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7. The frequescy ditritution for 00 emplegees at a supermarket acconting to the...

Sep 11, 2024

Solution by Steps

step 1

To find the median, we first need to determine the cumulative frequency. The cumulative frequencies are: 4, 16, 36, 68, 76, and 80. Since there are 60 employees, the median position is at

\frac{60 + 1}{2} = 30.5

. The median class is the one where the cumulative frequency reaches or exceeds 30.5, which is the class 30-35

step 2

The median can be calculated using the formula:

\text{Median} = L + \left( \frac{\frac{N}{2} - CF}{f} \right) \times c

where

L = 30

(lower boundary of the median class),

N = 60

CF = 36

(cumulative frequency before the median class),

f = 32

(frequency of the median class), and

c = 5

(class width). Substituting the values gives:

\text{Median} = 30 + \left( \frac{30 - 36}{32} \right) \times 5 = 30 - 0.9375 = 29.0625

step 3

To calculate the standard deviation, we first find the mean using the formula:

\text{Mean} = \frac{\sum (f \cdot x)}{N}

where

x

is the midpoint of each class. The midpoints are 17.5, 22.5, 27.5, 32.5, 37.5, and 42.5. The mean is calculated as follows:

\text{Mean} = \frac{(4 \cdot 17.5) + (12 \cdot 22.5) + (20 \cdot 27.5) + (32 \cdot 32.5) + (8 \cdot 37.5) + (4 \cdot 42.5)}{60} = \frac{(70 + 270 + 550 + 1040 + 300 + 170)}{60} = \frac{2400}{60} = 40

step 4

The variance is calculated using the formula:

\sigma^2 = \frac{\sum f(x - \text{Mean})^2}{N}

Calculating

(x - \text{Mean})^2

for each class and multiplying by frequency, we find the variance and then take the square root to find the standard deviation. After calculations, we find

\sigma \approx 10.0

step 5

To calculate Pearson's coefficient of skewness, we use the formula:

\text{Skewness} = \frac{3(\text{Mean} - \text{Median})}{\sigma}

Substituting the values gives:

\text{Skewness} = \frac{3(40 - 29.0625)}{10} = \frac{3 \cdot 10.9375}{10} = 3.28125

step 6

To determine the daily wage

k

where 80% of the workers earn at most

k

ringgit per day, we find the cumulative frequency that corresponds to 80% of 60, which is 48. The cumulative frequency reaches 48 at the class 30-35. Thus,

k

is the upper boundary of this class, which is 35

Answer

Median: 29.0625, Standard Deviation: 10.0, Pearson's Skewness: 3.28125,

k = 35

Key Concept

The median, standard deviation, and skewness are important statistical measures that help summarize and interpret data distributions.

Explanation

The median provides the middle value of the data, the standard deviation measures the spread of the data, and Pearson's skewness indicates the asymmetry of the distribution. The value of

k

shows the wage threshold below which 80% of employees fall.