Question

1. the percentage of students with less than $10 in their possession is closest to

a. 30%
b. 35%
c. 45%
d. 60%

1. which of the following statements about this distribution can be concluded from the graph?

a. the distribution is left - skewed.
b. the median is between $10 and $20.
c. the standard deviation of the distribution is more than $60.
d. the mean is less than the median.

1. the mean speed of vehicles in the \cars only\ lanes of the new jersey turnpike is 68 mph. the mean speed of vehicles in the \any vehicle\ lanes is 64 mph. what must be true about the mean speed of all vehicles on the turnpike, assuming these are the only types of lanes?

a. it could be any number from 64 to 68 mph.
b. it must be larger than the median speed.
c. it must be larger than 66 mph.
d. it must be 66 mph.

Explanation:

Step1: Calculate total frequency

Total frequency \(=60 + 40+20 + 5+1+1=127\)

Step2: Calculate frequency of students with less than $10

Frequency of students with less than $10 is \(60\)

Step3: Calculate percentage

Percentage \(=\frac{60}{127}\times100\approx 47.24\%\) which is closest to \(45\%\)

Step1: Analyze skewness

The distribution is right - skewed (tail on the right), so a is wrong.

Step2: Analyze median

Total frequency \(n = 127\). The median is the \(\frac{n + 1}{2}=64^{th}\) value. Cumulative frequency: first bar (\(0 - 10\)) has \(60\), second bar (\(10 - 20\)) has \(40\). So median is in \(10 - 20\)

Step3: Analyze standard deviation

The data ranges from \(0\) to \(110\), but most of the data (\(60 + 40+20=120\) out of \(127\)) is within \(0 - 30\). Standard deviation is not more than \(60\), so c is wrong.

Step4: Analyze mean and median

For a right - skewed distribution, mean \(>\) median, so d is wrong.

Step1: Use formula for combined mean

Let \(n_1\) be the number of cars in “cars only” lanes and \(n_2\) be the number of cars in “any vehicle” lanes. The combined mean \(\bar{x}=\frac{n_1\times68 + n_2\times64}{n_1 + n_2}\)

Step2: Simplify the expression

\(\bar{x}=64+\frac{4n_1}{n_1 + n_2}\). Since \(n_1,n_2>0\), \(64<\bar{x}<68\)

Answer:

c. \(45\%\)