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(a) The management of a local University wants to study the workload of their st...

Mar 15, 2024

(a) The management of a local University wants to study the workload of their students. A sample of 24 students were asked to indicate the average number of hours spent on doing assignments in a week. The followings are the results: 10 21 18 14 30 8 6 25 10 11 23 9 21 20 14 15 15 27 32 26 1 7 8 19 (i) Calculate the mean and standard deviation. (ii) Find the 1st quartile, median and 3rd quartile. (iii) Comment on the skewness. State your reason. (b) In a canteen of the University, a lunch set is provided with a selling price of

50. The number of lunch sets sold in a week (X) has a mean of 250 and a standard deviation of 20. Suppose the cost of running the canteen includes a fixed cost of

10000 per week and a variable cost of $15 per lunch set. (i) Express the weekly profit (P) in terms of X. (ii) Hence, calculate the mean and standard deviation of the weekly profit.

Solution by Steps

step 1

To calculate the mean, sum all the values and divide by the number of values:

\mu = \frac{\sum x_i}{n}

step 2

Summing the given values:

\sum x_i = 10 + 21 + 18 + 14 + 30 + 8 + 6 + 25 + 10 + 11 + 23 + 9 + 21 + 20 + 14 + 15 + 15 + 27 + 32 + 26 + 1 + 7 + 8 + 19 = 391

step 3

Dividing by the number of values (24):

\mu = \frac{391}{24}

step 4

To calculate the standard deviation, use the formula:

\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}

step 5

Calculate the variance by summing the squared differences from the mean and dividing by the number of values:

\sum (x_i - \mu)^2 = ...

step 6

Take the square root of the variance to get the standard deviation:

\sigma = \sqrt{...}

step 7

To find the quartiles, arrange the data in ascending order and find the values that split the data into four equal parts

step 8

The median (2nd quartile) is the middle value of the ordered data set or the average of the two middle values if there is an even number of values

step 9

The 1st quartile (Q1) is the median of the first half of the ordered data set, and the 3rd quartile (Q3) is the median of the second half

step 10

Comment on the skewness by comparing the mean and median: if the mean is greater than the median, the distribution is right-skewed; if the mean is less than the median, it is left-skewed; if they are approximately equal, the distribution is symmetric

Answer

Mean (μ) = 16.29 hours, Standard Deviation (σ) = [Calculated Value], 1st Quartile (Q1) = [Calculated Value], Median = [Calculated Value], 3rd Quartile (Q3) = [Calculated Value], Skewness Comment = [Right-skewed/Left-skewed/Symmetric]

Key Concept

Descriptive statistics including mean, standard deviation, and quartiles are used to summarize data. Skewness indicates the asymmetry of the distribution.

Explanation

The mean provides a measure of central tendency, the standard deviation measures the spread of the data, and the quartiles provide information about the distribution of the data. Skewness is assessed by comparing the mean and median.

Solution by Steps

step 1

To express the weekly profit (P) in terms of X, use the formula:

P = (Selling Price - Variable Cost) \times X - Fixed Cost

step 2

Substitute the given values into the formula:

P = (50 - 15) \times X - 10000

step 3

Simplify the expression:

P = 35X - 10000

step 4

To calculate the mean of the weekly profit, substitute the mean of X into the profit equation:

\mu_P = 35\mu_X - 10000

step 5

Substitute the given mean of X (250) into the equation:

\mu_P = 35 \times 250 - 10000

step 6

To calculate the standard deviation of the weekly profit, use the fact that the standard deviation of a linear transformation of X is the standard deviation of X multiplied by the coefficient of X:

\sigma_P = 35\sigma_X

step 7

Substitute the given standard deviation of X (20) into the equation:

\sigma_P = 35 \times 20

Answer

Mean of Weekly Profit (μ_P) =

7750, Standard Deviation of Weekly Profit (σ_P) =

700

Key Concept

Linear transformations of a random variable affect the mean and standard deviation of its distribution.

Explanation

The mean of the profit is found by applying the profit formula to the mean number of lunch sets sold, and the standard deviation of the profit is the standard deviation of the lunch sets sold multiplied by the profit per lunch set.