A survey of 100 households gave the following results for weekly income $y$ - Edexcel - A-Level Maths Statistics - Question 5 - 2013 - Paper 1

The median represents the middle value of the data set. For 100 households, the median position is at the 50th value. Given the cumulative frequency, the median class is found where cumulative frequency reaches 50. By using linear interpolation:

1. Determine the cumulative frequency just before and at the class interval containing the median.
2. Apply the formula: $ext{Median} = L + \left(\frac{\frac{N}{2} - CF}{f}\right) \times c$ where:

• L = lower boundary of median class
• N = total frequency
• CF = cumulative frequency before median class
• f = frequency of median class
• c = class width

We calculate as per obtained values to get a final estimated value.

To estimate the mean ($\mu$), we use: $\mu = \frac{\Sigma (f \cdot x)}{N}$ where:

• f = frequency
• x = mid-point
• N = total number of frequencies = 100.

After calculating the sums by multiplying each frequency with its respective mid-point and dividing it by total frequencies, we arrive at a value for the mean.

For the standard deviation ($\sigma$), we use: $\sigma = \sqrt{\frac{\Sigma f (x - \mu)^2}{N}}$ This involves finding the squared differences, multiplying by frequency, summing, and subsequently dividing.

The measure of skewness is given by: $\text{Skewness} = \frac{3(\text{mean} - \text{median})}{\text{standard deviation}}$ Using our previously calculated values for mean, median, and standard deviation, we substitute:

• mean = {calculated_mean}
• median = {calculated_median}
• std. dev. = {calculated_std_dev}. This will yield a skewness value, which we interpret as indicates the direction of skewness based on its sign.

To find this probability, we analyze the given normal distribution approximating our discrete frequencies. Using the z-score, we find: $P(a < X < b) = P(Z < \frac{b - \mu}{\sigma}) - P(Z < \frac{a - \mu}{\sigma})$ Substituting in our values leads to finding $P(240 < X < 400)$, thus yielding an accurate probability.

Katie’s suggestion to model the weekly income with a normal distribution can be validated if the skewness obtained from part (d) was close to 0, indicating symmetry. However, using our probability from part (e), if it's significantly skewed (positive or negative), it suggests this model might not accurately reflect our data; thus, we reassess or look for a better fitting model.