CALCU-7. The frequescy ditritution for 00 emplegee

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.

Solution by Steps

step 1

To find the probability that a student is either majoring in Medicine or doing a minor in Information Technology, we first calculate the number of students majoring in Medicine (100) and those minoring in Information Technology (210). We also need to subtract the overlap, which is the number of students majoring in Medicine and minoring in Information Technology (60). Thus, the total is:

100 + 210 - 60 = 250

. The probability is then given by

\frac{250}{400} = \frac{5}{8}

step 2

For the second question, we need to find the number of non-Medicine students who do a minor in Language. The total number of students minoring in Language is 140, and the number of Medicine students minoring in Language is 30. Therefore, the number of non-Medicine students minoring in Language is

140 - 30 = 110

. The probability is

\frac{110}{400} = \frac{11}{40}

step 3

For the third question, we need to find the probability that a student chooses a minor in Statistics given that they are an Economics student. The number of Economics students is 70, and the number of Economics students minoring in Statistics is 10. Thus, the conditional probability is

\frac{10}{70} = \frac{1}{7}

step 4

For the fourth question, we need to find the probability that a student is neither an Engineering student who does a minor in Statistics nor an Economics student who does a minor in Language. The number of Engineering students minoring in Statistics is 30, and the number of Economics students minoring in Language is 30. The total number of students in these categories is

30 + 30 = 60

. Therefore, the probability is

\frac{400 - 60}{400} = \frac{340}{400} = \frac{17}{20}

[1] Answer

A

[2] Answer

B

[3] Answer

C

[4] Answer

D

Key Concept

Probability Calculation

Explanation

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of outcomes. Conditional probability considers the probability of an event given that another event has occurred.

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