19,400 miles.  Use the Empirical Rule to estimate the percentage of vehicle mileage expectancies that are more than 188,600 miles.

# Question

A random sample of small business stock prices has a sample mean of x¯=\$54.82 and sample standard deviation of s=\$8.95.  Use the Empirical Rule to estimate the percentage of small business stock prices that are more than \$81.67.

# Question

A random sample of hybrid vehicle fuel consumptions has a sample mean of x¯=53.2 mpg and sample standard deviation of s=4.8 mpg.  Use the Empirical Rule to estimate the percentage of hybrid vehicle fuel consumptions that are less than 43.6 mpg.

# Question

Patients coming to a medical clinic have a mean weight of 207.6 pounds, with a standard deviation of 22.6 pounds. The distribution of weights is not assumed to be symmetric.

Between what two weights does Chebyshev’s Theorem guarantees that we will find at least 95% of the patients?

Round your answers to the nearest tenth. Enter the bounds in ascending order.

# Question

Smoking males in a given area have a mean life expectancy of 68.5 years, with a standard deviation of 5.3 years. The distribution of life expectancy is not assumed to be symmetric.

Between what two life expectancies does Chebyshev’s Theorem guarantee that we will find at least 89% of smoking males?

Round your answers to the nearest tenth. Enter the bounds in ascending order.

# Question

Returning to the sample of adult women, with a sample mean height of x¯=64.3 inches and sample standard deviation of s=2.4 inches,  use the Empirical Rule to estimate the percentage of heights that are less than 61.9 inches.

# Question

For the same random sample of adult women, with a sample mean height of x¯=64.3 inches and sample standard deviation of s=2.4 inches,  use the Empirical Rule to determine the approximate percent of heights that lie between 59.5 inches and 69.1 inches.

# Question

Suppose a random sample of adult women has a sample mean height of x¯=64.3 inches, with a sample standard deviation of s=2.4 inches. Since height distribution are generally symmetric and bell-shaped, we can apply the Empirical Rule.

Between what two heights are approximately 99.7% of the data?

# Solution:

Question

At a carnival, contestants are asked to keep rolling a pair of dice until they roll snake eyes.  The number of rolls needed has a mean of 36 rolls, with a standard deviation of 5.4 rolls. The distribution of the number of rolls needed is not assumed to be symmetric.

Between what two numbers of rolls does Chebyshev’s Theorem guarantee that we will find at least 75% of the contestants?

Round your answers to the nearest tenth. Enter the bounds in ascending order.

1:     25.2 rolls.

2:     46.8 rolls

# Question

According to Chebyshev’s Theorem, for any distribution, at least what proportion of data are within k=2.5 standard deviations of the mean?