Questions: Airport Parking The number of short-term parking spaces at 15 airports is shown. Find the mean. According to Chebychev's Theorem, what portion (in terms of percentage) of the data for any dataset will be within one-and-a-quarter standard deviations of the mean? Find the z-score for the value below, which comes from a distribution with mean 18.7 and standard deviation 1.3. x=17.4 z=?

Solution Steps

1. Finding the Mean: To find the mean of the short-term parking spaces at the 15 airports, sum all the given values and then divide by the number of values (15).

2. Chebyshev's Theorem: According to Chebyshev's Theorem, for any dataset, the proportion of data within \( k \) standard deviations of the mean is at least \( 1 - \frac{1}{k^2} \). For \( k = 1.25 \), calculate this proportion and convert it to a percentage.

3. Calculating the z-score: The z-score is calculated using the formula \( z = \frac{x - \mu}{\sigma} \), where \( x \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

To find the mean of the short-term parking spaces at the 15 airports, sum all the given values and divide by the number of values (15).

\[ \text{Mean} = \frac{750 + 3400 + 1962 + 700 + 203 + 900 + 8662 + 260 + 1479 + 5905 + 9239 + 690 + 9822 + 1131 + 2516}{15} = 3174.6 \]

According to Chebyshev's Theorem, for any dataset, the proportion of data within \( k \) standard deviations of the mean is at least \( 1 - \frac{1}{k^2} \). For \( k = 1.25 \):

\[ \text{Proportion} = 1 - \frac{1}{(1.25)^2} = 0.36 \]

Convert this proportion to a percentage:

\[ \text{Percentage} = 0.36 \times 100 = 36\% \]

The z-score is calculated using the formula \( z = \frac{x - \mu}{\sigma} \), where \( x = 17.4 \), \( \mu = 18.7 \), and \( \sigma = 1.3 \).

\[ z = \frac{17.4 - 18.7}{1.3} = -1.0000 \]

Final Answer