The age in years of the residents of two hotels are shown in the back to back stem and leaf diagram below - Edexcel - A-Level Maths Statistics - Question 2 - 2008 - Paper 2

To find the lower quartile (Q1), median (Q2), and upper quartile (Q3):

1. Lower Quartile (Q1): This is the 25th percentile. Given our ordered ages, Q1 can be calculated by finding the median of the first half of the data. Here, we find that Q1 = 45.

2. Median (Q2): This is the middle value of the ordered data. (If there’s an even number, the median is the average of the two middle numbers.) For our data, Q2 = 50.5.

3. Upper Quartile (Q3): This is the 75th percentile, which is determined by the median of the second half of the data. Ultimately, Q3 = 63.

To compute the mean age of the residents, use the formula:

ar{x} = \frac{\sum_{i=1}^{n} x_i}{n}

Where:

• $\sum_{i=1}^{n} x_i$ is the sum of all ages and
• $n$ is the number of residents.

After adding the ages, we get a total of 1469 years and there are 28 residents, therefore:

$\bar{x} = \frac{1469}{28} \approx 52.46.$

The standard deviation can be calculated using the formula:

$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$

Given that $\sum x^{2} = 81 213$, we can plug our values into the equation:

First, calculate the variance:

1. $n = 28$.
2. Mean found = 52.46.
3. Variance = $\frac{81 213 - 28 \times (52.46)^2}{28 - 1} = 10.80$

So the standard deviation, $s$, is:

$s \approx \sqrt{10.80} \approx 3.29.$

The skewness can be calculated using:

$skewness = \frac{mean - mode}{standard deviation}$

Substituting our previously calculated values:

a. Mean = 52.46 b. Mode = 50 c. Standard deviation = 3.29

does give:

$skewness = \frac{52.46 - 50}{3.29} \approx 0.75.$

The comparison of age distributions between the Abbey Hotel and the Balmoral Hotel can be summarized as follows:

• Abbey Hotel: The mode of the Abbey Hotel was found to be 39 and the standard deviation was 12.7.

• Balmoral Hotel: The mode was 50 and had a standard deviation of approximately 3.29.

It is evident that the Balmoral Hotel residents are generally older, and the data appears to be less spread out compared to the Abbey Hotel residents, suggesting a more consistent age among those residing at the Balmoral Hotel.